Comparison and Limitations

The master watchmaker keeps three timepieces on her bench: a marine chronometer for absolute accuracy, a minute repeater for darkness, and a split-seconds chronograph for measuring intervals. She does not ask which is best. She asks which is right for the task at hand. The answer depends on what you need to measure, how quickly, and how much uncertainty you can tolerate.

This chapter places Balance Wheel in the full landscape of neural and classical methods for population-genetic inference. We compare it against the other two Complications, trace its connections to every relevant Timepiece, enumerate its limitations honestly, and provide a decision tree for choosing the right tool.

The Three Complications

Each Complication operates at a different level of the data hierarchy and serves a different purpose. They are not competing approaches – they are complementary tools for different questions.

The three Complications compared
Property	Mainspring	Escapement	Balance Wheel
Metaphor	Learn to assemble the watch from blueprints	Understand the physics of timekeeping	Feel the aggregate pressure of evolution
Data input	Raw genotypes \(\mathbf{D} \in \{0,1\}^{n \times L}\)	Raw genotypes \(\mathbf{D} \in \{0,1\}^{n \times L}\)	Site Frequency Spectrum \(\mathbf{D} \in \mathbb{Z}^{n-1}\)
Training signal	Simulated ARGs (msprime)	Coalescent likelihood (analytical)	Exact SFS from moments / dadi
Simulations needed	Yes (millions of ARGs)	No	No (just moments evaluations)
What it infers	Full ARG + \(N_e(t)\)	Genealogy + \(N_e(t)\)	Demographic parameters only
Resolution	Per-site, per-sample	Per-site, per-sample	Population-level (SFS)
Speed at inference	~1 second (forward pass)	~10–30 minutes (ELBO optimization)	~0.1 ms per SFS + seconds for posterior
Closest Timepiece	tsinfer + tsdate	PSMC + ARGweaver	dadi + moments
Best for	High-throughput ARG inference	Deep analysis of one dataset	Demographic model fitting + comparison

Each operates at a different level of the data hierarchy:

Mainspring: sequence \(\to\) ARG \(\to\) demography (most detailed, needs simulations)
Escapement: sequence \(\to\) coalescent times \(\to\) demography (no simulations, per-dataset)
Balance Wheel: SFS \(\to\) demography (fastest, most practical for demographic inference)

The information hierarchy

Moving from Mainspring to Balance Wheel, we trade information for speed. The raw genotype matrix contains all the information: haplotype structure, LD, spatial patterns, allele frequencies. The SFS retains only allele frequencies. This is a massive compression – and under the Poisson Random Field model, it is lossless for demographic inference. But for questions about recombination, selection, or genealogy structure, the SFS is insufficient. Choose the Complication that matches the question, not the one that processes the most data.

Connection to Every Timepiece

Balance Wheel does not exist in isolation. Every major design decision traces to a mathematical insight from a Timepiece.

Design principles and their Timepiece origins
Timepiece	What Balance Wheel borrows	How it appears
dadi	The Wright-Fisher diffusion PDE	Balance Wheel learns to approximate the PDE solution. The teacher (dadi) solves the PDE exactly; the student (neural network) reproduces the result without solving the PDE. The diffusion theory provides the mathematical foundation that makes the SFS a smooth function of \(\Theta\).
moments	The ODE system for SFS entries	moments is the primary teacher during training. Its ODE integrator computes the exact expected SFS for each training example. Balance Wheel distills this computation into a neural network.
PSMC	Piecewise-constant \(N_e(t)\) parameterization	Balance Wheel extends PSMC’s piecewise-constant representation to continuous \(N_e(t)\) via neural splines, while retaining the piecewise-constant option as the simplest case.
momi2	Coalescent SFS computation for multi-population models	momi2 serves as an alternative teacher for complex multi-population topologies where moments becomes expensive. Its tensor machinery inspires the factored output approach.
Gamma-SMC	Gamma distributions for parameter posteriors	The Bayesian posterior inference via HMC produces posterior distributions that are often well-approximated by gamma distributions, providing a connection to Gamma-SMC’s analytical posteriors.
phlash	Differentiable inference engine	The core idea of replacing a classical inference engine with a differentiable neural alternative. phlash pioneered this for the SMC likelihood; Balance Wheel does it for the SFS computation.
SMC++	Continuous-time demographic inference	The motivation for continuous \(N_e(t)\) comes from SMC++’s demonstration that piecewise-constant models are limiting.
msprime	Kingman coalescent theory	The prior distributions on demographic parameters are grounded in coalescent expectations (e.g., expected TMRCA for pair of lineages).

