diff --git a/.gitignore b/.gitignore
index f076c58..3a9297e 100644
--- a/.gitignore
+++ b/.gitignore
@@ -224,3 +224,9 @@ __marimo__/
 # Benchmarking data and results
 benchmarking/data/
 benchmarking/results/
+
+# Verification output
+verification/tmp__NOBACKUP__/
+
+# Experiment results
+experiments/*/results/
diff --git a/README.md b/README.md
index f04677d..da9ad37 100644
--- a/README.md
+++ b/README.md
@@ -25,7 +25,9 @@ submodule, so clone with `--recurse-submodules` or run
 module `_tsls`). It is based on the tsinfer engine:
 
 - It copies from every node in the tree at a given position (any node
-  with a parent or a child); isolated nodes cannot be copied.
+  with a parent or a child); isolated nodes cannot be copied. Copying a
+  node from one site to the next without a recombination requires it to
+  be in the tree over the whole interval between them.
 - It supports arbitrary topologies (multiple roots, no special root node).
 - It provides the Viterbi algorithm (`AncestorMatcher.find_path`, with
   the same interface as tsinfer-hmm) and the forward and backward
@@ -39,7 +41,8 @@ Paths match tsinfer-hmm except for ties. To match tsinfer's convention
 that its ultimate root node 0 is never copied, remove node 0's edges
 before building the `MatcherIndexes`; sc2ts ARGs already have an
 isolated node 0. The `benchmarking/` directory compares the engines on
-the sc2ts ARG.
+the sc2ts ARG, and `verification/` runs statistical validations of tsls on
+simulated ARGs (plots, not pass/fail tests).
 
