Comparison and Limitations

The master watchmaker tests every complication against the simplest reference: a well-regulated pocket watch. If the complication cannot tell time as accurately as the basic movement, it belongs on the shelf, not on the wrist.

This chapter places Escapement in the landscape of population-genetic inference. We compare it systematically against every relevant method, trace the design principles borrowed from each Timepiece, enumerate five honest limitations, and describe the hybrid pipeline where Mainspring’s speed meets Escapement’s statistical rigor.

Comparison Against All Methods

Escapement vs. all inference methods
Method	Simulations needed?	Full posterior?	Scales to biobank?	Continuous \(N_e\)?	Joint topology?	Per-dataset time	Statistical guarantee
PSMC	No	No (point est.)	No (2 haplotypes)	Piecewise-constant	No (fixed HMM)	Minutes	MLE convergence
ARGweaver	No	Yes (MCMC)	No (~10 samples)	Piecewise-constant	Yes	Hours–days	Asymptotic exactness
SINGER	No	Yes (Gibbs)	No (~20 samples)	GP prior	Yes	Hours	Ergodicity
tsinfer + tsdate	No	Partial (times only)	Yes (millions)	No (\(N_e\) assumed)	tsinfer: heuristic	Minutes	tsdate: ELBO bound
Gamma-SMC	No	Yes (2 haplotypes)	No (2 haplotypes)	Piecewise-constant	No	Seconds	Analytical posteriors
phlash	No	Yes (SVGD)	Partial (pairs)	Neural spline	No	Minutes–hours	Score function est.
Mainspring	Yes (training)	Approximate	50–100 samples	Normalizing flow	Yes	Seconds	None
Escapement	No	Approximate (VI)	20–50 samples	Piecewise / spline / GP	Yes (Gumbel-SM)	10–30 min	ELBO bound

Reading the table

No method dominates all columns. PSMC is fast and needs no simulations but sees only two haplotypes. ARGweaver gives the full posterior but takes days. tsinfer scales to millions but surrenders posterior uncertainty. Escapement occupies a specific niche: simulation-free, joint topology-time inference with principled uncertainty, at the cost of per-dataset optimization time and limited sample size.

Design Principles Borrowed from Each Timepiece

Escapement is not built in a vacuum. Every major design decision traces to a mathematical insight from a Timepiece.

Design principles and their Timepiece origins
Timepiece	Principle borrowed	How it appears in Escapement
PSMC	SMC factorization	The coalescent prior factors approximately across local trees (Section 3 of Variational Inference Without Simulations). Without this, the prior over full ARGs is intractable.
tsdate	Gamma/log-normal posteriors for coalescence times	Branch lengths are parameterized as log-normal distributions, inspired by tsdate’s variational gamma posteriors. The entropy has a closed-form expression.
tsinfer	Attention as the Li & Stephens copying model	The sample attention in the encoder mirrors tsinfer’s copying probabilities: high attention weight between samples \(i\) and \(j\) indicates recent common ancestry.
Gamma-SMC	Continuous time, no grid	Coalescence times are continuous (log-normal), not discretized into intervals. This avoids the discretization artifacts of PSMC.
phlash	Continuous \(N_e(t)\) with flexible parameterization	\(N_e(t)\) can be parameterized as a neural spline or GP, enabling smooth demographic trajectories beyond piecewise-constant.
ARGweaver	Poisson mutation model on edges	The mutation log-likelihood uses the same Poisson model as ARGweaver’s emission probabilities.
SINGER	GP prior on demographic parameters	The optional GP parameterization of \(N_e(t)\) is inspired by SINGER’s GP prior on branch lengths.
msprime	Kingman coalescent prior	The coalescent log-prior is the exponential (constant \(N_e\)) or piecewise-exponential (variable \(N_e\)) distribution derived in msprime’s coalescent theory chapter.
SMC++	Distinguished lineage structure for scaling	While Escapement does not use a distinguished lineage, the insight that a single lineage’s coalescence history is informative motivates the per-sample branch-length predictions.
moments / dadi	SFS as a consistency check	The optional SFS auxiliary loss (inherited from Mainspring) can be added to the ELBO as a physics-informed regularizer.