What Balance Wheel Cannot Do

Four fundamental limitations, stated without euphemism.

1. Inherits the SFS’s Limitations

The SFS discards:

Linkage disequilibrium (LD). Two-locus statistics, haplotype blocks, and LD decay patterns are invisible in the SFS. Recent selective sweeps that leave strong LD signatures but modest SFS distortions will be missed.
Haplotype structure. The SFS counts allele frequencies but not which alleles co-occur on the same haplotype. Admixture events that create characteristic haplotype patterns (e.g., long blocks of introgressed sequence) cannot be detected from the SFS alone.
Spatial information. The genomic positions of SNPs are irrelevant to the SFS. Recombination rate variation, which produces spatial patterns of diversity, is invisible.

\[\text{SFS} = f(\text{allele frequencies}) \neq g(\text{haplotype structure})\]

For questions about selection, recombination, or genealogy structure, use Escapement or Mainspring.

2. Teacher Quality Ceiling

Balance Wheel’s accuracy is bounded by the teacher’s accuracy. If moments has numerical issues for extreme parameter values – very large or very small population sizes, very short epochs, very high migration rates – the neural network will inherit those issues or extrapolate unpredictably.

Teacher accuracy in boundary cases
Scenario	Teacher behavior	Neural network behavior
\(N_e < 100\)	moments may lose precision (drift dominated)	May extrapolate poorly
\(N_e > 10^6\)	moments may be slow (many ODE steps)	May interpolate well if trained in range
Epoch duration < 0.001 coalescent units	moments may miss the effect	May smooth over the epoch
Migration rate \(m > 1\)	moments may be numerically unstable	Will reflect the instability

Mitigation. Validate the neural predictions against the teacher in the specific parameter regime of interest. If moments gives unreliable results in some regime, filter those examples from the training set and acknowledge the limitation.

3. Generalization to Unseen Topologies

The multi-population version is trained on a distribution of population tree topologies. If the true topology is:

More complex than the training distribution (e.g., 6 populations when training covered up to 4).
Structurally different (e.g., reticulate admixture when training used only tree-like splits).
Extreme (e.g., very asymmetric topologies not represented in the prior).

the network may produce inaccurate joint SFS predictions. Unlike moments, which can compute the SFS for any topology that can be specified, the neural network is limited to the topologies it has seen during training.

Mitigation. Ensure the training topology distribution covers the models of interest. For novel topologies, generate a focused training set and fine-tune the network. Always validate against moments for the specific topology being analyzed.

4. Not a Replacement for Full-Likelihood

For a single dataset analyzed once with a well-specified model, running moments directly is more trustworthy than running Balance Wheel. The classical solver gives the exact expected SFS; the neural approximation introduces an error (typically < 1%, but still an error). Balance Wheel’s advantages emerge only when:

You need thousands of likelihood evaluations (HMC, model comparison, bootstrap).
The model has \(k \geq 3\) populations (moments becomes slow).
You need continuous \(N_e(t)\) (moments requires piecewise-constant).
You need full posterior distributions (moments gives only the MLE).

For a two-population model analyzed once to find the MLE, moments is simpler, more transparent, and more trustworthy.