 The git history records exactly what was copied from where. Each
 directory was first added verbatim, then changed in separate commits to
diff --git a/experiments/thresholding/README.md b/experiments/thresholding/README.md
new file mode 100644
index 0000000..2c41db1
--- /dev/null
+++ b/experiments/thresholding/README.md
@@ -0,0 +1,144 @@
+# Thresholding experiments
+
+`thresholding.py` measures how well different ways of limiting the number of
+stored likelihood values work in the tsls engine. It also measures what each
+one costs on the tasks from `verification/`.
+
+```
+uv sync
+uv run python thresholding.py --quick -t 4    # about 2 minutes
+uv run python thresholding.py -t 4            # full grid; about an hour
+uv run python thresholding.py --analyse-only  # redo plots and summary
+```
+
+Results, plots and `summary.md` are written to `results/`, which is not
+tracked by git. The sc2ts scenario needs the sub-ARGs and queries from
+`benchmarking/data`; if they are missing, the scenario is skipped.
+
+## Methods
+
+Every method is applied after the values are normalised at each site:
+by the maximum for Viterbi, and by the sum for forward/backward. A "pruned"
+node is given the value `likelihood_threshold`, which by default is the
+smallest positive double. It then merges into its parent, and can only be
+reached again by recombination.
+
+- **`floor`** is the existing `likelihood_threshold`: any value below `t` is
+  set to `t`.
+- **`relative`** prunes nodes with a value below `f` times the maximum. Use
+  `pruning=("relative", f)` (experimental option).
+- **`top_n`** prunes all but the `n` nodes with the largest values. Nodes tied
+  with the n-th value are kept. Use `pruning=("top_n", n)` (experimental option).
+- **`lossless_floor`** is `floor` at `t* = (ρ/n) / (1 − ρ + ρ/n)`. In Viterbi,
+  any node below `t*` loses to recombination at the next site, so this floor
+  changes no Viterbi likelihoods. With `weight_by_n=False`, as sc2ts uses,
+  `t* = ρ`.
+
+`AncestorMatcher(count_distinct=True)` also records the mean number of
+**distinct** values per site in Viterbi (`mean_num_distinct_values`). This
+is compared with the number of stored nodes (`mean_traceback_size`).
+
+## Scenarios
+
+- **threading**: a held-out sample from an msprime simulation, matched as in
+  `verification.py` (`HeldOutThreading`). The HMM uses sc2ts parameters
+  (k = 3, `weight_by_n=False`). For Viterbi we measure the loss in path
+  log-likelihood relative to exact, how often the path is unchanged, and the
+  accuracy of copying the true parent. For forward/backward we measure the
+  posterior total-variation distance from exact and the posterior on the
+  true parent.
+- **calibration**: queries simulated from the HMM itself, as in
+  `verification.py` (`ModelCalibration`). For forward/backward we measure
+  the reliability error, and 90% credible-set coverage minus the expected
+  coverage.
+- **sc2ts**: queries from `benchmarking/` on sc2ts sub-ARGs, using the tsls
+  benchmark engine settings. We measure the loss in Viterbi path
+  log-likelihood, how often the path is unchanged, time, and stored values.
+  At 10k samples we also measure the forward/backward posterior
+  total-variation distance, at 100 evenly spaced sites.
+
+## Findings (quick run)
+
+These numbers come from `--quick`, which uses:
+- 3 replicates of threading, with 50 samples only;
+- 3 ARGs per calibration setting;
+- 3 sc2ts queries on the 10k sub-ARG (an exact copy, and recombinants with 2
+  and 5 extra mutations);
+- a reduced grid: t and f in {1e-20, 1e-8, 1e-4, 1e-2}, and n in
+  {10, 100, 1000}.
+
+Treat them as indicative. The full grid, with 50–1000 samples and sc2ts
+sub-ARGs up to 100k, was not run to completion.
+
+### 1. For Viterbi, compression limits storage more than thresholding does
+
+On the 10k sc2ts sub-ARG, exact Viterbi stores 312–1,248 nodes per site,
+but only **4–6 distinct values** per site. With 50 simulated samples, it
+stores 9.4 nodes for 4.4 distinct values.
+
+The tsinfer-style compression only removes a node's value if it equals its
+parent's. Siblings with the same value are kept separately, so most stored
+nodes repeat a few values. Thresholding changes values, not structure. The
+best any method achieved without loss was about a 2× smaller traceback:
+
+| sc2ts 10k, Viterbi | traceback vs exact | time vs exact | log-likelihood loss | identical paths |
+|---|---|---|---|---|
+| `lossless_floor` | 1.00 | 0.96 | 0 | 3/3 |
+| `top_n` = 10 | 0.52 | 0.61 | 0 | 3/3 |
+| `relative` = 1e-4 | 1.00 | 1.04 | 0 | 3/3 |
+| `floor` = 1e-4 | 0.93 | 0.90 | 0.06 | 1/3 |
+| `floor` = 1e-2 | 0.44 | 0.36 | 1.5 | 1/3 |
+| `relative` = 1e-2 | 0.44 | 0.34 | 5.9 | 1/3 |
+
+Merging equal-valued siblings, which is closer to tskit's parsimony step but
+cheaper, looks like the larger opportunity.
+
+### 2. For Viterbi, a floor above `t*` is harmful, and pruning should be used instead
+
+A floor raises weak nodes up to `t`. Once `t` is above the recombination
+value, staying on a raised node beats switching, so stale copying targets
+survive. Pruning sets weak nodes close to 0 instead, so they lose to
+recombination as they should.
+
+| threading (50 samples), Viterbi | traceback vs exact | log-likelihood loss | accuracy drop |
+|---|---|---|---|
+| `floor` = 1e-4 | 0.92 | 200 | 0.34 |
+| `relative` = 1e-4 | 0.92 | 0 | 0 |
+| `top_n` = 10 | 0.98 | 0 | 0 |
+| `lossless_floor` (2.4e-6) | 0.99 | 0 | 0 |
+
+The lossless floor explains why every threshold up to 1e-6 was free in
+earlier runs: with sc2ts settings, `t* = ρ ≈ 2.4e-6`.
+
+### 3. For forward/backward, every method distorts the posteriors, in different directions
+
+- **`floor`** works by adding probability to every node, so posteriors no
+  longer sum to 1 (they are off by about 0.99 at 50 samples). It is still
+  the least biased of the three.
+  - On sc2ts 10k, `floor` = 1e-4 cut the stored forward values by about
+    200× and the forward-backward time to under 1%. The posterior
+    total-variation distance from exact was 0.02.
+  - With 50 samples the same setting gave a total-variation distance of
+    0.07, and calibration was as good as exact.
+  - `floor` = 1e-2 made the posteriors under-confident: 90% coverage was
+    9 points above expected.
+- **`relative`** and **`top_n`** make the posteriors over-confident:
+  - `relative` = 1e-4 gave a total-variation distance of 0.44 with 50
+    samples;
+  - `top_n` = 10 gave 0.13 with 50 samples and 0.68 on sc2ts;
+  - 90% coverage was 7–28 points below expected.
+
+  With `top_n` = 100 on sc2ts, more values were stored than exact (1.23×),
+  because pruned values stop compressing as before.
+
+## Recommendations
+
+- **Viterbi:** use the lossless floor `t*` as the default threshold; it
+  costs nothing. If more speed is needed, use `relative` or `top_n` rather
+  than a floor above `t*`.
+- **Forward/backward:** treat any thresholding as approximate. A floor
+  around 1e-4 gives large speedups on sc2ts with small posterior error.
+  Check the posterior total-variation distance for the intended use.
+- **Next step:** improve compression by merging siblings with equal values,
+  rather than adding more thresholding. Use `count_distinct=True` to measure
+  the potential gain.
diff --git a/experiments/thresholding/pyproject.toml b/experiments/thresholding/pyproject.toml
new file mode 100644
index 0000000..bd4d4ff
--- /dev/null
+++ b/experiments/thresholding/pyproject.toml
@@ -0,0 +1,27 @@
+[project]
+name = "tsls-thresholding-experiments"
+version = "0.0.0"
+description = "Experiments on likelihood thresholding and pruning in tsls."
+requires-python = ">=3.11"
+dependencies = [
+    "matplotlib",
+    "msprime",
+    "numpy>=2",
+    "pandas",
+    "tabulate",
+    "tqdm",
+    "tskit>=1.0.0",
+    "tsls",
+]
+
+[tool.uv]
+package = false
+
+[tool.uv.sources]
+tsls = { path = "../../tsls", editable = true }
+
+[tool.ruff]
+line-length = 89
+
+[tool.ruff.lint.isort]
+known-first-party = ["dense", "tsls", "verification"]
diff --git a/experiments/thresholding/thresholding.py b/experiments/thresholding/thresholding.py
new file mode 100644
index 0000000..c1b7aef
--- /dev/null
+++ b/experiments/thresholding/thresholding.py
@@ -0,0 +1,948 @@
+"""
+Experiments on the effectiveness of thresholding and pruning in the tsls
+engine.
+
+After normalisation at each site, each method reduces the number of stored
+likelihood values in a different way:
+
+- floor: values below t are set to t (the likelihood_threshold option);
+- relative: nodes with values below f times the maximum are pruned;
+- top_n: all but the n nodes with the largest values are pruned;
+- lossless_floor: a floor at t* = (rho / n) / (1 - rho + rho / n). In Viterbi,
+  any node below t* is beaten by recombination at the next site, so this
+  floor does not change the Viterbi likelihoods.
+
+Pruned nodes get the value likelihood_threshold (by default the smallest
+positive double), so they can then only be reached by recombination.
+
+Scenarios:
+
+- threading: match a held-out sample from an msprime simulation, as in
+  verification.py (HeldOutThreading), at several sample sizes;
+- calibration: queries simulated from the HMM, as in verification.py
+  (ModelCalibration), to see whether pruning biases the posteriors;
+- sc2ts: queries from benchmarking/ on sub-ARGs of the sc2ts ARG.
+
+Run with --quick for a small version. Results and plots are written to
+results/.
+"""
+
+import argparse
+import concurrent.futures
+import functools
+import logging
+import pathlib
+import sys
+import time
+
+import matplotlib
+
+matplotlib.use("Agg")
+
+import numpy as np
+import pandas as pd
+import tqdm
+from matplotlib import pyplot as plt
+
+ROOT = pathlib.Path(__file__).resolve().parents[2]
+sys.path.insert(0, str(ROOT / "verification"))
+import tsls
+import verification as v
+
+logger = logging.getLogger(__name__)
+
+RESULTS = pathlib.Path(__file__).resolve().parent / "results"
+SC2TS_DATA = ROOT / "benchmarking" / "data"
+
+FLOOR_GRID = [1e-30, 1e-20, 1e-12, 1e-10, 1e-8, 1e-6, 1e-4, 1e-3, 1e-2]
+TOP_N_GRID = [10, 30, 100, 300, 1000, 3000]
+QUICK_FLOOR_GRID = [1e-20, 1e-8, 1e-4, 1e-2]
+QUICK_TOP_N_GRID = [10, 100, 1000]
+METHODS = ["floor", "relative", "top_n"]
+CALIBRATION_BINS = np.linspace(0, 1, 21)
+
+
+def method_grid(quick, rho, n, viterbi=True):
+    """
+    Return the list of (method, parameter, matcher kwargs) tuples to run.
+    """
+    floors = QUICK_FLOOR_GRID if quick else FLOOR_GRID
+    top_ns = QUICK_TOP_N_GRID if quick else TOP_N_GRID
+    grid = [("exact", np.nan, {})]
+    grid += [("floor", t, {"likelihood_threshold": t}) for t in floors]
+    grid += [("relative", f, {"pruning": ("relative", f)}) for f in floors]
+    grid += [("top_n", k, {"pruning": ("top_n", k)}) for k in top_ns]
+    if viterbi:
+        t_star = (rho / n) / (1 - rho + rho / n)
+        # Just below t*, so that pruned nodes lose the tie with recombination
+        grid.append(("lossless_floor", t_star, {"likelihood_threshold": t_star * 0.999}))
+    return grid
+
+
+def path_log_likelihood(h, num_segments, matched, mu, rho):
+    """
+    Return the log likelihood of a Viterbi path with weight_by_n=False (so
+    staying has probability 1 and each switch rho), from the engine's output.
+    """
+    non_missing = h != -1
+    num_mismatches = np.sum(h[non_missing] != matched[non_missing])
+    return (
+        (num_segments - 1) * np.log(rho)
+        + num_mismatches * np.log(mu)
+        + (len(h) - num_mismatches) * np.log(1 - (v.NUM_ALLELES - 1) * mu)
+    )
+
+
+def run_viterbi(am, ts, h):
+    """
+    Return the (left, right, parent, path, matched, seconds) tuple, with the
+    segments in left-to-right order and path the node copied at each site.
+    """
+    m = ts.num_sites
+    matched = np.zeros(m, dtype=np.int8)
+    before = time.perf_counter()
+    left, right, parent = am.find_path(h, 0, m, matched)
+    seconds = time.perf_counter() - before
+    order = np.argsort(left)
+    left, right, parent = left[order], right[order], parent[order]
+    segment = np.searchsorted(left, ts.sites_position, side="right") - 1
+    return left, right, parent, parent[segment], matched, seconds
+
+
+def mean_distinct_values(cm):
+    counts = [
+        len(np.unique(cm.values[cm.offsets[j] : cm.offsets[j + 1]]))
+        for j in range(cm.num_sites)
+    ]
+    return float(np.mean(counts))
+
+
+def run_forward_backward(am, h):
+    """
+    Return the (fm, bm, seconds) tuple.
+    """
+    before = time.perf_counter()
+    fm = am.forward_matrix(h)
+    bm = am.backward_matrix(h, fm.normalisation_factor)
+    return fm, bm, time.perf_counter() - before
+
+
+def normalise_rows(P):
+    return P / np.sum(P, axis=1)[:, None]
+
+
+def total_variation(P, Q):
+    """
+    Mean total variation distance between the rows of P and Q.
+    """
+    return 0.5 * float(np.mean(np.sum(np.abs(P - Q), axis=1)))
+
+
+# ---------------------------------------------------------------------------
+# Scenario 1: threading a held-out sample
+# ---------------------------------------------------------------------------
+
+
+def threading_case(num_samples, seed):
+    """
+    Return (ts, h, true_parent, near) for a held-out sample, as in
+    verification.threading_replicate: ts is the ARG of the other samples
+    with the same node IDs, true_parent[j] is the held-out sample's parent at
+    site j and near[j] the nodes within 1 edge of it in ts.
+    """
+    params = dict(v.THREADING_DEFAULTS)
+    full_ts = v.simulate_arg(
+        num_samples + 1,
+        params["sequence_length"],
+        params["recombination_rate"],
+        params["mutation_rate"],
+        seed,
+    )
+    samples = full_ts.samples()
+    query = samples[-1]
+    ts = full_ts.simplify(
+        samples[:-1],
+        keep_unary=True,
+        filter_nodes=False,
+        filter_sites=False,
+        update_sample_flags=False,
+    )
+    h = full_ts.genotype_matrix(
+        samples=[query], isolated_as_missing=False, alleles=v.ALLELES
+    )[:, 0].astype(np.int8)
+    true_parent = np.zeros(ts.num_sites, dtype=np.int32)
+    for tree in full_ts.trees():
+        for site in tree.sites():
+            true_parent[site.id] = tree.parent(query)
+    near = [None] * ts.num_sites
+    for tree in ts.trees():
+        for site in tree.sites():
+            p = true_parent[site.id]
+            nodes = [p, *tree.children(p)]
+            if tree.parent(p) != -1:
+                nodes.append(tree.parent(p))
+            near[site.id] = nodes
+    return ts, h, true_parent, near
+
+
+def mean_breakpoint_distance(true_switch, inferred_switch):
+    if len(true_switch) == 0 or len(inferred_switch) == 0:
+        return np.nan
+    return float(np.mean(v.nearest_distances(true_switch, inferred_switch)))
+
+
+def threading_distances(ts, path, true_parent):
+    distance = np.zeros(ts.num_sites, dtype=int)
+    for tree in ts.trees():
+        parent = tree.parent_array
+        for site in tree.sites():
+            distance[site.id] = v.tree_distance(
+                parent, path[site.id], true_parent[site.id]
+            )
+    return distance
+
+
+def run_threading_task(num_samples, seed, quick):
+    ts, h, true_parent, near = threading_case(num_samples, seed)
+    m = ts.num_sites
+    mu, rho = v.hmm_params(v.THREADING_DEFAULTS["k"])
+    mi = tsls.MatcherIndexes(ts, alleles=v.ALLELES)
+    true_switch = np.where(true_parent[1:] != true_parent[:-1])[0] + 1
+    rows = []
+    exact_posterior = None
+    exact = None
+    for method, parameter, kwargs in method_grid(quick, rho, 1):
+        am = v.make_matcher(
+            ts,
+            mu,
+            rho,
+            weight_by_n=False,
+            count_distinct=True,
+            matcher_indexes=mi,
+            **kwargs,
+        )
+        left, right, parent, path, matched, seconds = run_viterbi(am, ts, h)
+        distance = threading_distances(ts, path, true_parent)
+        inferred_switch = np.where(path[1:] != path[:-1])[0] + 1
+        row = {
+            "num_samples": num_samples,
+            "case": seed,
+            "method": method,
+            "parameter": parameter,
+            "num_sites": m,
+            "num_nodes": ts.num_nodes,
+            "viterbi_time": seconds,
+            "traceback_size": am.mean_traceback_size,
+            "distinct_values": am.mean_num_distinct_values,
+            "total_memory": am.total_memory,
+            "log_likelihood": path_log_likelihood(h, len(parent), matched, mu, rho),
+            "exact_parent": np.mean(path == true_parent),
+            "within_2": np.mean((distance >= 0) & (distance <= 2)),
+            "breakpoint_distance": mean_breakpoint_distance(
+                true_switch, inferred_switch
+            ),
+        }
+        if exact is None:
+            exact = (left, right, parent)
+        row["identical_path"] = all(
+            np.array_equal(x, y) for x, y in zip(exact, (left, right, parent))
+        )
+        if method != "lossless_floor":
+            fm, bm, seconds = run_forward_backward(am, h)
+            P = fm.decode() * bm.decode()
+            mass = np.sum(P, axis=1)
+            P = normalise_rows(P)
+            if exact_posterior is None:
+                exact_posterior = (P, fm.log_likelihood)
+            row.update(
+                {
+                    "fb_time": seconds,
+                    "forward_values": len(fm.values) / m,
+                    "forward_distinct_values": mean_distinct_values(fm),
+                    "forward_log_likelihood": fm.log_likelihood,
+                    "posterior_mass_error": float(np.max(np.abs(mass - 1))),
+                    "posterior_tv": total_variation(P, exact_posterior[0]),
+                    "posterior_true": float(np.mean(P[np.arange(m), true_parent])),
+                    "posterior_near": float(
+                        np.mean([np.sum(P[j, near[j]]) for j in range(m)])
+                    ),
+                }
+            )
+        rows.append(row)
+    return rows
+
+
+# ---------------------------------------------------------------------------
+# Scenario 2: calibration under the model
+# ---------------------------------------------------------------------------
+
+CALIBRATION_SETTINGS = {"default": (5, 0.01), "high_recombination": (25, 0.01)}
+
+
+def run_calibration_task(setting, seed, quick):
+    """
+    Simulate an ARG and several paths from the model's prior, and return
+    per-method sums for the reliability diagram and 90% credible sets.
+    """
+    switches, mu = CALIBRATION_SETTINGS[setting]
+    rng = np.random.default_rng(seed)
+    ts = v.simulate_arg(50, 5e5, 1e-8, 1e-8, seed)
+    A, present, continuous = v.node_states(ts)
+    m = ts.num_sites
+    n = ts.num_nodes
+    rho = switches / m
+    mi = tsls.MatcherIndexes(ts, alleles=v.ALLELES)
+    queries = []
+    for _ in range(3 if quick else 5):
+        path = v.sample_prior_path(present, continuous, np.full(m, rho), n, rng)
+        queries.append((path, v.mutate_along_path(A, path, mu, rng)))
+    rows = []
+    for method, parameter, kwargs in method_grid(quick, rho, n, viterbi=False):
+        am = v.make_matcher(ts, mu, rho, weight_by_n=True, matcher_indexes=mi, **kwargs)
+        num_bins = len(CALIBRATION_BINS) - 1
+        count = np.zeros(num_bins)
+        total_p = np.zeros(num_bins)
+        num_true = np.zeros(num_bins)
+        covered = 0
+        set_mass = 0
+        values = 0
+        for path, h in queries:
+            fm, bm, _ = run_forward_backward(am, h)
+            values += len(fm.values) / m
+            P = normalise_rows(fm.decode() * bm.decode())
+            is_true = np.zeros_like(P, dtype=bool)
+            is_true[np.arange(m), path] = True
+            keep = P > 1e-4
+            index = np.clip(np.digitize(P[keep], CALIBRATION_BINS) - 1, 0, num_bins - 1)
+            count += np.bincount(index, minlength=num_bins)
+            total_p += np.bincount(index, weights=P[keep], minlength=num_bins)
+            num_true += np.bincount(index, weights=is_true[keep], minlength=num_bins)
+            cumulative = np.cumsum(-np.sort(-P, axis=1), axis=1)
+            p_true = P[np.arange(m), path][:, None]
+            tied = np.abs(P - p_true) <= 1e-9 * p_true
+            num_greater = np.sum((P > p_true) & ~tied, axis=1)
+            size = np.sum(cumulative < 0.9, axis=1) + 1
+            covered += np.sum(np.clip((size - num_greater) / np.sum(tied, axis=1), 0, 1))
+            set_mass += np.sum(cumulative[np.arange(m), size - 1])
+        row = {
+            "setting": setting,
+            "case": seed,
+            "method": method,
+            "parameter": parameter,
+            "num_site_queries": m * len(queries),
+            "covered_90": covered,
+            "set_mass_90": set_mass,
+            "forward_values": values / len(queries),
+        }
+        for j in range(num_bins):
+            row[f"count_{j}"] = count[j]
+            row[f"total_p_{j}"] = total_p[j]
+            row[f"num_true_{j}"] = num_true[j]
+        rows.append(row)
+    return rows
+
+
+# ---------------------------------------------------------------------------
+# Scenario 3: sc2ts sub-ARGs
+# ---------------------------------------------------------------------------
+
+SC2TS_SIZES = {10000: 30, 30000: 30, 100000: 10}
+SC2TS_SAMPLED_SITES = 100
+
+
+@functools.lru_cache(maxsize=1)
+def load_sc2ts(size):
+    """
+    Return the (ts, matcher_indexes, H, kind, k) tuple for the sc2ts sub-ARG
+    of the specified size, with the vestigial 0 -> 1 edge removed so that
+    node 0 is isolated (as in benchmarking/engines.py).
+    """
+    import tskit
+
+    ts = tskit.load(SC2TS_DATA / "subargs" / f"sc2ts_bench_n{size}.trees")
+    tables = ts.dump_tables()
+    tables.edges.keep_rows(tables.edges.parent != 0)
+    ts = tables.tree_sequence()
+    mi = tsls.MatcherIndexes(ts, alleles=v.ALLELES)
+    query_dir = SC2TS_DATA / "queries_scaling"
+    H = np.load(query_dir / "queries.npz")["H"]
+    df = pd.read_csv(query_dir / "queries.csv")
+    return ts, mi, H, df.kind.values, df.k.values
+
+
+def decode_at_sites(cm, sites):
+    """
+    Return the dense (len(sites), num_nodes) array of values at the
+    specified sorted sites, decoding only those sites.
+    """
+    ts = cm.ts