What Escapement Cannot Do

Five fundamental limitations, stated without euphemism.

1. Model Misspecification

Escapement assumes the neutral coalescent with recombination. The ELBO is a lower bound on \(\log P(\mathbf{D} \mid \theta)\) only under this model. If the true data-generating process includes:

Natural selection (sweeps, background selection, balancing selection)
Population structure (migration, admixture, isolation-by-distance)
Gene conversion (non-crossover recombination)
Sequencing artifacts (errors, missing data, batch effects)

then the coalescent model is wrong, and the ELBO optimizes toward the best genealogy under a wrong model. The inferred \(N_e(t)\) will absorb some of these effects (e.g., background selection appears as reduced \(N_e\)), but others (e.g., population structure) may produce qualitatively misleading results.

\[P_{\text{true}}(\mathbf{D}) \neq \int P(\mathbf{D} \mid \tau, \mu) \cdot P(\tau \mid N_e, \rho)\, d\tau\]

Mitigation. Compare the inferred \(N_e(t)\) against results from PSMC or SMC++. If they disagree qualitatively, model misspecification is likely. Run the inference on different genomic regions; selection produces region-specific patterns while neutral demography is genome-wide.

2. Slower Per-Dataset

Escapement requires 1,000–10,000 gradient steps per dataset. At ~10 ms per step on a GPU, this is 10–100 seconds for simple cases and 10–30 minutes for complex demography. For a study requiring inference on 1,000 genomic windows, Escapement alone would take days.

Method	Per-dataset time	1000 datasets	Bottleneck
PSMC	~5 minutes	~3 days	EM convergence
Mainspring	~1 second	~17 minutes	Forward pass
Escapement	~15 minutes	~10 days	ELBO optimization
Hybrid (Mainspring → Escapement)	~3 minutes	~2 days	Warm-started ELBO

Mitigation. Use the hybrid pipeline (Mainspring warm-start) to reduce per-dataset time from 15 minutes to 3 minutes. Parallelize across GPUs for multi-window analysis.

3. Approximate Posterior

The variational posterior \(q(\tau)\) is a mean-field approximation: topology, branch lengths, and breakpoints are approximately independent. The true posterior has strong correlations:

Adjacent trees share most of their topology (recombination modifies one lineage, not the whole tree).
Branch lengths and topology are correlated (star-like trees have short internal branches).
Breakpoints and topology changes are deterministically linked.

The ELBO provides a lower bound on the log-evidence, but the gap can be significant. A structured variational family (e.g., autoregressive over positions) would reduce the gap at the cost of slower sampling.

Mitigation. Use multiple random restarts and select the run with the highest ELBO. Compare the variational posterior against MCMC samples from ARGweaver or SINGER on a small subset to assess the quality of the approximation.

4. Topology Inference Is Hard

The space of tree topologies is discrete and exponentially large. For \(n = 20\) samples, the number of labeled binary trees is \((2 \cdot 20 - 3)!! = 37!! \approx 2 \times 10^{22}\). The Gumbel-softmax provides gradients through this discrete space, but the optimization landscape has many local optima.

In practice, the topology is the last component to converge and the most sensitive to initialization. Errors in the topology propagate into errors in branch lengths and \(N_e(t)\).

Mitigation. Warm-start from Mainspring or tsinfer. Use multiple restarts. Monitor the topology entropy during training – if it remains high after annealing, the topology is uncertain and the results should be interpreted cautiously.

5. Breakpoint Detection

Recombination breakpoints are modeled as independent Bernoulli variables at each position. This local model has two weaknesses:

Closely spaced breakpoints (e.g., gene conversion tracts, which are typically 50–1000 bp) produce pairs of breakpoints that the Bernoulli model treats independently, potentially missing the paired structure.
The SMC approximation assumes that breakpoints are well-separated and that recombination modifies one lineage at a time. In regions of very high recombination, this assumption breaks down.

Mitigation. Post-process the inferred breakpoints by merging closely spaced events. Use a recombination map as a prior rather than assuming uniform \(\rho\).