+    N = ts.num_nodes
+    D = np.zeros((len(sites), N))
+    position = ts.sites_position[sites]
+    for tree in ts.trees():
+        start, stop = np.searchsorted(position, tree.interval)
+        if start == stop:
+            continue
+        parent = tree.parent_array
+        in_tree = (parent[:N] != -1) | (tree.num_children_array[:N] > 0)
+        for k in range(start, stop):
+            site_nodes, site_values = cm.get_site(sites[k])
+            value = np.full(N + 1, np.nan)
+            value[site_nodes] = site_values
+            ancestor = parent.copy()
+            unset = np.where(in_tree & np.isnan(value[:N]))[0]
+            while len(unset) > 0:
+                value[unset] = value[ancestor[unset]]
+                ancestor[unset] = parent[ancestor[unset]]
+                unset = unset[np.isnan(value[unset])]
+            D[k] = np.where(in_tree, value[:N], 0)
+    return D
+
+
+def sc2ts_query_ids(size, quick):
+    num_queries = 3 if quick else SC2TS_SIZES[size]
+    return np.unique(np.linspace(0, 109, num_queries).astype(int))
+
+
+def run_sc2ts_task(size, query_id, quick):
+    ts, mi, H, kind, k = load_sc2ts(size)
+    h = H[query_id]
+    m = ts.num_sites
+    mu, rho = v.hmm_params(3)
+    with_forward = size == min(SC2TS_SIZES)
+    sites = np.unique(np.linspace(0, m - 1, SC2TS_SAMPLED_SITES).astype(int))
+    rows = []
+    exact = None
+    exact_posterior = None
+    for method, parameter, kwargs in method_grid(quick, rho, 1):
+        am = v.make_matcher(
+            ts,
+            mu,
+            rho,
+            weight_by_n=False,
+            count_distinct=True,
+            matcher_indexes=mi,
+            **kwargs,
+        )
+        left, right, parent, _, matched, seconds = run_viterbi(am, ts, h)
+        if exact is None:
+            exact = (left, right, parent)
+        row = {
+            "size": size,
+            "case": query_id,
+            "kind": kind[query_id],
+            "k": k[query_id],
+            "method": method,
+            "parameter": parameter,
+            "viterbi_time": seconds,
+            "traceback_size": am.mean_traceback_size,
+            "distinct_values": am.mean_num_distinct_values,
+            "total_memory": am.total_memory,
+            "log_likelihood": path_log_likelihood(h, len(parent), matched, mu, rho),
+            "num_mismatches": int(np.sum(matched != h)),
+            "identical_path": all(
+                np.array_equal(x, y) for x, y in zip(exact, (left, right, parent))
+            ),
+        }
+        if with_forward and method != "lossless_floor":
+            fm, bm, seconds = run_forward_backward(am, h)
+            P = normalise_rows(decode_at_sites(fm, sites) * decode_at_sites(bm, sites))
+            if exact_posterior is None:
+                exact_posterior = P
+            row.update(
+                {
+                    "fb_time": seconds,
+                    "forward_values": len(fm.values) / m,
+                    "forward_log_likelihood": fm.log_likelihood,
+                    "posterior_tv": total_variation(P, exact_posterior),
+                }
+            )
+        rows.append(row)
+    return rows
+
+
+# ---------------------------------------------------------------------------
+# Running and analysis
+# ---------------------------------------------------------------------------
+
+
+def make_tasks(scenarios, quick):
+    tasks = []
+    if "threading" in scenarios:
+        sizes = [50] if quick else [50, 200, 1000]
+        replicates = 3 if quick else 10
+        for n in sizes:
+            for seed in range(1, replicates + 1):
+                tasks.append(("threading", run_threading_task, (n, seed, quick)))
+    if "calibration" in scenarios:
+        for setting in CALIBRATION_SETTINGS:
+            for seed in range(1, (3 if quick else 10) + 1):
+                tasks.append(
+                    ("calibration", run_calibration_task, (setting, seed, quick))
+                )
+    if "sc2ts" in scenarios:
+        if not (SC2TS_DATA / "subargs").exists():
+            logger.warning(f"No sc2ts data in {SC2TS_DATA}: skipping sc2ts scenario")
+        else:
+            sizes = [10000] if quick else list(SC2TS_SIZES)
+            for size in sizes:
+                for query_id in sc2ts_query_ids(size, quick):
+                    tasks.append(("sc2ts", run_sc2ts_task, (size, int(query_id), quick)))
+    return tasks
+
+
+def run_task(task):
+    scenario, func, args = task
+    return scenario, func(*args)
+
+
+def run(scenarios, quick, num_processes):
+    tasks = make_tasks(scenarios, quick)
+    results = {scenario: [] for scenario in scenarios}
+    progress = tqdm.tqdm(total=len(tasks))
+    if num_processes <= 1:
+        for task in tasks:
+            scenario, rows = run_task(task)
+            results[scenario].extend(rows)
+            progress.update()
+    else:
+        with concurrent.futures.ProcessPoolExecutor(num_processes) as executor:
+            futures = [executor.submit(run_task, task) for task in tasks]
+            for future in concurrent.futures.as_completed(futures):
+                scenario, rows = future.result()
+                results[scenario].extend(rows)
+                progress.update()
+    progress.close()
+    RESULTS.mkdir(exist_ok=True)
+    for scenario, rows in results.items():
+        if len(rows) > 0:
+            pd.DataFrame(rows).to_csv(RESULTS / f"{scenario}.csv", index=False)
+
+
+def add_relative_columns(df, group, columns):
+    """
+    Add, for each column, a column of its value relative to the exact method
+    in the same group and case (ratio_<col>), and the difference
+    (diff_<col> = method - exact).
+    """
+    exact = df[df.method == "exact"].set_index([group, "case"])
+    for column in columns:
+        if column not in df:
+            continue
+        base = exact[column].reindex(pd.MultiIndex.from_frame(df[[group, "case"]]))
+        with np.errstate(divide="ignore", invalid="ignore"):
+            df[f"ratio_{column}"] = df[column].values / base.values
+        df[f"diff_{column}"] = df[column].values - base.values
+    return df
+
+
+def summarise(df, group, x, losses):
+    """
+    Return the mean of x and the losses for each (group, method, parameter).
+    """
+    columns = [x, *losses]
+    return (
+        df.groupby([group, "method", "parameter"], dropna=False)[columns]
+        .mean()
+        .reset_index()
+    )
+
+
+def plot_frontiers(summary, group, x, losses, xlabel, name):
+    """
+    For each value of group, plot each loss against x (relative storage),
+    with one curve per method across its parameters.
+    """
+    groups = sorted(summary[group].unique())
+    fig, axes = plt.subplots(
+        len(losses),
+        len(groups),
+        figsize=(4.5 * len(groups), 3.5 * len(losses)),
+        squeeze=False,
+    )
+    for col, g in enumerate(groups):
+        sub = summary[summary[group] == g]
+        for row, (loss, label) in enumerate(losses):
+            ax = axes[row, col]
+            for method in METHODS:
+                d = sub[sub.method == method].sort_values(x)
+                ax.plot(d[x], d[loss], marker="o", label=method)
+                for _, r in d.iterrows():
+                    ax.annotate(
+                        f"{r.parameter:g}", (r[x], r[loss]), fontsize=6, alpha=0.6
+                    )
+            lossless = sub[sub.method == "lossless_floor"]
+            if len(lossless) > 0 and not np.all(np.isnan(lossless[loss])):
+                ax.plot(
+                    lossless[x],
+                    lossless[loss],
+                    marker="*",
+                    markersize=12,
+                    linestyle="none",
+                    label="lossless_floor",
+                )
+            ax.set_xscale("log")
+            ax.set_xlabel(xlabel)
+            ax.set_ylabel(label)
+            ax.set_title(f"{group}={g}")
+            ax.axvline(1, color="grey", linestyle=":")
+        axes[0, col].legend(fontsize=8)
+    fig.tight_layout()
+    fig.savefig(RESULTS / f"{name}.png", dpi=100)
+    plt.close(fig)
+
+
+def best_within_budget(summary, group, x, budgets):
+    """
+    For each group, method and budget, the smallest storage ratio among the
+    parameters that meet the budget.
+    """
+    rows = []
+    for (g, method), sub in summary.groupby([group, "method"]):
+        if method == "exact":
+            continue
+        row = {group: g, "method": method}
+        for name, condition in budgets.items():
+            ok = sub[condition(sub)]
+            if len(ok) == 0:
+                row[name] = "-"
+            else:
+                best = ok.loc[ok[x].idxmin()]
+                row[name] = f"{best[x]:.3f} ({best.parameter:g})"
+        rows.append(row)
+    return pd.DataFrame(rows)
+
+
+def analyse_threading(lines):
+    df = pd.read_csv(RESULTS / "threading.csv")
+    df = add_relative_columns(
+        df,
+        "num_samples",
+        [
+            "traceback_size",
+            "viterbi_time",
+            "log_likelihood",
+            "exact_parent",
+            "within_2",
+            "forward_values",
+            "fb_time",
+            "posterior_true",
+            "forward_log_likelihood",
+        ],
+    )
+    df["log_likelihood_loss"] = -df.diff_log_likelihood
+    df["accuracy_drop"] = -df.diff_exact_parent
+    df["within_2_drop"] = -df.diff_within_2
+    df["posterior_true_drop"] = -df.diff_posterior_true
+    df["identical_path"] = df.identical_path.astype(float)
+    viterbi = summarise(
+        df,
+        "num_samples",
+        "ratio_traceback_size",
+        [
+            "log_likelihood_loss",
+            "accuracy_drop",
+            "within_2_drop",
+            "identical_path",
+            "ratio_viterbi_time",
+            "breakpoint_distance",
+            "traceback_size",
+            "distinct_values",
+        ],
+    )
+    viterbi.to_csv(RESULTS / "threading_viterbi_summary.csv", index=False)
+    plot_frontiers(
+        viterbi,
+        "num_samples",
+        "ratio_traceback_size",
+        [
+            ("log_likelihood_loss", "Viterbi path log-likelihood loss"),
+            ("accuracy_drop", "Drop in fraction of sites copying parent"),
+            ("ratio_viterbi_time", "Viterbi time relative to exact"),
+        ],
+        "Mean traceback size relative to exact",
+        "threading_viterbi",
+    )
+    fb = summarise(
+        df[df.method != "lossless_floor"],
+        "num_samples",
+        "ratio_forward_values",
+        ["posterior_tv", "posterior_true_drop", "posterior_mass_error", "ratio_fb_time"],
+    )
+    fb.to_csv(RESULTS / "threading_fb_summary.csv", index=False)
+    plot_frontiers(
+        fb,
+        "num_samples",
+        "ratio_forward_values",
+        [
+            ("posterior_tv", "Posterior total variation from exact"),
+            ("posterior_true_drop", "Drop in posterior on the true parent"),
+            ("ratio_fb_time", "Forward-backward time relative to exact"),
+        ],
+        "Mean forward values per site relative to exact",
+        "threading_forward_backward",
+    )
+    budgets = {
+        "no loss": lambda s: (s.log_likelihood_loss < 1e-9) & (s.identical_path == 1),
+        "loss < 0.1": lambda s: s.log_likelihood_loss < 0.1,
+        "accuracy drop < 0.01": lambda s: s.accuracy_drop < 0.01,
+    }
+    lines += [
+        "## Threading a held-out sample: Viterbi",
+        "",
+        (
+            "Smallest mean traceback size relative to exact (parameter) within "
+            "each loss budget."
+        ),
+        "",
+        best_within_budget(
+            viterbi, "num_samples", "ratio_traceback_size", budgets
+        ).to_markdown(index=False),
+        "",
+        "Exact traceback size and distinct values per site:",
+        "",
+        viterbi[viterbi.method == "exact"][
+            ["num_samples", "traceback_size", "distinct_values"]
+        ].to_markdown(index=False),
+        "",
+    ]
+    budgets = {
+        "TV < 0.001": lambda s: s.posterior_tv < 0.001,
+        "TV < 0.01": lambda s: s.posterior_tv < 0.01,
+        "TV < 0.05": lambda s: s.posterior_tv < 0.05,
+    }
+    lines += [
+        "## Threading a held-out sample: forward-backward",
+        "",
+        (
+            "Smallest mean number of forward values per site relative to exact "
+            "(parameter) within each budget on the posterior total variation "
+            "distance from exact."
+        ),
+        "",
+        best_within_budget(
+            fb, "num_samples", "ratio_forward_values", budgets
+        ).to_markdown(index=False),
+        "",
+    ]
+    exact = df[df.method == "exact"]
+    lines += [
+        "Exact forward values and distinct values per site:",
+        "",
+        exact.groupby("num_samples")[["forward_values", "forward_distinct_values"]]
+        .mean()
+        .reset_index()
+        .to_markdown(index=False),
+        "",
+    ]
+
+
+def analyse_calibration(lines):
+    df = pd.read_csv(RESULTS / "calibration.csv")
+    num_bins = len(CALIBRATION_BINS) - 1
+    rows = []
+    for (setting, method, parameter), sub in df.groupby(
+        ["setting", "method", "parameter"], dropna=False
+    ):
+        count = np.array([sub[f"count_{j}"].sum() for j in range(num_bins)])
+        total_p = np.array([sub[f"total_p_{j}"].sum() for j in range(num_bins)])
+        num_true = np.array([sub[f"num_true_{j}"].sum() for j in range(num_bins)])
+        nonempty = count > 0
+        error = np.abs(num_true[nonempty] - total_p[nonempty]) / count[nonempty]
+        sites = sub.num_site_queries.sum()
+        rows.append(
+            {
+                "setting": setting,
+                "method": method,
+                "parameter": parameter,
+                "reliability_error": np.sum(error * count[nonempty])
+                / np.sum(count[nonempty]),
+                "coverage_90": sub.covered_90.sum() / sites,
+                "expected_coverage_90": sub.set_mass_90.sum() / sites,
+                "forward_values": sub.forward_values.mean(),
+            }
+        )
+    summary = pd.DataFrame(rows)
+    summary["coverage_excess"] = summary.coverage_90 - summary.expected_coverage_90
+    exact = summary[summary.method == "exact"].set_index("setting")
+    summary["ratio_forward_values"] = (
+        summary.forward_values / exact.forward_values.reindex(summary.setting).values
+    )
+    summary.to_csv(RESULTS / "calibration_summary.csv", index=False)
+    plot_frontiers(
+        summary,
+        "setting",
+        "ratio_forward_values",
+        [
+            ("reliability_error", "Mean |observed - predicted| (reliability)"),
+            ("coverage_excess", "90% set coverage - expected"),
+        ],
+        "Mean forward values per site relative to exact",
+        "calibration",
+    )
+    lines += [
+        "## Calibration under the model",
+        "",
+        (
+            "Mean absolute calibration error (weighted over reliability bins) and "
+            "90% credible-set coverage minus expected coverage. Exact values give "
+            "the noise level."
+        ),
+        "",
+        summary[
+            [
+                "setting",
+                "method",
+                "parameter",
+                "ratio_forward_values",
+                "reliability_error",
+                "coverage_excess",
+            ]
+        ].to_markdown(index=False, floatfmt=".4g"),
+        "",
+    ]
+
+
+def analyse_sc2ts(lines):
+    df = pd.read_csv(RESULTS / "sc2ts.csv")
+    df = add_relative_columns(
+        df,
+        "size",
+        [
+            "traceback_size",
+            "viterbi_time",
+            "total_memory",
+            "log_likelihood",
+            "forward_values",
+            "fb_time",
+            "forward_log_likelihood",
+        ],
+    )
+    df["log_likelihood_loss"] = -df.diff_log_likelihood
+    df["identical_path"] = df.identical_path.astype(float)
+    exact_copy = (df.kind == "copy") & (df.k == 0)
+    df["exact_copy_failed"] = np.where(
+        exact_copy, (df.num_mismatches > 0).astype(float), np.nan
+    )
+    viterbi = summarise(
+        df,
+        "size",
+        "ratio_traceback_size",
+        [
+            "log_likelihood_loss",
+            "identical_path",
+            "ratio_viterbi_time",
+            "ratio_total_memory",
+            "exact_copy_failed",
+            "traceback_size",
+            "distinct_values",
+            "viterbi_time",
+        ],
+    )
+    viterbi.to_csv(RESULTS / "sc2ts_viterbi_summary.csv", index=False)
+    plot_frontiers(
+        viterbi,
+        "size",
+        "ratio_traceback_size",
+        [
+            ("log_likelihood_loss", "Viterbi path log-likelihood loss"),
+            ("identical_path", "Fraction of paths identical to exact"),
+            ("ratio_viterbi_time", "Viterbi time relative to exact"),
+        ],
+        "Mean traceback size relative to exact",
+        "sc2ts_viterbi",
+    )
+    budgets = {
+        "no loss": lambda s: (s.log_likelihood_loss < 1e-9) & (s.identical_path == 1),
+        "loss < 0.1": lambda s: s.log_likelihood_loss < 0.1,
+        "loss < 1": lambda s: s.log_likelihood_loss < 1,
+    }
+    lines += [
+        "## sc2ts sub-ARGs: Viterbi",
+        "",
+        (
+            "Smallest mean traceback size relative to exact (parameter) within "
+            "each loss budget."
+        ),
+        "",
+        best_within_budget(viterbi, "size", "ratio_traceback_size", budgets).to_markdown(
+            index=False
+        ),
+        "",
+        "Exact traceback size, distinct values per site and time:",
+        "",
+        viterbi[viterbi.method == "exact"][
+            ["size", "traceback_size", "distinct_values", "viterbi_time"]
+        ].to_markdown(index=False),
+        "",
+        "Time relative to exact for each method's settings:",
+        "",
+        viterbi[viterbi.method != "exact"][
+            [
+                "size",
+                "method",
+                "parameter",
+                "ratio_traceback_size",
+                "ratio_viterbi_time",
+                "log_likelihood_loss",
+                "identical_path",
+            ]
+        ].to_markdown(index=False, floatfmt=".4g"),
+        "",
+    ]
+    if "posterior_tv" in df:
+        fb = summarise(
+            df[df.method != "lossless_floor"].dropna(subset=["posterior_tv"]),
+            "size",
+            "ratio_forward_values",
+            ["posterior_tv", "ratio_fb_time"],
+        )
+        fb.to_csv(RESULTS / "sc2ts_fb_summary.csv", index=False)
+        lines += [
+            "## sc2ts sub-ARGs: forward-backward (10k)",
+            "",
+            fb.to_markdown(index=False, floatfmt=".4g"),
+            "",
+        ]
+
+
+def analyse(scenarios):
+    lines = ["# Thresholding experiments: summary", ""]
+    for scenario, func in [
+        ("threading", analyse_threading),
+        ("calibration", analyse_calibration),
+        ("sc2ts", analyse_sc2ts),
+    ]:
+        if scenario in scenarios and (RESULTS / f"{scenario}.csv").exists():
+            func(lines)
+    (RESULTS / "summary.md").write_text("\n".join(lines))
+    logger.info(f"Wrote {RESULTS / 'summary.md'}")
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__.split("\n\n")[0])
+    parser.add_argument(
+        "--scenario",
+        "-s",
+        action="append",
+        choices=["threading", "calibration", "sc2ts"],
+        help="Scenarios to run (default: all)",
+    )
+    parser.add_argument("--quick", action="store_true", help="Run a small version")
+    parser.add_argument("--num-processes", "-t", type=int, default=1)
+    parser.add_argument(
+        "--analyse-only", action="store_true", help="Only analyse existing results"
+    )
+    args = parser.parse_args()
+    logging.basicConfig(level=logging.INFO, format="%(asctime)s %(message)s")
+    logging.getLogger("matplotlib").setLevel(logging.WARNING)
+    scenarios = args.scenario
+    if scenarios is None:
+        scenarios = ["threading", "calibration", "sc2ts"]
+    if not args.analyse_only:
+        run(scenarios, args.quick, args.num_processes)
+    analyse(scenarios)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/experiments/thresholding/uv.lock b/experiments/thresholding/uv.lock
new file mode 100644
index 0000000..de307aa
--- /dev/null
+++ b/experiments/thresholding/uv.lock
@@ -0,0 +1,1205 @@
+version = 1
+revision = 3
+requires-python = ">=3.11"
+resolution-markers = [
+    "python_full_version >= '3.14' and sys_platform == 'win32'",
+    "python_full_version >= '3.14' and sys_platform == 'emscripten'",
+    "python_full_version >= '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform == 'win32'",
+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform == 'emscripten'",
+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+    "python_full_version < '3.12' and sys_platform == 'win32'",
+    "python_full_version < '3.12' and sys_platform == 'emscripten'",
+    "python_full_version < '3.12' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+]
+
+[[package]]
+name = "attrs"
+version = "26.1.0"
+source = { registry = "https://pypi.org/simple" }
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+[package.metadata.requires-dev]
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diff --git a/tsls/_tslsmodule.c b/tsls/_tslsmodule.c
index dc622b0..5af9b2f 100644
--- a/tsls/_tslsmodule.c
+++ b/tsls/_tslsmodule.c
@@ -269,8 +269,12 @@ AncestorMatcher_init(AncestorMatcher *self, PyObject *args, PyObject *kwds)
     int err;
     int extended_checks = 0;
     int weight_by_n = 1;
+    int pruning_mode = TSLS_PRUNE_NONE;
+    double pruning_parameter = 0;
+    int count_distinct = 0;
     static char *kwlist[] = { "matcher_indexes", "recombination", "mismatch",
-        "extended_checks", "likelihood_threshold", "weight_by_n", NULL };
+        "extended_checks", "likelihood_threshold", "weight_by_n", "pruning_mode",
+        "pruning_parameter", "count_distinct", NULL };
     MatcherIndexes *matcher_indexes = NULL;
     PyObject *recombination = NULL;
     PyObject *mismatch = NULL;
@@ -282,9 +286,10 @@ AncestorMatcher_init(AncestorMatcher *self, PyObject *args, PyObject *kwds)
 
     self->ancestor_matcher = NULL;
     self->matcher_indexes = NULL;
-    if (!PyArg_ParseTupleAndKeywords(args, kwds, "O!OO|idi", kwlist, &MatcherIndexesType,
-            &matcher_indexes, &recombination, &mismatch, &extended_checks,
-            &likelihood_threshold, &weight_by_n)) {
+    if (!PyArg_ParseTupleAndKeywords(args, kwds, "O!OO|idiidi", kwlist,
+            &MatcherIndexesType, &matcher_indexes, &recombination, &mismatch,
+            &extended_checks, &likelihood_threshold, &weight_by_n, &pruning_mode,
+            &pruning_parameter, &count_distinct)) {
         goto out;
     }
     self->matcher_indexes = matcher_indexes;
@@ -326,6 +331,9 @@ AncestorMatcher_init(AncestorMatcher *self, PyObject *args, PyObject *kwds)
     if (!weight_by_n) {
         flags |= TSLS_DISABLE_WEIGHT_BY_N;
     }
+    if (count_distinct) {
+        flags |= TSLS_COUNT_DISTINCT;
+    }
     err = ancestor_matcher_alloc(self->ancestor_matcher,
         self->matcher_indexes->matcher_indexes, PyArray_DATA(recombination_array),
         PyArray_DATA(mismatch_array), likelihood_threshold, flags);
@@ -333,6 +341,12 @@ AncestorMatcher_init(AncestorMatcher *self, PyObject *args, PyObject *kwds)
         handle_library_error(err);
         goto out;
     }
+    err = ancestor_matcher_set_pruning(
+        self->ancestor_matcher, pruning_mode, pruning_parameter);
+    if (err != 0) {
+        handle_library_error(err);
+        goto out;
+    }
     ret = 0;
 out:
     Py_XDECREF(recombination_array);
@@ -632,6 +646,20 @@ AncestorMatcher_get_mean_traceback_size(AncestorMatcher *self, void *closure)
     return ret;
 }
 
+static PyObject *
+AncestorMatcher_get_mean_num_distinct_values(AncestorMatcher *self, void *closure)
+{
+    PyObject *ret = NULL;
+
+    if (AncestorMatcher_check_state(self) != 0) {
+        goto out;
+    }
+    ret = Py_BuildValue(
+        "d", ancestor_matcher_get_mean_num_distinct_values(self->ancestor_matcher));
+out:
+    return ret;
+}
+
 static PyObject *
 AncestorMatcher_get_total_memory(AncestorMatcher *self, void *closure)
 {
@@ -657,6 +685,10 @@ AncestorMatcher_get_total_memory(AncestorMatcher *self, void *closure)
 static PyGetSetDef AncestorMatcher_getsetters[] = {
     { "mean_traceback_size", (getter) AncestorMatcher_get_mean_traceback_size, NULL,
         "The mean size of the traceback per site." },
+    { "mean_num_distinct_values",
+        (getter) AncestorMatcher_get_mean_num_distinct_values, NULL,
+        "The mean number of distinct likelihood values per site in the last "
+        "Viterbi run (if count_distinct is set)." },
     { "total_memory", (getter) AncestorMatcher_get_total_memory, NULL,
         "The total amount of memory used by this matcher." },
     { NULL } /* Sentinel */
diff --git a/tsls/lib/err.c b/tsls/lib/err.c
index 5c6b64d..ba426d9 100644
--- a/tsls/lib/err.c
+++ b/tsls/lib/err.c
@@ -54,6 +54,9 @@ tsls_strerror(int err)
         case TSLS_ERR_BAD_NORMALISATION_FACTOR:
             ret = "Forward normalisation factors must be > 0";
             break;
+        case TSLS_ERR_BAD_PRUNING:
+            ret = "Bad pruning: relative factor must be in [0, 1] and top_n >= 1";
+            break;
     }
     return ret;
 }
diff --git a/tsls/lib/err.h b/tsls/lib/err.h
index 71f11cc..d37c03b 100644
--- a/tsls/lib/err.h
+++ b/tsls/lib/err.h
@@ -16,6 +16,7 @@
 #define TSLS_ERR_BAD_COORDINATE                             -12
 #define TSLS_ERR_BAD_SITE_RANGE                             -13
 #define TSLS_ERR_BAD_NORMALISATION_FACTOR                   -14
+#define TSLS_ERR_BAD_PRUNING                                -15
 // clang-format on
 