The Hybrid: Mainspring → Escapement Pipeline

The most powerful workflow combines both Complications:

┌────────────────────────────┐
│  OBSERVED GENOTYPE MATRIX  │
│  D ∈ {0,1}^{n × L}        │
└─────────────┬──────────────┘
              │
              v
┌────────────────────────────┐
│  STEP 1: MAINSPRING        │    ~1 second
│  Fast amortized inference  │
│  → initial ARG + N_e(t)    │
└─────────────┬──────────────┘
              │ warm-start
              v
┌────────────────────────────┐
│  STEP 2: ESCAPEMENT        │    ~3 minutes (warm-started)
│  Likelihood-based refine   │
│  → calibrated posterior    │
│  → refined N_e(t)          │
│  → ELBO diagnostic         │
└─────────────┬──────────────┘
              │
              v
┌────────────────────────────┐
│  OUTPUT                     │
│  Posterior over genealogies │
│  N_e(t) with uncertainty   │
│  ELBO as model fit metric  │
└────────────────────────────┘

This pipeline is analogous to the classical tsinfer → tsdate workflow, but fully neural:

Classical vs. neural two-stage pipelines
Stage	Classical (tsinfer → tsdate)	Neural (Mainspring → Escapement)
Stage 1	tsinfer: Li & Stephens ancestor matching → topology	Mainspring: neural encoder → full ARG
Stage 2	tsdate: variational gamma EP → node times	Escapement: neural variational posterior → ELBO optimization
Topology	Fixed from tsinfer	Refined by Escapement
Demography	Assumed known	Jointly inferred
Uncertainty	Gamma posteriors on times	Full variational posterior (topology + times + breaks)
Loss function	EP messages (hand-derived)	ELBO (auto-differentiated)

The hybrid pipeline preserves each approach’s strengths:

From Mainspring: fast initialization, learned representations, topology structure.
From Escapement: principled objective (ELBO), no simulation dependency at refinement time, joint topology-time-demography inference, built-in model diagnostic.

The watchmaker’s grande complication

In horology, a grande complication combines multiple complications into a single movement: perpetual calendar, minute repeater, split-seconds chronograph. Each complication reinforces the others – the chronograph needs the calendar to timestamp events; the repeater needs the chronograph to mark elapsed time.

The Mainspring → Escapement pipeline is the grande complication of this book. Mainspring provides the fast, broad inference. Escapement provides the principled, per-dataset refinement. Together, they combine the amortized economics of simulation-based training with the statistical rigor of likelihood-based inference.

Neither is sufficient alone. Mainspring without Escapement has no guarantees. Escapement without Mainspring starts from scratch and may never converge. Together, they keep time that the watchmaker can trust.

When to Use What

A practical decision guide:

Scenario	Recommended approach
Screening 1,000 genomic windows for demographic events	Mainspring alone (speed is paramount)
Careful \(N_e(t)\) inference from 30 samples	Hybrid: Mainspring → Escapement
Single diploid genome, well-characterized species	PSMC (interpretable, proven, fast enough)
Provably correct posterior samples from the ARG	ARGweaver (no shortcut to exactness)
Biobank-scale tree sequence (>10,000 samples)	tsinfer + tsdate (only methods that scale)
Multi-population split times	SMC++ or dadi
Need ELBO diagnostic to check model fit	Escapement (only neural method with a principled fit metric)
No GPU available	PSMC, tsinfer + tsdate, or moments
Teaching and understanding	The Timepieces, always (the whole point of this book)

The honest summary

Escapement is most valuable when (1) you have a moderately sized dataset (20–50 samples), (2) you care about posterior uncertainty, (3) you want to infer demography jointly with the genealogy, and (4) you are willing to spend 10–30 minutes per dataset on a GPU. For any single axis – speed, scalability, posterior quality, interpretability – there is a classical method that does better. Escapement’s contribution is occupying a point in the trade-off space that no classical method reaches: simulation-free, neural, joint, principled, and fast enough.

The Timepieces are the foundation. Mainspring and Escapement are complications. Use the simplest tool that answers your question. And always check the results against a method you trust – whether that is PSMC, tsdate, or a simulation study you run yourself.