 #ifdef __GNUC__
diff --git a/tsls/lib/matcher.c b/tsls/lib/matcher.c
index 186c0a7..f4967ae 100644
--- a/tsls/lib/matcher.c
+++ b/tsls/lib/matcher.c
@@ -20,7 +20,9 @@
  *
  * When a node enters the tree it gets the value 0, and so it can only be
  * reached by recombination. When a node leaves the tree, the probability
- * of continuing from it is only the probability of recombination.
+ * of continuing from it is only the probability of recombination. So a node
+ * can only be copied from one site to the next without recombination if it
+ * is in the tree over the whole interval between them.
  */
 
 #include "tsls.h"
@@ -543,6 +545,7 @@ ancestor_matcher_alloc(ancestor_matcher_t *self,
     self->likelihood_cache = malloc((N + 1) * sizeof(*self->likelihood_cache));
     self->likelihood_nodes = malloc((N + 1) * sizeof(*self->likelihood_nodes));
     self->likelihood_nodes_tmp = malloc((N + 1) * sizeof(*self->likelihood_nodes_tmp));
+    self->values_tmp = malloc((N + 1) * sizeof(*self->values_tmp));
 
     if (self->recombination_rate == NULL || self->mismatch_rate == NULL
         || self->traceback == NULL || self->max_likelihood_node == NULL
@@ -551,7 +554,7 @@ ancestor_matcher_alloc(ancestor_matcher_t *self,
         || self->num_inheriting == NULL || self->recombination_required == NULL
         || self->likelihood == NULL || self->likelihood_cache == NULL
         || self->likelihood_nodes == NULL || self->likelihood_nodes_tmp == NULL
-        || self->allelic_state == NULL) {
+        || self->values_tmp == NULL || self->allelic_state == NULL) {
         ret = TSLS_ERR_NO_MEMORY;
         goto out;
     }
@@ -594,6 +597,7 @@ ancestor_matcher_free(ancestor_matcher_t *self)
     tsi_safe_free(self->likelihood_cache);
     tsi_safe_free(self->likelihood_nodes);
     tsi_safe_free(self->likelihood_nodes_tmp);
+    tsi_safe_free(self->values_tmp);
     tsi_safe_free(self->allelic_state);
     tsi_safe_free(self->max_likelihood_node);
     tsi_safe_free(self->traceback);
@@ -601,6 +605,110 @@ ancestor_matcher_free(ancestor_matcher_t *self)
     return 0;
 }
 
+int
+ancestor_matcher_set_pruning(ancestor_matcher_t *self, int mode, double parameter)
+{
+    int ret = 0;
+
+    if (mode == TSLS_PRUNE_NONE) {
+        parameter = 0;
+    } else if (mode == TSLS_PRUNE_RELATIVE) {
+        if (!(parameter >= 0 && parameter <= 1)) {
+            ret = TSLS_ERR_BAD_PRUNING;
+            goto out;
+        }
+    } else if (mode == TSLS_PRUNE_TOP_N) {
+        if (!(parameter >= 1)) {
+            ret = TSLS_ERR_BAD_PRUNING;
+            goto out;
+        }
+    } else {
+        ret = TSLS_ERR_BAD_PRUNING;
+        goto out;
+    }
+    self->pruning_mode = mode;
+    self->pruning_parameter = parameter;
+out:
+    return ret;
+}
+
+static int
+cmp_double(const void *a, const void *b)
+{
+    const double x = *(const double *) a;
+    const double y = *(const double *) b;
+    return (x > y) - (x < y);
+}
+
+/* Copy the current likelihood values into values_tmp, sorted. */
+static void
+ancestor_matcher_sort_values(ancestor_matcher_t *self)
+{
+    int j;
+
+    for (j = 0; j < self->num_likelihood_nodes; j++) {
+        self->values_tmp[j] = self->likelihood[self->likelihood_nodes[j]];
+    }
+    qsort(self->values_tmp, (size_t) self->num_likelihood_nodes, sizeof(double),
+        cmp_double);
+}
+
+/* Apply the experimental pruning, giving pruned nodes the value
+ * likelihood_threshold. The values must be normalised. */
+static void
+ancestor_matcher_prune(ancestor_matcher_t *self)
+{
+    double *restrict L = self->likelihood;
+    const tsk_id_t *restrict L_nodes = self->likelihood_nodes;
+    const int num_likelihood_nodes = self->num_likelihood_nodes;
+    double max_L, cutoff;
+    int j, n;
+
+    if (self->pruning_mode == TSLS_PRUNE_NONE || num_likelihood_nodes == 0) {
+        return;
+    }
+    if (self->pruning_mode == TSLS_PRUNE_RELATIVE) {
+        max_L = 0;
+        for (j = 0; j < num_likelihood_nodes; j++) {
+            max_L = TSK_MAX(max_L, L[L_nodes[j]]);
+        }
+        cutoff = self->pruning_parameter * max_L;
+    } else {
+        n = (int) self->pruning_parameter;
+        if (num_likelihood_nodes <= n) {
+            return;
+        }
+        ancestor_matcher_sort_values(self);
+        /* Keep values >= the nth largest (ties at the cutoff are kept) */
+        cutoff = self->values_tmp[num_likelihood_nodes - n];
+    }
+    for (j = 0; j < num_likelihood_nodes; j++) {
+        if (L[L_nodes[j]] < cutoff) {
+            L[L_nodes[j]] = self->likelihood_threshold;
+        }
+    }
+}
+
+/* Count the number of distinct likelihood values at this site. */
+static void
+ancestor_matcher_count_distinct(ancestor_matcher_t *self)
+{
+    int j;
+    size_t num_distinct;
+
+    if (!(self->flags & TSLS_COUNT_DISTINCT) || self->num_likelihood_nodes == 0) {
+        return;
+    }
+    ancestor_matcher_sort_values(self);
+    num_distinct = 1;
+    for (j = 1; j < self->num_likelihood_nodes; j++) {
+        if (self->values_tmp[j] != self->values_tmp[j - 1]) {
+            num_distinct++;
+        }
+    }
+    self->total_distinct_values += num_distinct;
+}
+
 static inline void
 ancestor_matcher_add_likelihood(ancestor_matcher_t *self, tsk_id_t u, double value)
 {
@@ -1177,6 +1285,8 @@ ancestor_matcher_process_site_viterbi(
         u = L_nodes[j];
         L[u] = TSK_MAX(L[u] / max_L, self->likelihood_threshold);
     }
+    ancestor_matcher_prune(self);
+    ancestor_matcher_count_distinct(self);
     ret = ancestor_matcher_store_traceback(self, site);
     if (ret != 0) {
         goto out;
@@ -1232,6 +1342,7 @@ ancestor_matcher_process_site_forward(ancestor_matcher_t *self, const tsk_id_t s
         u = L_nodes[j];
         L[u] = TSK_MAX(L[u] / normalisation_factor, self->likelihood_threshold);
     }
+    ancestor_matcher_prune(self);
     *previous_total = ancestor_matcher_total_likelihood(self);
     ret = compressed_matrix_store_site(output, site, normalisation_factor,
         self->num_likelihood_nodes, L_nodes, L);
@@ -1279,6 +1390,7 @@ ancestor_matcher_process_site_backward(ancestor_matcher_t *self, const tsk_id_t
             L[u] = TSK_MAX(
                 x / transition->normalisation_factor, self->likelihood_threshold);
         }
+        ancestor_matcher_prune(self);
     }
     ancestor_matcher_insert_mutation_likelihoods(self, site);
     ret = compressed_matrix_store_site(output, site, forward_normalisation_factor[site],
@@ -1502,6 +1614,7 @@ ancestor_matcher_reset_traceback(ancestor_matcher_t *self)
         tsi_blkalloc_reset(&self->traceback_allocator);
     }
     self->total_traceback_size = 0;
+    self->total_distinct_values = 0;
 out:
     return ret;
 }
@@ -1590,6 +1703,15 @@ ancestor_matcher_get_mean_traceback_size(ancestor_matcher_t *self)
     return (double) self->total_traceback_size / ((double) self->num_sites);
 }
 
+double
+ancestor_matcher_get_mean_num_distinct_values(ancestor_matcher_t *self)
+{
+    if (self->num_sites == 0) {
+        return 0;
+    }
+    return (double) self->total_distinct_values / ((double) self->num_sites);
+}
+
 size_t
 ancestor_matcher_get_total_memory(ancestor_matcher_t *self)
 {
diff --git a/tsls/lib/tsls.h b/tsls/lib/tsls.h
index acd666d..6f12fe4 100644
--- a/tsls/lib/tsls.h
+++ b/tsls/lib/tsls.h
@@ -36,6 +36,15 @@
 /* Matcher flags */
 #define TSLS_EXTENDED_CHECKS     (1 << 0)
 #define TSLS_DISABLE_WEIGHT_BY_N (1 << 1)
+#define TSLS_COUNT_DISTINCT      (1 << 2)
+
+/* Experimental pruning modes. After normalisation at each site, pruned
+ * nodes are given the value likelihood_threshold. */
+#define TSLS_PRUNE_NONE     0
+/* Prune nodes with likelihood < parameter * max likelihood */
+#define TSLS_PRUNE_RELATIVE 1
+/* Prune all but the parameter nodes with the largest likelihoods */
+#define TSLS_PRUNE_TOP_N    2
 
 typedef int8_t allele_t;
 typedef tsk_id_t coordinate_t;
@@ -118,6 +127,8 @@ typedef struct {
     size_t num_sites;
     /* Input LS model rates */
     double likelihood_threshold;
+    int pruning_mode;
+    double pruning_parameter;
     double *recombination_rate;
     double *mismatch_rate;
     /* The quintuply linked tree */
@@ -145,6 +156,11 @@ typedef struct {
     node_state_list_t *traceback;
     tsi_blkalloc_t traceback_allocator;
     size_t total_traceback_size;
+    /* The total number of distinct likelihood values over sites, if
+     * TSLS_COUNT_DISTINCT is set. */
+    size_t total_distinct_values;
+    /* Scratch buffer for values */
+    double *values_tmp;
     size_t traceback_block_size;
     size_t traceback_realloc_size;
 } ancestor_matcher_t;
@@ -163,6 +179,7 @@ size_t compressed_matrix_get_total_values(const compressed_matrix_t *self);
 int ancestor_matcher_alloc(ancestor_matcher_t *self,
     const matcher_indexes_t *matcher_indexes, const double *recombination_rate,
     const double *mismatch_rate, double likelihood_threshold, int flags);
+int ancestor_matcher_set_pruning(ancestor_matcher_t *self, int mode, double parameter);
 int ancestor_matcher_free(ancestor_matcher_t *self);
 int ancestor_matcher_find_path(ancestor_matcher_t *self, tsk_id_t start, tsk_id_t end,
     const allele_t *haplotype, allele_t *matched_haplotype, size_t *path_length,
@@ -173,6 +190,7 @@ int ancestor_matcher_backward(ancestor_matcher_t *self, const allele_t *haplotyp
     const double *forward_normalisation_factor, compressed_matrix_t *output);
 int ancestor_matcher_print_state(ancestor_matcher_t *self, FILE *out);
 double ancestor_matcher_get_mean_traceback_size(ancestor_matcher_t *self);
+double ancestor_matcher_get_mean_num_distinct_values(ancestor_matcher_t *self);
 size_t ancestor_matcher_get_total_memory(ancestor_matcher_t *self);
 
 #define tsi_safe_free(pointer)                                                          \
diff --git a/tsls/tests/dense.py b/tsls/tests/dense.py
index 82a66d3..90b1141 100644
--- a/tsls/tests/dense.py
+++ b/tsls/tests/dense.py
@@ -3,10 +3,13 @@
 tree sequence, used as a reference for the tree-based engine.
 
 The states at a site are the nodes in the tree at that site (nodes with a
-parent or at least one child). Nodes entering the tree have probability
-zero, and so can only be reached by recombination. The probability of
-leaving a node that leaves the tree is only the probability of
-recombination.
+parent or at least one child). We can only continue copying a node from one
+site to the next if it is in the tree over the whole interval between them
+(in every tree from the tree of the previous site to the tree of this one);
+otherwise reaching it again requires a recombination. Nodes entering the
+tree have probability zero, and so can only be reached by recombination.
+Recombination redistributes the total over all states at the previous site,
+including nodes that are no longer in the tree.
 """
 
 import numpy as np
@@ -14,21 +17,24 @@
 
 def node_states(ts, alleles=None):
     """
-    Return the (A, present) tuple of (num_sites, num_nodes) arrays, where
-    A[j, u] is the allele index of node u at site j and present[j, u] is
-    True if node u is in the tree at site j. If alleles is None, the ancestral
-    state is 0 and every derived state is 1.
+    Return the (A, present, continuous) tuple of (num_sites, num_nodes)
+    arrays, where A[j, u] is the allele index of node u at site j,
+    present[j, u] is True if node u is in the tree at site j, and
+    continuous[j, u] is True if node u is in every tree between site j - 1
+    and site j (continuous[0] = present[0]). If alleles is None, the
+    ancestral state is 0 and every derived state is 1.
     """
     A = np.zeros((ts.num_sites, ts.num_nodes), dtype=np.int8)
     present = np.zeros((ts.num_sites, ts.num_nodes), dtype=bool)
+    tree_index = np.zeros(ts.num_sites, dtype=int)
+    in_tree = np.zeros((ts.num_trees, ts.num_nodes), dtype=bool)
 
     def encode(state, default):
         return default if alleles is None else alleles.index(state)
 
     for tree in ts.trees():
-        in_tree = [
-            tree.parent(u) != -1 or tree.num_children(u) > 0 for u in range(ts.num_nodes)
-        ]
+        for u in range(ts.num_nodes):
+            in_tree[tree.index, u] = tree.parent(u) != -1 or tree.num_children(u) > 0
         for site in tree.sites():
             state = {}
             for mutation in site.mutations:
@@ -39,8 +45,13 @@ def encode(state, default):
                 while v != -1 and v not in state:
                     v = tree.parent(v)
                 A[site.id, u] = ancestral if v == -1 else state[v]
-                present[site.id, u] = in_tree[u]
-    return A, present
+            present[site.id] = in_tree[tree.index]
+            tree_index[site.id] = tree.index
+    continuous = present.copy()
+    for j in range(1, ts.num_sites):
+        trees = in_tree[tree_index[j - 1] : tree_index[j] + 1]
+        continuous[j] = np.all(trees, axis=0)
+    return A, present, continuous
 
 
 def emission(A_site, h_site, mu, num_alleles):
@@ -48,7 +59,7 @@ def emission(A_site, h_site, mu, num_alleles):
     return np.where(match, 1 - (num_alleles - 1) * mu, mu)
 
 
-def forward(A, present, h, rho, mu, num_alleles, n, threshold=0):
+def forward(A, present, continuous, h, rho, mu, num_alleles, n, threshold=0):
     """
     Return the (F, c) tuple of normalised forward probabilities and
     normalisation factors.
@@ -58,11 +69,8 @@ def forward(A, present, h, rho, mu, num_alleles, n, threshold=0):
     c = np.zeros(m)
     f = present[0].astype(float)
     for j in range(m):
-        # Recombination redistributes the total over all states at the
-        # previous site, including nodes that are not in the tree at this
-        # site.
         total = np.sum(f)
-        f = np.where(present[j], f, 0)
+        f = np.where(continuous[j], f, 0)
         f = (f * (1 - rho[j]) + rho[j] / n * total) * emission(
             A[j], h[j], mu[j], num_alleles
         )
@@ -73,7 +81,7 @@ def forward(A, present, h, rho, mu, num_alleles, n, threshold=0):
     return F, c
 
 
-def backward(A, present, h, rho, mu, num_alleles, n, c, threshold=0):
+def backward(A, present, continuous, h, rho, mu, num_alleles, n, c, threshold=0):
     """
     Return the backward probabilities, scaled by the forward normalisation
     factors c.
@@ -84,7 +92,7 @@ def backward(A, present, h, rho, mu, num_alleles, n, c, threshold=0):
     total = 0
     for j in range(m - 1, -1, -1):
         if j < m - 1:
-            b = np.where(present[j], b, 0)
+            b = np.where(continuous[j + 1], b, 0)
             b = ((1 - rho[j + 1]) * b + rho[j + 1] * total / n) / c[j + 1]
             b = np.where(present[j], np.maximum(b, threshold), 0)
         B[j] = b
@@ -93,7 +101,7 @@ def backward(A, present, h, rho, mu, num_alleles, n, c, threshold=0):
     return B
 
 
-def viterbi_log_likelihood(A, present, h, rho, mu, num_alleles, n):
+def viterbi_log_likelihood(A, present, continuous, h, rho, mu, num_alleles, n):
     """
     Return the log likelihood of the most likely path, where staying on a
     node has probability (1 - rho + rho / n) and switching to a given node
@@ -103,7 +111,7 @@ def viterbi_log_likelihood(A, present, h, rho, mu, num_alleles, n):
     v = present[0].astype(float)
     log_likelihood = 0
     for j in range(m):
-        v = np.where(present[j], v, 0)
+        v = np.where(continuous[j], v, 0)
         v = np.maximum(v * (1 - rho[j] + rho[j] / n), rho[j] / n)
         v = np.where(present[j], v * emission(A[j], h[j], mu[j], num_alleles), 0)
         max_v = np.max(v)
@@ -112,15 +120,17 @@ def viterbi_log_likelihood(A, present, h, rho, mu, num_alleles, n):
     return log_likelihood
 
 
-def path_log_likelihood(A, present, h, rho, mu, num_alleles, n, path):
+def path_log_likelihood(A, present, continuous, h, rho, mu, num_alleles, n, path):
     """
     Return the log likelihood of the specified path (the node copied at
-    each site) under the same model as viterbi_log_likelihood.
+    each site) under the same model as viterbi_log_likelihood. Copying the
+    same node at consecutive sites counts as a switch if the node is not in
+    the tree over the whole interval between them.
     """
     log_likelihood = 0
     for j, u in enumerate(path):
         assert present[j, u]
-        if j == 0 or path[j - 1] == u:
+        if j == 0 or (path[j - 1] == u and continuous[j, u]):
             p_t = 1 - rho[j] + rho[j] / n
         else:
             p_t = rho[j] / n
diff --git a/tsls/tests/test_matching.py b/tsls/tests/test_matching.py
index 05c6fec..59a8928 100644
--- a/tsls/tests/test_matching.py
+++ b/tsls/tests/test_matching.py
@@ -178,11 +178,54 @@ def isolated_root_ts():
     return finalise(tables)
 
 
+def gap_ts():
+    # Node 3 is in the left and right trees but not in the middle tree, which
+    # has no sites, so copying node 3 across the middle tree requires a
+    # recombination. Node 4 is only in the middle tree, and so is never a
+    # state at a site.
+    #
+    # 2.00┊    5    ┊    5    ┊    5    ┊
+    #     ┊  ┏━┻┓   ┊  ┏━┻┓   ┊  ┏━┻┓   ┊
+    # 1.50┊  ┃  ┃   ┊  4  ┃   ┊  ┃  ┃   ┊
+    #     ┊  ┃  ┃   ┊ ┏┻┓ ┃   ┊  ┃  ┃   ┊
+    # 1.00┊  3  ┃   ┊ ┃ ┃ ┃   ┊  3  ┃   ┊
+    #     ┊ ┏┻┓ ┃   ┊ ┃ ┃ ┃   ┊ ┏┻┓ ┃   ┊
+    # 0.00┊ 0 1 2   ┊ 0 1 2   ┊ 0 1 2   ┊
+    #     0         4         8        12
+    tables = tskit.TableCollection(sequence_length=12)
+    for t in [0, 0, 0, 1, 1.5, 2]:
+        tables.nodes.add_row(flags=tskit.NODE_IS_SAMPLE if t == 0 else 0, time=t)
+    for left, right, parent, child in [
+        (0, 4, 3, 0),
+        (0, 4, 3, 1),
+        (8, 12, 3, 0),
+        (8, 12, 3, 1),
+        (4, 8, 4, 0),
+        (4, 8, 4, 1),
+        (0, 4, 5, 3),
+        (8, 12, 5, 3),
+        (4, 8, 5, 4),
+        (0, 12, 5, 2),
+    ]:
+        tables.edges.add_row(left, right, parent, child)
+    add_mutations(
+        tables,
+        [
+            (1, "A", [(3, "C")]),
+            (3, "G", [(0, "T"), (2, "A")]),
+            (9, "T", [(3, "G"), (1, "A")]),
+            (11, "C", [(5, "A"), (0, "G")]),
+        ],
+    )
+    return finalise(tables)
+
+
 EXAMPLES = {
     "single_tree": single_tree_ts,
     "two_trees": two_tree_ts,
     "forest": forest_ts,
     "isolated_root": isolated_root_ts,
+    "gap": gap_ts,
 }
 ALLELES = ACGT + ["-"]
 PARAMS = [(1e-2, 1e-2), (0.2, 0.05), (1e-4, 1e-3)]
@@ -241,13 +284,13 @@ def path_by_site(ts, left, right, parent):
 @pytest.fixture(scope="module", params=list(EXAMPLES))
 def example(request):
     ts = EXAMPLES[request.param]()
-    A, present = dense.node_states(ts, ALLELES)
-    return ts, A, present
+    A, present, continuous = dense.node_states(ts, ALLELES)
+    return ts, A, present, continuous
 
 
 class TestExamples:
     def test_node_states(self, example):
-        ts, A, present = example
+        ts, A, present, continuous = example
         G = ts.genotype_matrix(isolated_as_missing=False, alleles=tuple(ALLELES))
         np.testing.assert_array_equal(A[:, ts.samples()], G)
         assert np.any(np.bincount(ts.mutations_site) > 1)
@@ -257,7 +300,7 @@ def test_node_states(self, example):
 @pytest.mark.parametrize(("rho", "mu"), PARAMS)
 class TestAgainstDense:
     def test_viterbi(self, example, weight_by_n, rho, mu):
-        ts, A, present = example
+        ts, A, present, continuous = example
         am, n = make_matcher(ts, rho, mu, weight_by_n)
         m = ts.num_sites
         rho_a = np.full(m, rho)
@@ -269,33 +312,35 @@ def test_viterbi(self, example, weight_by_n, rho, mu):
             np.testing.assert_array_equal(match, A[np.arange(m), path])
             assert np.all(present[np.arange(m), path])
             value = dense.path_log_likelihood(
-                A, present, h, rho_a, mu_a, len(ALLELES), n, path
+                A, present, continuous, h, rho_a, mu_a, len(ALLELES), n, path
             )
             optimum = dense.viterbi_log_likelihood(
-                A, present, h, rho_a, mu_a, len(ALLELES), n
+                A, present, continuous, h, rho_a, mu_a, len(ALLELES), n
             )
             assert value == pytest.approx(optimum, rel=1e-9)
 
     def test_forward_backward(self, example, weight_by_n, rho, mu):
-        ts, A, present = example
+        ts, A, present, continuous = example
         am, n = make_matcher(ts, rho, mu, weight_by_n)
         m = ts.num_sites
         rho_a = np.full(m, rho)
         mu_a = np.full(m, mu)
         for h in queries(ts, A, present):
             fm = am.forward_matrix(h)
-            F, c = dense.forward(A, present, h, rho_a, mu_a, len(ALLELES), n)
+            F, c = dense.forward(A, present, continuous, h, rho_a, mu_a, len(ALLELES), n)
             np.testing.assert_allclose(fm.normalisation_factor, c, rtol=1e-12)
             np.testing.assert_allclose(fm.decode(), F, rtol=1e-12, atol=1e-300)
             assert fm.log_likelihood == pytest.approx(np.sum(np.log(c)))
 
             bm = am.backward_matrix(h, fm.normalisation_factor)
-            B = dense.backward(A, present, h, rho_a, mu_a, len(ALLELES), n, c)
+            B = dense.backward(
+                A, present, continuous, h, rho_a, mu_a, len(ALLELES), n, c
+            )
             np.testing.assert_allclose(bm.decode(), B, rtol=1e-12, atol=1e-300)
             np.testing.assert_allclose(np.sum(F * B, axis=1), 1)
 
     def test_likelihood_threshold(self, example, weight_by_n, rho, mu):
-        ts, A, present = example
+        ts, A, present, continuous = example
         threshold = 1e-3
         am, n = make_matcher(ts, rho, mu, weight_by_n, likelihood_threshold=threshold)
         m = ts.num_sites
@@ -304,12 +349,29 @@ def test_likelihood_threshold(self, example, weight_by_n, rho, mu):
         for h in queries(ts, A, present):
             fm = am.forward_matrix(h)
             F, c = dense.forward(
-                A, present, h, rho_a, mu_a, len(ALLELES), n, threshold=threshold
+                A,
+                present,
+                continuous,
+                h,
+                rho_a,
+                mu_a,
+                len(ALLELES),
+                n,
+                threshold=threshold,
             )
             np.testing.assert_allclose(fm.decode(), F, rtol=1e-12)
             bm = am.backward_matrix(h, fm.normalisation_factor)
             B = dense.backward(
-                A, present, h, rho_a, mu_a, len(ALLELES), n, c, threshold=threshold
+                A,
+                present,
+                continuous,
+                h,
+                rho_a,
+                mu_a,
+                len(ALLELES),
+                n,
+                c,
+                threshold=threshold,
             )
             np.testing.assert_allclose(bm.decode(), B, rtol=1e-12)
             match = np.zeros(m, dtype=np.int8)
@@ -319,19 +381,38 @@ def test_likelihood_threshold(self, example, weight_by_n, rho, mu):
 
 
 class TestViterbiPaths:
+    def test_gap_requires_recombination(self):
+        # Copying node 3 across the middle tree, where it is not in the tree,
+        # needs two switches, so a node in every tree is preferred.
+        ts = gap_ts()
+        A, _, _ = dense.node_states(ts, ALLELES)
+        am, _ = make_matcher(ts, 1e-4, 1e-2, True)
+        h = A[:, 3]
+        match = np.zeros(ts.num_sites, dtype=np.int8)
+        _, _, parent = am.find_path(h, 0, ts.num_sites, match)
+        assert 3 not in parent
+        # With a high recombination rate, we copy node 3 in two segments.
+        am, _ = make_matcher(ts, 0.3, 1e-3, True)
+        left, right, parent = am.find_path(h, 0, ts.num_sites, match)
+        # Breakpoints are at site positions: the first site after the gap.
+        assert list(parent) == [3, 3]
+        assert list(left) == [9, 0]
+        np.testing.assert_array_equal(match, h)
+
     @pytest.mark.parametrize("name", list(EXAMPLES))
     def test_exact_copy(self, name):
         # Each node that is in every tree and has a unique haplotype is
         # copied exactly with no recombination.
         ts = EXAMPLES[name]()
-        A, present = dense.node_states(ts, ALLELES)
+        A, _, continuous = dense.node_states(ts, ALLELES)
         am, _ = make_matcher(ts, 1e-4, 1e-3, True)
         m = ts.num_sites
+        always = np.all(continuous, axis=0)
         for u in range(ts.num_nodes):
             h = A[:, u]
-            if not np.all(present[:, u]):
+            if not always[u]:
                 continue
-            num_same = np.sum(np.all(A == h[:, None], axis=0) & np.all(present, axis=0))
+            num_same = np.sum(np.all(A == h[:, None], axis=0) & always)
             if num_same > 1:
                 continue
             match = np.zeros(m, dtype=np.int8)
@@ -344,7 +425,7 @@ def test_exact_copy(self, name):
     def test_two_tree_recombinant(self):
         # Copy node 4 (only in the left tree) then node 7 (only in the right)
         ts = two_tree_ts()
-        A, _ = dense.node_states(ts, ALLELES)
+        A, _, _ = dense.node_states(ts, ALLELES)
         h = np.concatenate([A[:3, 4], A[3:, 7]])
         am, _ = make_matcher(ts, 1e-2, 1e-3, True)
         match = np.zeros(ts.num_sites, dtype=np.int8)
@@ -357,7 +438,7 @@ def test_two_tree_recombinant(self):
 
     def test_isolated_node_never_copied(self):
         ts = isolated_root_ts()
-        A, present = dense.node_states(ts, ALLELES)
+        A, present, _ = dense.node_states(ts, ALLELES)
         assert not np.any(present[:, 0])
         am, _ = make_matcher(ts, 0.2, 0.05, True)
         for h in queries(ts, A, present):
@@ -367,7 +448,7 @@ def test_isolated_node_never_copied(self):
 
     def test_start_end(self):
         ts = two_tree_ts()
-        A, _ = dense.node_states(ts, ALLELES)
+        A, _, _ = dense.node_states(ts, ALLELES)
         h = A[:, 0].copy()
         h[0] = -1
         h[-1] = -1
@@ -466,7 +547,7 @@ def test_reuse_matcher(self):
         # Running the algorithms repeatedly on the same matcher gives the
         # same results.
         ts = two_tree_ts()
-        A, present = dense.node_states(ts, ALLELES)
+        A, present, _ = dense.node_states(ts, ALLELES)
         am, _ = make_matcher(ts, 0.2, 0.05, True)
         h = queries(ts, A, present)[-1]
         m = ts.num_sites
@@ -491,3 +572,95 @@ def test_compressed_matrix(self):
         assert len(nodes) == len(values) > 0
         D = fm.decode([0, 6])
         assert D.shape == (ts.num_sites, 2)
+
+
+class TestPruning:
+    def run_all(self, ts, h, **kwargs):
+        m = ts.num_sites
+        mi = tsls.MatcherIndexes(ts, alleles=ALLELES)
+        am = tsls.AncestorMatcher(
+            mi, np.full(m, 0.05), np.full(m, 0.02), extended_checks=True, **kwargs
+        )
+        match = np.zeros(m, dtype=np.int8)
+        path = am.find_path(h, 0, m, match)
+        fm = am.forward_matrix(h)
+        bm = am.backward_matrix(h, fm.normalisation_factor)
+        return am, path, match, fm.decode(), bm.decode()
+
+    @pytest.mark.parametrize("name", list(EXAMPLES))
+    @pytest.mark.parametrize("pruning", [("relative", 0), ("top_n", 1000)])
+    def test_weak_pruning_is_exact(self, name, pruning):
+        ts = EXAMPLES[name]()
+        A, present, _ = dense.node_states(ts, ALLELES)
+        for h in queries(ts, A, present):
+            _, path1, match1, F1, B1 = self.run_all(ts, h)
+            _, path2, match2, F2, B2 = self.run_all(ts, h, pruning=pruning)
+            for x, y in zip(path1, path2):
+                np.testing.assert_array_equal(x, y)
+            np.testing.assert_array_equal(match1, match2)
+            np.testing.assert_array_equal(F1, F2)
+            np.testing.assert_array_equal(B1, B2)
+
+    @pytest.mark.parametrize("pruning", [("relative", 0.5), ("top_n", 1)])
+    def test_strong_pruning_reduces_traceback(self, pruning):
+        ts = two_tree_ts()
+        A, present, _ = dense.node_states(ts, ALLELES)
+        h = queries(ts, A, present)[-2]
+        am1, *_ = self.run_all(ts, h)
+        size1 = am1.mean_traceback_size
+        am2, *_ = self.run_all(ts, h, pruning=pruning)
+        assert am2.mean_traceback_size < size1
+
+    @pytest.mark.parametrize("name", list(EXAMPLES))
+    @pytest.mark.parametrize("weight_by_n", [True, False])
+    def test_lossless_floor(self, name, weight_by_n):
+        # Values below t = (rho / n) / (1 - rho + rho / n) are beaten by
+        # recombination at the next site, so flooring at t does not change
+        # the Viterbi likelihood.
+        ts = EXAMPLES[name]()
+        A, present, continuous = dense.node_states(ts, ALLELES)
+        rho, mu = 0.05, 0.02
+        n = ts.num_nodes if weight_by_n else 1
+        threshold = (rho / n) / (1 - rho + rho / n)
+        m = ts.num_sites
+        am, _ = make_matcher(
+            ts, rho, mu, weight_by_n, likelihood_threshold=threshold * (1 - 1e-12)
+        )
+        rho_a = np.full(m, rho)
+        mu_a = np.full(m, mu)
+        for h in queries(ts, A, present):
+            match = np.zeros(m, dtype=np.int8)
+            path = path_by_site(ts, *am.find_path(h, 0, m, match))
+            value = dense.path_log_likelihood(
+                A, present, continuous, h, rho_a, mu_a, len(ALLELES), n, path
+            )
+            optimum = dense.viterbi_log_likelihood(
+                A, present, continuous, h, rho_a, mu_a, len(ALLELES), n
+            )
+            assert value == pytest.approx(optimum, rel=1e-9)
+
+    def test_count_distinct(self):
+        ts = two_tree_ts()
+        A, present, _ = dense.node_states(ts, ALLELES)
+        h = queries(ts, A, present)[-2]
+        am, *_ = self.run_all(ts, h, count_distinct=True)
+        assert 0 < am.mean_num_distinct_values <= am.mean_traceback_size
+        am, *_ = self.run_all(ts, h)
+        assert am.mean_num_distinct_values == 0
+
+    @pytest.mark.parametrize(
+        "pruning", [("relative", 2), ("relative", -1), ("top_n", 0), ("top_n", 0.5)]
+    )
+    def test_bad_pruning(self, pruning):
+        ts = single_tree_ts()
+        mi = tsls.MatcherIndexes(ts, alleles=ALLELES)
+        m = ts.num_sites
+        with pytest.raises(tsls.LibraryError, match="pruning"):
+            tsls.AncestorMatcher(mi, np.zeros(m), np.zeros(m), pruning=pruning)
+
+    def test_unknown_pruning_mode(self):
+        ts = single_tree_ts()
+        mi = tsls.MatcherIndexes(ts, alleles=ALLELES)
+        m = ts.num_sites
+        with pytest.raises(ValueError, match="Unknown pruning"):
+            tsls.AncestorMatcher(mi, np.zeros(m), np.zeros(m), pruning=("x", 1))
diff --git a/tsls/tsls/matching.py b/tsls/tsls/matching.py
index 1fa0c26..feb0872 100644
--- a/tsls/tsls/matching.py
+++ b/tsls/tsls/matching.py
@@ -106,23 +106,34 @@ def decode(self, nodes=None):
         """
         Return the dense (num_sites, len(nodes)) array of values for the
         specified nodes (default: all nodes). Nodes that are not in the tree
-        at a site have value 0. This is intended for small examples.
+        at a site have value 0.
         """
+        N = self.ts.num_nodes
         if nodes is None:
-            nodes = np.arange(self.ts.num_nodes)
+            nodes = np.arange(N)
         nodes = np.asarray(nodes, dtype=np.int32)
         D = np.zeros((self.num_sites, len(nodes)))
         for tree in self.ts.trees():
+            # The parent array has an extra entry for the virtual root, which
+            # is the parent of the roots (index -1) and never has a value.
+            parent = tree.parent_array
+            in_tree = (parent[:N] != -1) | (tree.num_children_array[:N] > 0)
             for site in tree.sites():
                 site_nodes, site_values = self.get_site(site.id)
-                value = dict(zip(site_nodes.tolist(), site_values.tolist()))
-                for j, u in enumerate(nodes.tolist()):
-                    if tree.parent(u) == -1 and tree.num_children(u) == 0:
-                        continue
-                    v = u
-                    while v not in value:
-                        v = tree.parent(v)
-                    D[site.id, j] = value[v]
+                value = np.full(N + 1, np.nan)
+                value[site_nodes] = site_values
+                # Each node takes the value of its nearest ancestor with a
+                # value: at each step, copy values from the ancestor and move
+                # up a level, so this takes as many steps as the tree is deep.
+                ancestor = parent.copy()
+                unset = np.where(in_tree & np.isnan(value[:N]))[0]
+                while len(unset) > 0:
+                    if np.any(ancestor[unset] == -1):
+                        raise ValueError("Node in tree with no stored ancestor")
+                    value[unset] = value[ancestor[unset]]
+                    ancestor[unset] = parent[ancestor[unset]]
+                    unset = unset[np.isnan(value[unset])]
+                D[site.id] = np.where(in_tree[nodes], value[nodes], 0)
         return D
 
 
@@ -131,8 +142,18 @@ class AncestorMatcher(_tsls.AncestorMatcher):
     Runs the Viterbi, forward and backward algorithms for haplotypes against
     the tree sequence in the specified MatcherIndexes, using the specified
     per-site recombination and mismatch probabilities.
+
+    After normalisation at each site, values below likelihood_threshold are
+    set to it. The experimental ``pruning`` option also sets the values of
+    some nodes to likelihood_threshold: either ``("relative", f)``, for
+    nodes with values less than f times the maximum, or ``("top_n", n)``,
+    for all but the n nodes with the largest values. If ``count_distinct``
+    is True, the mean number of distinct values per site in Viterbi is
+    recorded in ``mean_num_distinct_values``.
     """
 
+    PRUNING_MODES = {"relative": 1, "top_n": 2}
+
     def __init__(
         self,
         matcher_indexes,
@@ -142,8 +163,16 @@ def __init__(
         likelihood_threshold=None,
         weight_by_n=None,
         extended_checks=False,
+        pruning=None,
+        count_distinct=False,
     ):
-        kwargs = {"extended_checks": extended_checks}
+        kwargs = {"extended_checks": extended_checks, "count_distinct": count_distinct}
+        if pruning is not None:
+            mode, parameter = pruning
+            if mode not in self.PRUNING_MODES:
+                raise ValueError(f"Unknown pruning mode {mode!r}")
+            kwargs["pruning_mode"] = self.PRUNING_MODES[mode]
+            kwargs["pruning_parameter"] = parameter
         if likelihood_threshold is not None:
             kwargs["likelihood_threshold"] = likelihood_threshold
         if weight_by_n is not None:
diff --git a/verification/README.md b/verification/README.md
new file mode 100644
index 0000000..193281d
--- /dev/null
+++ b/verification/README.md
@@ -0,0 +1,83 @@
+# Statistical validation of tsls
+
+`verification.py` runs empirical validations of the tsls HMM engine on
+ARGs simulated with msprime, and writes plots and CSV files. It is
+modelled on msprime's `verification.py`.
+
+These are not unit tests. They don't make hard assertions, and they are
+not run in CI. They are meant to give you a feel for how the engine
+behaves statistically, and to catch problems that only show up on
+realistic ARGs.
+
+```
+uv sync
+uv run python verification.py --list          # list the tests
+uv run python verification.py -t 4            # run everything on 4 processes
+uv run python verification.py test_calibration_default
+uv run python verification.py -c HeldOutThreading
+```
+
+Output goes to `tmp__NOBACKUP__/<class>/<test>/` (`-d` changes the base
+directory). Each test takes a few minutes on one core.
+
+## Calibration under the model (`ModelCalibration`)
+
+These tests simulate queries from the HMM itself:
+
+1. Sample a copying path exactly from the engine's prior over paths on an
+   msprime ARG. This uses a backward pass with no data, then forward
+   sampling, so it allows for nodes entering and leaving the tree.
+2. Add mismatches using the engine's emission model.
+
+Because the data come from the model, the forward-backward posteriors
+should be calibrated.
+
+- **`reliability.png`**: (site, node) pairs are binned by posterior
+  probability. The plot shows the fraction in each bin where the node is
+  the true one, with 95% intervals. The points should lie on y = x.
+- **`credible_sets.png`**: the smallest set of nodes reaching each
+  posterior level should contain the true node at about that rate. Set
+  sizes show how uncertain the copying target is. Nodes with identical
+  haplotypes over a region can't be told apart.
+- **`decoding.png`** (from the `decoding_accuracy` tests): per-site accuracy
+  of Viterbi and posterior-max decoding, the mean posterior of the true
+  node, and Viterbi against true switch counts, as μ or ρ varies.
+
+With `weight_by_n=False` the prior over paths is an unnormalised measure.
+We sample from it exactly, so the posteriors should still be calibrated.
+
+## Held-out sample threading (`HeldOutThreading`)
+
+A sample is held out of an msprime simulation, and its haplotype is
+matched against the ARG of the other samples. The "truth" at each site is
+the held-out sample's parent in the full ARG: the node where its lineage
+joins the others, which is the best node to copy. These tests need no
+assumptions about the HMM. They show how well the copying path tracks the
+true genealogy as the mutation rate, recombination rate, sample size and
+HMM ρ vary. The HMM uses sc2ts-style parameters (`k` mismatches cost
+about one recombination, with `weight_by_n=False`).
+
+- **`threading.png`** shows four panels:
+  - the fraction of sites where the Viterbi node is the parent, or within
+    2 edges of it;
+  - the posterior mass on the parent;
+  - the time error of the Viterbi node;
+  - the number of switches, true and Viterbi.
+- **`breakpoints.png`** shows the distance from each true switch of the
+  parent to the nearest Viterbi switch, and true against Viterbi switch
+  counts.
+
+## Numerics and approximations (`Numerics`)
+
+- **`test_likelihood_threshold`** shows what the likelihood threshold costs
+  and gains: the loss in Viterbi path likelihood, the distance of the
+  posteriors from the exact ones, traceback size, time and threading
+  accuracy.
+- **`test_posterior_normalisation`** shows Σ_u F·B, which should be 1 at
+  every site, and the forward log-likelihood per site, along sequences
+  with up to about 10⁴ sites.
+- **`test_dense_agreement_random_args`** compares the engine with the
+  dense reference (`tsls/tests/dense.py`) on many small random ARGs with
+  random parameters. Errors should be around machine precision.
+- **`test_weight_by_n`** compares threading with `weight_by_n` on and off,
+  with the same per-node switching ratio.
diff --git a/verification/pyproject.toml b/verification/pyproject.toml
new file mode 100644
index 0000000..cf3d4de
--- /dev/null
+++ b/verification/pyproject.toml
@@ -0,0 +1,26 @@
+[project]
+name = "tsls-verification"
+version = "0.0.0"
+description = "Statistical validation of the tsls Li and Stephens HMM engine."
+requires-python = ">=3.11"
+dependencies = [
+    "matplotlib",
+    "msprime",
+    "numpy>=2",
+    "pandas",
+    "tqdm",
+    "tskit>=1.0.0",
+    "tsls",
+]
+
+[tool.uv]
+package = false
+
+[tool.uv.sources]
+tsls = { path = "../tsls", editable = true }
+
+[tool.ruff]
+line-length = 89
+
+[tool.ruff.lint.isort]
+known-first-party = ["dense", "tsls"]
diff --git a/verification/uv.lock b/verification/uv.lock
new file mode 100644
index 0000000..e5adb25
--- /dev/null
+++ b/verification/uv.lock
@@ -0,0 +1,1194 @@
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+revision = 3
+requires-python = ">=3.11"
+resolution-markers = [
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+    "python_full_version >= '3.14' and sys_platform == 'emscripten'",
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+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform == 'win32'",
+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform == 'emscripten'",
+    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+    "python_full_version < '3.12' and sys_platform == 'win32'",
+    "python_full_version < '3.12' and sys_platform == 'emscripten'",
+    "python_full_version < '3.12' and sys_platform != 'emscripten' and sys_platform != 'win32'",
+]
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+]
+
+[[package]]
+name = "contourpy"
+version = "1.3.3"
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diff --git a/verification/verification.py b/verification/verification.py
new file mode 100644
index 0000000..257e06b
--- /dev/null
+++ b/verification/verification.py
@@ -0,0 +1,1136 @@
+"""
+Statistical validation of the tsls Li and Stephens HMM engine.
+
+This is not a test suite: it runs empirical validations on simulated ARGs
+and writes plots that help a user get a feel for the statistical
+performance of the engine, rather than making hard assertions. It is
+modelled on msprime's verification.py.
+
+Tests are organised into classes of similar tests, and each test is
+usually a call to a general method with different parameters. Tests must
+be independent and not depend on any shared state within the class, other
+than self.output_dir, which is guaranteed to exist when the test method is
+called. The output directory is <output-dir>/<class name>/<test name>.
+Each test writes one or more plots and a CSV file of the numbers behind
+them. Test method names must be unique across all classes.
+
+To list the available tests:
+
+    uv run python verification.py --list
+
+Run with no arguments to run all tests, with test names to run specific
+tests, or with -c to run all the tests in a class. Use -t to run tests in
+parallel processes.
+"""
+
+import argparse
+import concurrent.futures
+import inspect
+import logging
+import pathlib
+import sys
+import time
+
+import matplotlib
+
+matplotlib.use("Agg")
+
+import msprime
+import numpy as np
+import pandas as pd
+import tqdm
+from matplotlib import pyplot as plt
+
+import tsls
+
+sys.path.insert(
+    0, str(pathlib.Path(__file__).resolve().parent.parent / "tsls" / "tests")
+)
+import dense
+
+logger = logging.getLogger(__name__)
+
+ALLELES = ("A", "C", "G", "T")
+NUM_ALLELES = len(ALLELES)
+
+
+# ---------------------------------------------------------------------------
+# Helpers
+# ---------------------------------------------------------------------------
+
+
+def simulate_arg(
+    num_samples,
+    sequence_length,
+    recombination_rate,
+    mutation_rate,
+    seed,
+    population_size=1e4,
+):
+    """
+    Simulate a haploid ARG with JC69 mutations (so sites can have several
+    mutations and alleles).
+    """
+    ts = msprime.sim_ancestry(
+        num_samples,
+        ploidy=1,
+        population_size=population_size,
+        sequence_length=sequence_length,
+        recombination_rate=recombination_rate,
+        random_seed=seed,
+    )
+    return msprime.sim_mutations(
+        ts, rate=mutation_rate, model=msprime.JC69(), random_seed=seed
+    )
+
+
+def node_states(ts):
+    """
+    Return the (A, present, continuous) tuple of (num_sites, num_nodes)
+    arrays, where A[j, u] is the allele index of node u at site j,
+    present[j, u] is True if node u is in the tree (has a parent or a child)
+    at site j, and continuous[j, u] is True if node u is in every tree
+    between site j - 1 and site j. This is the vectorised equivalent of
+    dense.node_states.
+    """
+    N = ts.num_nodes
+    A = ts.genotype_matrix(
+        samples=np.arange(N, dtype=np.int32), isolated_as_missing=False, alleles=ALLELES
+    ).astype(np.int8)
+    present = np.zeros((ts.num_sites, N), dtype=bool)
+    continuous = np.zeros((ts.num_sites, N), dtype=bool)
+    position = ts.sites_position
+    # Nodes that have been in every tree since the last site
+    since_last_site = np.ones(N, dtype=bool)
+    for tree in ts.trees():
+        start, stop = np.searchsorted(position, tree.interval)
+        in_tree = (tree.parent_array[:N] != -1) | (tree.num_children_array[:N] > 0)
+        since_last_site &= in_tree
+        if stop > start:
+            present[start:stop] = in_tree
+            continuous[start] = since_last_site
+            continuous[start + 1 : stop] = in_tree
+            since_last_site = in_tree.copy()
+    if ts.num_sites > 0:
+        continuous[0] = present[0]
+    return A, present, continuous
+
+
+def hmm_params(k, num_alleles=NUM_ALLELES):
+    """
+    Return the (mu, rho) tuple for which k mismatches have about the same
+    likelihood as one recombination, with weight_by_n=False (as in sc2ts and
+    benchmarking/engines.py).
+    """
+    mu = 0.0125
+    denom = mu**k + (1 - (num_alleles - 1) * mu) ** k
+    rho = mu**k / denom
+    rho += rho * 0.1
+    return mu, rho
+
+
+def n_weighted_rho(rho, n):
+    """
+    Return the rho for weight_by_n=True with the same ratio of switching to a
+    given node to staying as rho with weight_by_n=False.
+    """
+    return rho / (rho + (1 - rho) / n)
+
+
+def make_matcher(
+    ts,
+    mu,
+    rho,
+    *,
+    weight_by_n=True,
+    likelihood_threshold=None,
+    pruning=None,
+    count_distinct=False,
+    matcher_indexes=None,
+):
+    m = ts.num_sites
+    if matcher_indexes is None:
+        matcher_indexes = tsls.MatcherIndexes(ts, alleles=ALLELES)
+    return tsls.AncestorMatcher(
+        matcher_indexes,
+        np.full(m, rho),
+        np.full(m, mu),
+        weight_by_n=weight_by_n,
+        likelihood_threshold=likelihood_threshold,
+        pruning=pruning,
+        count_distinct=count_distinct,
+    )
+
+
+def viterbi(am, ts, h):
+    """
+    Return the (path, matched) tuple, where path[j] is the node copied at
+    site j.
+    """
+    m = ts.num_sites
+    matched = np.zeros(m, dtype=np.int8)
+    left, _, parent = am.find_path(h.astype(np.int8), 0, m, matched)
+    index = np.argsort(left)
+    segment = np.searchsorted(left[index], ts.sites_position, side="right") - 1
+    return parent[index][segment], matched
+
+
+def posterior(am, h):
+    """
+    Return the (P, log_likelihood) tuple, where P is the dense
+    (num_sites, num_nodes) matrix of posterior probabilities.
+    """
+    fm = am.forward_matrix(h.astype(np.int8))
+    bm = am.backward_matrix(h.astype(np.int8), fm.normalisation_factor)
+    return fm.decode() * bm.decode(), fm.log_likelihood
+
+
+def iter_decoded_sites(cm):
+    """
+    Yield the (site, values) of the specified CompressedMatrix one site at a
+    time, without building the dense matrix (for long sequences). Uses the
+    same approach as CompressedMatrix.decode.
+    """
+    ts = cm.ts
+    N = ts.num_nodes
+    for tree in ts.trees():
+        parent = tree.parent_array
+        in_tree = (parent[:N] != -1) | (tree.num_children_array[:N] > 0)
+        for site in tree.sites():
+            site_nodes, site_values = cm.get_site(site.id)
+            value = np.full(N + 1, np.nan)
+            value[site_nodes] = site_values
+            ancestor = parent.copy()
+            unset = np.where(in_tree & np.isnan(value[:N]))[0]
+            while len(unset) > 0:
+                value[unset] = value[ancestor[unset]]
+                ancestor[unset] = parent[ancestor[unset]]
+                unset = unset[np.isnan(value[unset])]
+            yield site.id, np.where(in_tree, value[:N], 0)
+
+
+def sample_prior_path(present, continuous, rho, n, rng):
+    """
+    Sample a copying path exactly from the engine's prior over paths,
+    p(path) proportional to the product of A_j(u_{j-1}, u_j), where
+    A_j(u, v) = present_j(v) [(1 - rho_j) [u = v] continuous_j(u) + rho_j / n].
+    We first run a backward pass with no data and then sample forwards.
+    """
+    m, N = present.shape
+    beta = np.zeros((m, N))
+    b = present[m - 1].astype(float)
+    beta[m - 1] = b / np.sum(b)
+    for j in range(m - 1, 0, -1):
+        b_next = beta[j]
+        b = present[j - 1] * (
+            (1 - rho[j]) * continuous[j] * b_next + rho[j] / n * np.sum(b_next)
+        )
+        beta[j - 1] = b / np.sum(b)
+    path = np.zeros(m, dtype=np.int32)
+    w = present[0] * beta[0]
+    path[0] = rng.choice(N, p=w / np.sum(w))
+    for j in range(1, m):
+        u = path[j - 1]
+        w = present[j] * rho[j] / n * beta[j]
+        if continuous[j, u]:
+            w[u] += (1 - rho[j]) * beta[j, u]
+        path[j] = rng.choice(N, p=w / np.sum(w))
+    return path
+
+
+def mutate_along_path(A, path, mu, rng):
+    """
+    Return the haplotype obtained by copying the alleles along the path,
+    changing each to each of the other alleles with probability mu (the
+    engine's emission model).
+    """
+    m = len(path)
+    h = A[np.arange(m), path].copy()
+    change = rng.random(m) < (NUM_ALLELES - 1) * mu
+    offset = rng.integers(1, NUM_ALLELES, size=m)
+    h[change] = (h[change] + offset[change]) % NUM_ALLELES
+    return h.astype(np.int8)
+
+
+def tree_distance(parent, u, v):
+    """
+    Return the number of edges between u and v in the tree with the specified
+    parent array, or -1 if they are not connected.
+    """
+    depth = {}
+    x, d = u, 0
+    while x != -1:
+        depth[x] = d
+        x = parent[x]
+        d += 1
+    x, d = v, 0
+    while x != -1:
+        if x in depth:
+            return depth[x] + d
+        x = parent[x]
+        d += 1
+    return -1
+
+
+def num_switches(path):
+    return int(np.sum(path[1:] != path[:-1]))
+
+
+def nearest_distances(true_sites, inferred_sites):
+    """
+    Return the distance (in sites) from each of the true switch sites to the
+    nearest inferred switch site, or NaN if there are none.
+    """
+    if len(inferred_sites) == 0:
+        return np.full(len(true_sites), np.nan)
+    inferred_sites = np.sort(inferred_sites)
+    distances = []
+    for s in true_sites:
+        k = np.searchsorted(inferred_sites, s)
+        candidates = inferred_sites[max(k - 1, 0) : k + 1]
+        distances.append(np.min(np.abs(candidates - s)))
+    return np.array(distances, dtype=float)
+
+
+# ---------------------------------------------------------------------------
+# Test infrastructure
+# ---------------------------------------------------------------------------
+
+
+class Test:
+    """
+    The superclass of all test classes. self.output_dir is set and exists
+    when a test method is called.
+    """
+
+    output_dir = None
+
+    def savefig(self, fig, name):
+        path = self.output_dir / f"{name}.png"
+        fig.savefig(path, dpi=100, bbox_inches="tight")
+        plt.close(fig)
+        logger.info(f"Wrote {path}")
+
+    def save_csv(self, df, name):
+        path = self.output_dir / f"{name}.csv"
+        df.to_csv(path, index=False)
+        logger.info(f"Wrote {path}")
+
+
+def plot_sweep(ax, df, x, y, label, logx=False):
+    """
+    Plot the mean and standard deviation of y over replicates for each x.
+    """
+    summary = df.groupby(x)[y].agg(["mean", "std"]).reset_index()
+    ax.errorbar(
+        summary[x],
+        summary["mean"],
+        yerr=summary["std"],
+        marker="o",
+        capsize=3,
+        label=label,
+    )
+    if logx:
+        ax.set_xscale("log")
+        ax.minorticks_off()
+    ax.set_xticks(summary[x], [f"{v:g}" for v in summary[x]])
+
+
+# ---------------------------------------------------------------------------
+# Calibration under the model
+# ---------------------------------------------------------------------------
+
+
+class ModelCalibration(Test):
+    """
+    Queries are simulated from the HMM itself: we sample a copying path
+    exactly from the engine's prior on an msprime ARG and add mutations with
+    the engine's emission model. Forward-backward posteriors should then be
+    calibrated: of the (site, node) pairs given posterior p, a fraction p
+    should be the true node. With weight_by_n=False the prior is an
+    unnormalised measure over paths, but we sample from it exactly, so the
+    posteriors should still be calibrated.
+    """
+
+    num_args = 20
+    paths_per_arg = 10
+    num_samples = 50
+    sequence_length = 5e5
+    levels = (0.5, 0.8, 0.9, 0.95, 0.99)
+
+    def simulate_queries(
+        self, switches_per_sequence, mu, weight_by_n, seed, num_args, paths_per_arg
+    ):
+        """
+        Yield (ts, am, A, path, h) for queries simulated from the model, with
+        about the specified expected number of switches along the sequence.
+        """
+        rng = np.random.default_rng(seed)
+        for j in range(num_args):
+            ts = simulate_arg(
+                self.num_samples, self.sequence_length, 1e-8, 1e-8, seed + j + 1
+            )
+            A, present, continuous = node_states(ts)
+            n = ts.num_nodes if weight_by_n else 1
+            rho = switches_per_sequence / ts.num_sites
+            # With weight_by_n=False use the same probability of switching to
+            # a given node.
+            engine_rho = rho if weight_by_n else rho / ts.num_nodes
+            am = make_matcher(ts, mu, engine_rho, weight_by_n=weight_by_n)
+            rho_array = np.full(ts.num_sites, engine_rho)
+            for _ in range(paths_per_arg):
+                path = sample_prior_path(present, continuous, rho_array, n, rng)
+                h = mutate_along_path(A, path, mu, rng)
+                yield ts, am, A, path, h
+
+    def _run_calibration(self, switches_per_sequence, mu, weight_by_n=True, seed=1):
+        bins = np.linspace(0, 1, 21)
+        num_bins = len(bins) - 1
+        # Per-query counts in each bin, so that we can bootstrap over queries
+        # (sites within a query are strongly correlated).
+        query_count = []
+        query_total_p = []
+        query_num_true = []
+        covered = {level: [] for level in self.levels}
+        set_size = {level: [] for level in self.levels}
+        set_mass = {level: [] for level in self.levels}
+        for ts, am, _, path, h in self.simulate_queries(
+            switches_per_sequence,
+            mu,
+            weight_by_n,
+            seed,
+            self.num_args,
+            self.paths_per_arg,
+        ):
+            P, _ = posterior(am, h)
+            m = ts.num_sites
+            is_true = np.zeros_like(P, dtype=bool)
+            is_true[np.arange(m), path] = True
+            keep = P > 1e-4
+            index = np.clip(np.digitize(P[keep], bins) - 1, 0, num_bins - 1)
+            query_count.append(np.bincount(index, minlength=num_bins))
+            query_total_p.append(np.bincount(index, weights=P[keep], minlength=num_bins))
+            query_num_true.append(
+                np.bincount(index, weights=is_true[keep], minlength=num_bins)
+            )
+            # Credible sets: the smallest set of nodes with total posterior at
+            # least the level, at each site. Nodes with identical haplotypes
+            # have exactly equal posteriors, so we break ties at random: if
+            # the true node is tied with g nodes, r nodes have a larger
+            # posterior and the set has size s, the probability that the set
+            # contains the true node is (s - r) / g, clipped to [0, 1].
+            sorted_P = -np.sort(-P, axis=1)
+            cumulative = np.cumsum(sorted_P, axis=1)
+            p_true = P[np.arange(m), path][:, None]
+            tied = np.abs(P - p_true) <= 1e-9 * p_true
+            num_greater = np.sum((P > p_true) & ~tied, axis=1)
+            num_tied = np.sum(tied, axis=1)
+            for level in self.levels:
+                size = np.sum(cumulative < level, axis=1) + 1
+                covered[level].extend(np.clip((size - num_greater) / num_tied, 0, 1))
+                set_size[level].extend(size)
+                set_mass[level].extend(cumulative[np.arange(m), size - 1])
+
+        query_count = np.array(query_count)
+        query_total_p = np.array(query_total_p)
+        query_num_true = np.array(query_num_true)
+        count = np.sum(query_count, axis=0)
+        nonempty = count > 0
+        mean_p = np.sum(query_total_p, axis=0)[nonempty] / count[nonempty]
+        frequency = np.sum(query_num_true, axis=0)[nonempty] / count[nonempty]
+        # Bootstrap over queries for the uncertainty in the observed frequency
+        rng = np.random.default_rng(seed)
+        replicates = []
+        for _ in range(500):
+            sample = rng.integers(0, len(query_count), size=len(query_count))
+            c = np.sum(query_count[sample], axis=0)[nonempty]
+            t = np.sum(query_num_true[sample], axis=0)[nonempty]
+            replicates.append(np.where(c > 0, t / np.maximum(c, 1), np.nan))
+        lower, upper = np.nanpercentile(replicates, [2.5, 97.5], axis=0)
+        df = pd.DataFrame(
+            {
+                "mean_posterior": mean_p,
+                "observed_frequency": frequency,
+                "lower": lower,
+                "upper": upper,
+                "count": count[nonempty],
+            }
+        )
+        self.save_csv(df, "reliability")
+        fig, (ax1, ax2) = plt.subplots(
+            2, 1, figsize=(6, 8), height_ratios=[3, 1], sharex=True
+        )
+        ax1.plot([0, 1], [0, 1], color="grey", linestyle="--")
+        ax1.errorbar(
+            mean_p,
+            frequency,
+            yerr=[frequency - lower, upper - frequency],
+            marker="o",
+            capsize=3,
+        )
+        ax1.set_ylabel("Fraction that are the true node (95% bootstrap CI)")
+        ax1.set_title("Reliability of posterior P(copy node u at site j)")
+        ax2.bar(
+            bins[:-1][nonempty], count[nonempty], width=np.diff(bins)[0], align="edge"
+        )
+        ax2.set_yscale("log")
+        ax2.set_xlabel("Posterior probability")
+        ax2.set_ylabel("Count")
+        self.savefig(fig, "reliability")
+
+        rows = []
+        for level in self.levels:
+            rows.append(
+                {
+                    "level": level,
+                    "coverage": np.mean(covered[level]),
+                    "mean_set_mass": np.mean(set_mass[level]),
+                    "mean_size": np.mean(set_size[level]),
+                    "median_size": np.median(set_size[level]),
+                }
+            )
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "credible_sets")
+        logger.info(f"\n{df}")
+        fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
+        ax1.plot([0.4, 1], [0.4, 1], color="grey", linestyle="--")
+        ax1.plot(df.level, df.coverage, marker="o", label="Coverage")
+        ax1.plot(
+            df.level,
+            df.mean_set_mass,
+            marker="x",
+            linestyle=":",
+            label="Mean posterior mass in set (expected coverage)",
+        )
+        ax1.legend()
+        ax1.set_xlabel("Nominal level")
+        ax1.set_ylabel("Fraction of sites where the set contains the true node")
+        ax1.set_title("Credible set coverage")
+        ax2.boxplot([set_size[level] for level in self.levels], showfliers=False)
+        ax2.set_xticks(range(1, len(self.levels) + 1), [str(x) for x in self.levels])
+        ax2.set_xlabel("Nominal level")
+        ax2.set_ylabel("Number of nodes in set")
+        ax2.set_title("Credible set size")
+        self.savefig(fig, "credible_sets")
+
+    def test_calibration_default(self):
+        self._run_calibration(switches_per_sequence=5, mu=0.01)
+
+    def test_calibration_high_recombination(self):
+        self._run_calibration(switches_per_sequence=25, mu=0.01)
+
+    def test_calibration_high_mutation(self):
+        self._run_calibration(switches_per_sequence=5, mu=0.05)
+
+    def test_calibration_no_weight_by_n(self):
+        self._run_calibration(switches_per_sequence=5, mu=0.01, weight_by_n=False)
+
+    def _run_decoding(self, name, values, make_params):
+        rows = []
+        for value in values:
+            switches, mu = make_params(value)
+            for ts, am, _, path, h in self.simulate_queries(
+                switches, mu, True, 2, num_args=10, paths_per_arg=5
+            ):
+                vpath, _ = viterbi(am, ts, h)
+                P, _ = posterior(am, h)
+                m = ts.num_sites
+                rows.append(
+                    {
+                        name: value,
+                        "viterbi_accuracy": np.mean(vpath == path),
+                        "posterior_max_accuracy": np.mean(np.argmax(P, axis=1) == path),
+                        "mean_posterior_true": np.mean(P[np.arange(m), path]),
+                        "true_switches": num_switches(path),
+                        "viterbi_switches": num_switches(vpath),
+                    }
+                )
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "decoding")
+        fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
+        plot_sweep(ax1, df, name, "viterbi_accuracy", "Viterbi", logx=True)
+        plot_sweep(ax1, df, name, "posterior_max_accuracy", "Posterior max", logx=True)
+        plot_sweep(
+            ax1, df, name, "mean_posterior_true", "Mean posterior of truth", logx=True
+        )
+        ax1.set_xlabel(name)
+        ax1.set_ylabel("Fraction of sites")
+        ax1.set_title("Decoding the true node")
+        ax1.legend()
+        for value, group in df.groupby(name):
+            ax2.scatter(
+                group.true_switches,
+                group.viterbi_switches,
+                label=f"{value:g}",
+                alpha=0.6,
+            )
+        top = max(df.true_switches.max(), df.viterbi_switches.max())
+        ax2.plot([0, top], [0, top], color="grey", linestyle="--")
+        ax2.set_xlabel("True number of switches")
+        ax2.set_ylabel("Viterbi number of switches")
+        ax2.legend(title=name)
+        self.savefig(fig, "decoding")
+
+    def test_decoding_accuracy_vs_mutation(self):
+        self._run_decoding("mu", [0.001, 0.003, 0.01, 0.03, 0.1], lambda mu: (5, mu))
+
+    def test_decoding_accuracy_vs_recombination(self):
+        self._run_decoding(
+            "switches_per_sequence", [1, 3, 10, 30, 100], lambda s: (s, 0.01)
+        )
+
+
+# ---------------------------------------------------------------------------
+# Held-out sample threading
+# ---------------------------------------------------------------------------
+
+
+def threading_replicate(
+    num_samples,
+    sequence_length,
+    recombination_rate,
+    mutation_rate,
+    k,
+    seed,
+    *,
+    weight_by_n=False,
+    likelihood_threshold=None,
+    compute_posterior=True,
+):
+    """
+    Simulate num_samples + 1 samples, hold out the last one, and match its
+    haplotype against the ARG of the others. Return the (metrics, details)
+    tuple, where metrics is a dict of per-replicate summaries.
+    """
+    full_ts = simulate_arg(
+        num_samples + 1, sequence_length, recombination_rate, mutation_rate, seed
+    )
+    samples = full_ts.samples()
+    query = samples[-1]
+    ts = full_ts.simplify(
+        samples[:-1],
+        keep_unary=True,
+        filter_nodes=False,
+        filter_sites=False,
+        update_sample_flags=False,
+    )
+    m = ts.num_sites
+    h = full_ts.genotype_matrix(
+        samples=[query], isolated_as_missing=False, alleles=ALLELES
+    )[:, 0].astype(np.int8)
+
+    # The true copying target at each site is the query's parent
+    true_parent = np.zeros(m, dtype=np.int32)
+    for tree in full_ts.trees():
+        for site in tree.sites():
+            true_parent[site.id] = tree.parent(query)
+
+    mu, rho = hmm_params(k)
+    if weight_by_n:
+        rho = n_weighted_rho(rho, ts.num_nodes)
+    am = make_matcher(
+        ts, mu, rho, weight_by_n=weight_by_n, likelihood_threshold=likelihood_threshold
+    )
+    before = time.perf_counter()
+    path, _ = viterbi(am, ts, h)
+    viterbi_time = time.perf_counter() - before
+
+    distance = np.zeros(m, dtype=int)
+    near = []
+    time_error = np.abs(ts.nodes_time[path] - ts.nodes_time[true_parent])
+    for tree in ts.trees():
+        parent = tree.parent_array
+        for site in tree.sites():
+            p = true_parent[site.id]
+            distance[site.id] = tree_distance(parent, path[site.id], p)
+            near.append(
+                [p, *tree.children(p)] + ([parent[p]] if parent[p] != -1 else [])
+            )
+
+    metrics = {
+        "exact": np.mean(path == true_parent),
+        "within_2": np.mean((distance >= 0) & (distance <= 2)),
+        "mean_distance": np.mean(distance[distance >= 0]),
+        "median_time_error": np.median(time_error),
+        "viterbi_time": viterbi_time,
+        "mean_traceback_size": am.mean_traceback_size,
+        "num_nodes": ts.num_nodes,
+        "num_sites": m,
+    }
+    P = None
+    if compute_posterior:
+        P, _ = posterior(am, h)
+        metrics["posterior_true"] = np.mean(P[np.arange(m), true_parent])
+        metrics["posterior_near"] = np.mean([np.sum(P[j, near[j]]) for j in range(m)])
+    true_switch = np.where(true_parent[1:] != true_parent[:-1])[0] + 1
+    inferred_switch = np.where(path[1:] != path[:-1])[0] + 1
+    metrics["true_switches"] = len(true_switch)
+    metrics["inferred_switches"] = len(inferred_switch)
+    details = {
+        "breakpoint_distance": nearest_distances(true_switch, inferred_switch),
+        "path": path,
+        "posterior": P,
+        "ts": ts,
+        "h": h,
+        "mu": mu,
+        "rho": rho,
+    }
+    return metrics, details
+
+
+THREADING_DEFAULTS = {
+    "num_samples": 50,
+    "sequence_length": 5e5,
+    "recombination_rate": 1e-8,
+    "mutation_rate": 1e-8,
+    "k": 3,
+}
+
+
+class HeldOutThreading(Test):
+    """
+    Match a held-out sample from an msprime simulation against the ARG of the
+    other samples. The truth at each site is the held-out sample's parent in
+    the full ARG: the node where its lineage joins the others, which is the
+    best node to copy. This needs no assumptions about the HMM.
+    """
+
+    num_replicates = 20
+
+    def _run(self, name, values, logx=True):
+        rows = []
+        breakpoint_rows = []
+        for value in values:
+            params = dict(THREADING_DEFAULTS)
+            params[name] = value
+            for j in range(self.num_replicates):
+                metrics, details = threading_replicate(**params, seed=j + 1)
+                metrics[name] = value
+                rows.append(metrics)
+                for d in details["breakpoint_distance"]:
+                    breakpoint_rows.append({name: value, "distance": d})
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "threading")
+        logger.info(f"\n{df.groupby(name).mean()}")
+
+        fig, axes = plt.subplots(2, 2, figsize=(11, 8))
+        ax = axes[0, 0]
+        plot_sweep(ax, df, name, "exact", "Viterbi node is the parent", logx)
+        plot_sweep(ax, df, name, "within_2", "Within 2 edges", logx)
+        ax.set_ylabel("Fraction of sites")
+        ax.set_title("Viterbi copying target")
+        ax.legend()
+        ax = axes[0, 1]
+        plot_sweep(ax, df, name, "posterior_true", "On the parent", logx)
+        plot_sweep(ax, df, name, "posterior_near", "Within 1 edge", logx)
+        ax.set_ylabel("Mean posterior mass")
+        ax.set_title("Posterior mass on the true parent")
+        ax.legend()
+        ax = axes[1, 0]
+        plot_sweep(ax, df, name, "median_time_error", "", logx)
+        ax.set_yscale("log")
+        ax.set_ylabel("Median |t(Viterbi node) - t(parent)| (generations)")
+        ax = axes[1, 1]
+        plot_sweep(ax, df, name, "true_switches", "True", logx)
+        plot_sweep(ax, df, name, "inferred_switches", "Viterbi", logx)
+        ax.set_ylabel("Number of switches")
+        ax.legend()
+        for ax in axes.flat:
+            ax.set_xlabel(name)
+        self.savefig(fig, "threading")
+
+        bp = pd.DataFrame(breakpoint_rows)
+        self.save_csv(bp, "breakpoints")
+        fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
+        groups = [g.distance.dropna().values for _, g in bp.groupby(name)]
+        labels = [f"{v:g}" for v, _ in bp.groupby(name)]
+        ax1.boxplot(groups, showfliers=False)
+        ax1.set_xticks(range(1, len(labels) + 1), labels)
+        ax1.set_xlabel(name)
+        ax1.set_ylabel("Sites to the nearest Viterbi switch")
+        ax1.set_title("Distance from each true switch to the nearest inferred")
+        for value, group in df.groupby(name):
+            ax2.scatter(
+                group.true_switches,
+                group.inferred_switches,
+                label=f"{value:g}",
+                alpha=0.6,
+            )
+        top = max(df.true_switches.max(), df.inferred_switches.max())
+        ax2.plot([0, top], [0, top], color="grey", linestyle="--")
+        ax2.set_xlabel("True switches")
+        ax2.set_ylabel("Viterbi switches")
+        ax2.legend(title=name)
+        self.savefig(fig, "breakpoints")
+
+    def test_threading_vs_mutation_rate(self):
+        self._run("mutation_rate", [2.5e-9, 5e-9, 1e-8, 2e-8, 4e-8])
+
+    def test_threading_vs_recombination_rate(self):
+        self._run("recombination_rate", [2.5e-9, 5e-9, 1e-8, 2e-8, 4e-8])
+
+    def test_threading_vs_sample_size(self):
+        self._run("num_samples", [10, 20, 50, 100, 200])
+
+    def test_threading_vs_hmm_rho(self):
+        self._run("k", [2, 3, 4, 5, 6], logx=False)
+
+
+# ---------------------------------------------------------------------------
+# Numerics and approximations
+# ---------------------------------------------------------------------------
+
+
+class Numerics(Test):
+    """
+    The numerical behaviour of the engine and the effect of its
+    approximations.
+    """
+
+    def test_likelihood_threshold(self):
+        """
+        The likelihood threshold sets normalised values below it to the
+        threshold, which bounds the number of distinct values (and so time and
+        memory) at the cost of exactness.
+        """
+        thresholds = [None, 1e-30, 1e-20, 1e-10, 1e-6, 1e-3]
+        rows = []
+        for j in range(20):
+            reference = None
+            for threshold in thresholds:
+                metrics, details = threading_replicate(
+                    **THREADING_DEFAULTS,
+                    seed=j + 1,
+                    likelihood_threshold=threshold,
+                )
+                ts = details["ts"]
+                A, present, continuous = node_states(ts)
+                m = ts.num_sites
+                value = dense.path_log_likelihood(
+                    A,
+                    present,
+                    continuous,
+                    details["h"],
+                    np.full(m, details["rho"]),
+                    np.full(m, details["mu"]),
+                    NUM_ALLELES,
+                    1,
+                    details["path"],
+                )
+                P = details["posterior"]
+                if reference is None:
+                    reference = (value, P)
+                metrics["threshold"] = "none" if threshold is None else f"{threshold:g}"
+                metrics["log_likelihood_loss"] = reference[0] - value
+                metrics["posterior_tv"] = 0.5 * np.mean(
+                    np.sum(np.abs(P - reference[1]), axis=1)
+                )
+                rows.append(metrics)
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "threshold")
+        labels = ["none"] + [f"{t:g}" for t in thresholds[1:]]
+        fig, axes = plt.subplots(2, 2, figsize=(11, 8))
+        for ax, column, ylabel in [
+            (axes[0, 0], "log_likelihood_loss", "Loss in Viterbi path log likelihood"),
+            (axes[0, 1], "posterior_tv", "Mean total variation from exact posterior"),
+            (axes[1, 0], "mean_traceback_size", "Mean traceback size"),
+            (axes[1, 1], "viterbi_time", "Viterbi time (s)"),
+        ]:
+            groups = [df[df.threshold == label][column].values for label in labels]
+            ax.boxplot(groups)
+            ax.set_xticks(range(1, len(labels) + 1), labels)
+            ax.set_xlabel("likelihood_threshold")
+            ax.set_ylabel(ylabel)
+        self.savefig(fig, "threshold")
+        fig, ax = plt.subplots(figsize=(6, 4))
+        groups = [df[df.threshold == label].exact.values for label in labels]
+        ax.boxplot(groups)
+        ax.set_xticks(range(1, len(labels) + 1), labels)
+        ax.set_xlabel("likelihood_threshold")
+        ax.set_ylabel("Fraction of sites where Viterbi copies the parent")
+        self.savefig(fig, "threshold_accuracy")
+
+    def test_posterior_normalisation(self):
+        """
+        With the forward normalisation factors, sum_u F(u) B(u) = 1 at every
+        site in exact arithmetic. Check for numerical drift along long
+        sequences.
+        """
+        rows = []
+        for sequence_length in [1e5, 3e5, 1e6, 3e6, 1e7, 2e7]:
+            for weight_by_n in [True, False]:
+                for j in range(3):
+                    ts = simulate_arg(20, sequence_length, 1e-8, 1e-8, j + 1)
+                    rng = np.random.default_rng(j)
+                    h = ts.genotype_matrix(
+                        samples=[0], isolated_as_missing=False, alleles=ALLELES
+                    )[:, 0]
+                    noise = rng.random(ts.num_sites) < 0.01
+                    h[noise] = rng.integers(0, NUM_ALLELES, size=np.sum(noise))
+                    mu, rho = hmm_params(3)
+                    if weight_by_n:
+                        rho = n_weighted_rho(rho, ts.num_nodes)
+                    am = make_matcher(ts, mu, rho, weight_by_n=weight_by_n)
+                    fm = am.forward_matrix(h.astype(np.int8))
+                    bm = am.backward_matrix(h.astype(np.int8), fm.normalisation_factor)
+                    max_error = 0
+                    for (_, f), (_, b) in zip(
+                        iter_decoded_sites(fm), iter_decoded_sites(bm)
+                    ):
+                        max_error = max(max_error, abs(np.sum(f * b) - 1))
+                    rows.append(
+                        {
+                            "num_sites": ts.num_sites,
+                            "num_nodes": ts.num_nodes,
+                            "weight_by_n": weight_by_n,
+                            "max_error": max_error,
+                            "log_likelihood_per_site": fm.log_likelihood / ts.num_sites,
+                        }
+                    )
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "normalisation")
+        fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
+        for weight_by_n, group in df.groupby("weight_by_n"):
+            label = f"weight_by_n={weight_by_n}"
+            ax1.scatter(group.num_sites, np.maximum(group.max_error, 1e-17), label=label)
+            ax2.scatter(group.num_sites, group.log_likelihood_per_site, label=label)
+        ax1.set_xscale("log")
+        ax1.set_yscale("log")
+        ax1.set_xlabel("Number of sites")
+        ax1.set_ylabel("max over sites of |sum F*B - 1|")
+        ax1.legend()
+        ax2.set_xscale("log")
+        ax2.set_xlabel("Number of sites")
+        ax2.set_ylabel("Forward log likelihood per site")
+        ax2.legend()
+        self.savefig(fig, "normalisation")
+
+    def test_dense_agreement_random_args(self):
+        """
+        Compare the engine with the dense implementation used in the unit tests
+        on many small random ARGs with random parameters.
+        """
+        rng = np.random.default_rng(1234)
+        rows = []
+        for j in range(200):
+            ts = simulate_arg(
+                int(rng.integers(3, 11)),
+                int(rng.integers(20, 61)),
+                rng.uniform(0, 0.05),
+                0.3,
+                j + 1,
+                population_size=1,
+            )
+            if ts.num_sites == 0:
+                continue
+            A, present, continuous = dense.node_states(ts, list(ALLELES))
+            rho = 10 ** rng.uniform(-4, -0.5)
+            mu = 10 ** rng.uniform(-3, np.log10(0.2))
+            weight_by_n = bool(rng.integers(0, 2))
+            threshold = [None, 1e-6, 1e-3][rng.integers(0, 3)]
+            am = make_matcher(
+                ts, mu, rho, weight_by_n=weight_by_n, likelihood_threshold=threshold
+            )
+            n = ts.num_nodes if weight_by_n else 1
+            m = ts.num_sites
+            rho_a = np.full(m, rho)
+            mu_a = np.full(m, mu)
+            dense_threshold = 0 if threshold is None else threshold
+            u = int(rng.integers(0, ts.num_samples))
+            h_random = rng.integers(0, NUM_ALLELES, size=m).astype(np.int8)
+            h_missing = A[:, u].copy()
+            h_missing[rng.random(m) < 0.2] = -1
+            for h in [A[:, u].copy(), h_random, h_missing]:
+                fm = am.forward_matrix(h)
+                bm = am.backward_matrix(h, fm.normalisation_factor)
+                F, c = dense.forward(
+                    A,
+                    present,
+                    continuous,
+                    h,
+                    rho_a,
+                    mu_a,
+                    NUM_ALLELES,
+                    n,
+                    dense_threshold,
+                )
+                B = dense.backward(
+                    A,
+                    present,
+                    continuous,
+                    h,
+                    rho_a,
+                    mu_a,
+                    NUM_ALLELES,
+                    n,
+                    c,
+                    dense_threshold,
+                )
+
+                def relative_error(x, y):
+                    return np.max(np.abs(x - y) / np.maximum(np.abs(y), 1e-200))
+
+                path, _ = viterbi(am, ts, h)
+                optimum = dense.viterbi_log_likelihood(
+                    A, present, continuous, h, rho_a, mu_a, NUM_ALLELES, n
+                )
+                value = dense.path_log_likelihood(
+                    A, present, continuous, h, rho_a, mu_a, NUM_ALLELES, n, path
+                )
+                rows.append(
+                    {
+                        "forward_error": relative_error(fm.decode(), F),
+                        "backward_error": relative_error(bm.decode(), B),
+                        "viterbi_gap": optimum - value if threshold is None else np.nan,
+                    }
+                )
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "dense_agreement")
+        logger.info(f"\n{df.describe()}")
+        fig, axes = plt.subplots(1, 3, figsize=(14, 4))
+        for ax, column in zip(axes, ["forward_error", "backward_error", "viterbi_gap"]):
+            values = np.abs(df[column].dropna().values)
+            ax.hist(np.log10(np.maximum(values, 1e-18)), bins=30)
+            ax.set_xlabel(f"log10 {column} (floored at -18)")
+            ax.set_ylabel("Count")
+        axes[2].set_title("Viterbi optimum - path (no threshold)")
+        self.savefig(fig, "dense_agreement")
+
+    def test_weight_by_n(self):
+        """
+        Compare threading with weight_by_n on and off, with the same ratio of
+        switching to a given node to staying (so the Viterbi paths are the
+        same up to ties); the posteriors differ because the models do.
+        """
+        rows = []
+        for k in [2, 3, 4, 5, 6]:
+            for weight_by_n in [True, False]:
+                for j in range(20):
+                    params = dict(THREADING_DEFAULTS)
+                    params["k"] = k
+                    metrics, _ = threading_replicate(
+                        **params, seed=j + 1, weight_by_n=weight_by_n
+                    )
+                    metrics["k"] = k
+                    metrics["weight_by_n"] = weight_by_n
+                    rows.append(metrics)
+        df = pd.DataFrame(rows)
+        self.save_csv(df, "weight_by_n")
+        fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
+        for weight_by_n, group in df.groupby("weight_by_n"):
+            label = f"weight_by_n={weight_by_n}"
+            plot_sweep(ax1, group, "k", "exact", label)
+            plot_sweep(ax2, group, "k", "posterior_true", label)
+        ax1.set_xlabel("k (mismatches per recombination)")
+        ax1.set_ylabel("Fraction of sites where Viterbi copies the parent")
+        ax1.legend()
+        ax2.set_xlabel("k (mismatches per recombination)")
+        ax2.set_ylabel("Mean posterior on the parent")
+        ax2.legend()
+        self.savefig(fig, "weight_by_n")
+
+
+# ---------------------------------------------------------------------------
+# Running tests
+# ---------------------------------------------------------------------------
+
+
+class TestInstance:
+    def __init__(self, test_class, method_name):
+        self.test_class = test_class
+        self.method_name = method_name
+
+    def run(self, basedir):
+        cls = globals()[self.test_class]
+        instance = cls()
+        instance.output_dir = pathlib.Path(basedir) / self.test_class / self.method_name
+        instance.output_dir.mkdir(parents=True, exist_ok=True)
+        logger.info(f"Running {self.method_name}")
+        getattr(instance, self.method_name)()
+
+
+def make_suite():
+    """
+    Return a dict mapping test method names to TestInstances.
+    """
+    tests = {}
+    for class_name, cls in inspect.getmembers(sys.modules[__name__], inspect.isclass):
+        if issubclass(cls, Test) and cls is not Test:
+            for name, _ in inspect.getmembers(cls, inspect.isfunction):
+                if name.startswith("test_"):
+                    if name in tests:
+                        raise ValueError(f"Duplicate test name {name}")
+                    tests[name] = TestInstance(class_name, name)
+    return tests
+
+
+def run_tests(tests, basedir, num_processes, show_progress):
+    progress = tqdm.tqdm(total=len(tests), disable=not show_progress)
+    if num_processes <= 1:
+        for test in tests:
+            test.run(basedir)
+            progress.update()
+    else:
+        with concurrent.futures.ProcessPoolExecutor(num_processes) as executor:
+            futures = [executor.submit(test.run, basedir) for test in tests]
+            for future in concurrent.futures.as_completed(futures):
+                exception = future.exception()
+                if exception is not None:
+                    logger.error(f"EXCEPTION: {exception}")
+                    for f in futures:
+                        f.cancel()
+                    raise exception
+                progress.update()
+    progress.close()
+
+
+def main():
+    suite = make_suite()
+    classes = sorted({test.test_class for test in suite.values()})
+    parser = argparse.ArgumentParser(description=__doc__.split("\n\n")[0])
+    parser.add_argument("tests", nargs="*", help="Run specific tests (see --list)")
+    parser.add_argument(
+        "--test-class", "-c", choices=classes, help="Run all the tests in this class"
+    )
+    parser.add_argument(
+        "--output-dir", "-d", default="tmp__NOBACKUP__", help="Base output directory"
+    )
+    parser.add_argument(
+        "--num-processes", "-t", type=int, default=1, help="Number of processes"
+    )
+    group = parser.add_mutually_exclusive_group()
+    group.add_argument("--quiet", "-q", action="store_true", help="Only log warnings")
+    group.add_argument("--debug", "-D", action="store_true", help="Debug logging")
+    parser.add_argument(
+        "--no-progress", "-n", action="store_true", help="Do not show a progress bar"
+    )
+    parser.add_argument("--list", "-l", action="store_true", help="List the tests")
+    args = parser.parse_args()
+
+    level = logging.INFO
+    if args.quiet:
+        level = logging.WARNING
+    if args.debug:
+        level = logging.DEBUG
+    logging.basicConfig(level=level, format="%(asctime)s %(levelname)s %(message)s")
+    logging.getLogger("matplotlib").setLevel(logging.WARNING)
+    logging.getLogger("msprime").setLevel(logging.WARNING)
+
+    if args.list:
+        for name, test in suite.items():
+            print(test.test_class, name, sep="\t")
+        return
+    if len(args.tests) > 0:
+        unknown = [name for name in args.tests if name not in suite]
+        if len(unknown) > 0:
+            parser.error(f"Unknown tests: {unknown}")
+        tests = [suite[name] for name in args.tests]
+    elif args.test_class is not None:
+        tests = [t for t in suite.values() if t.test_class == args.test_class]
+    else:
+        tests = list(suite.values())
+    run_tests(tests, args.output_dir, args.num_processes, not args.no_progress)
+
+
+if __name__ == "__main__":
+    main()