When to Use Which Complication

A decision tree for choosing the right tool:

START
  |
  v
What is your data?
  |
  ├── Raw genotypes (VCF, genotype matrix)
  |     |
  |     v
  |   Do you need the full ARG?
  |     |
  |     ├── Yes ──▶ Do you have GPU + training budget?
  |     |            |
  |     |            ├── Yes ──▶ MAINSPRING
  |     |            └── No  ──▶ tsinfer + tsdate
  |     |
  |     └── No ──▶ Do you need per-site uncertainty?
  |                 |
  |                 ├── Yes ──▶ ESCAPEMENT
  |                 └── No  ──▶ Compute SFS, then ──▶ (see SFS path)
  |
  └── Site Frequency Spectrum (SFS)
        |
        v
      How many populations?
        |
        ├── 1 or 2, simple model ──▶ moments / dadi (classical)
        |
        ├── 1 or 2, need posterior ──▶ BALANCE WHEEL + HMC
        |
        ├── 3+, any analysis ──▶ BALANCE WHEEL (only viable option)
        |
        └── Model comparison needed ──▶ BALANCE WHEEL + marginal likelihood

Summary: when to use each approach
Scenario	Recommended approach
Simple 2-pop split model, find MLE	moments directly
2-pop model, need posterior uncertainty	Balance Wheel + HMC
3+ populations, any analysis	Balance Wheel (moments too slow)
Continuous \(N_e(t)\) inference	Balance Wheel (moments needs piecewise-constant)
Model comparison (2-epoch vs. 3-epoch)	Balance Wheel + marginal likelihood
Need full ARG for downstream analysis	Mainspring
Need per-site genealogy uncertainty	Escapement
Screening 1,000 genomic windows for \(N_e(t)\)	Mainspring (speed)
Single dataset, maximal statistical rigor	Hybrid: Mainspring \(\to\) Escapement
Teaching and understanding demographic inference	dadi + moments (Timepieces)
No GPU available	moments or dadi

The honest summary

Balance Wheel occupies a specific niche: fast, differentiable SFS computation that enables Bayesian posterior inference and handles multi-population models where the classical solvers fail. It is not a replacement for dadi or moments – it is a neural accelerator for the computation they perform. When you need the MLE of a simple model, use moments. When you need the posterior of a complex model, use Balance Wheel. When you need the full ARG, use Mainspring or Escapement.

The three Complications form a hierarchy:

Balance Wheel: SFS \(\to\) demographic parameters (fastest, least detailed)
Escapement: genotypes \(\to\) genealogy + demography (per-dataset, principled)
Mainspring: genotypes \(\to\) full ARG + demography (amortized, most detailed)

And the Timepieces remain the foundation. Every neural approach in this book is built on the mathematical insights of the classical methods. Balance Wheel without dadi’s diffusion theory and moments’ ODE system would have no teacher. Escapement without the coalescent likelihood would have no loss function. Mainspring without msprime would have no training data.

Use the simplest tool that answers your question. And always check the results against a method you trust.

Full Comparison Against Classical Methods

For completeness, we place Balance Wheel alongside all relevant Timepieces:

Balance Wheel vs. all SFS-based methods
Property	dadi	moments	momi2	Balance Wheel	phlash	SMC++
Data input	SFS	SFS	SFS	SFS	Sequence pairs	Sequence pairs
Forward model	PDE solver	ODE integrator	Coalescent tensor	Neural MLP	SMC likelihood	SMC likelihood
Per-eval cost	~100 ms	~10 ms	~5 ms	~0.1 ms	~50 ms	~100 ms
Gradient	Finite diff	AD (ODE)	Analytic	Backprop	Score function	AD (ODE)
Multi-pop	Up to 3	Up to 3	Up to ~5	Up to 5+	2	2
Continuous \(N_e\)	No	No	No	Yes	Yes (spline)	Yes (spline)
Posterior	No (MLE only)	No (MLE only)	No (MLE only)	Yes (HMC)	Yes (SVGD)	No (MLE only)
Training cost	None	None	None	One-time	None	None
Needs GPU	No	No	No	Yes	Yes	No

The place of each method

dadi and moments are the gold standard for SFS-based demographic inference. They are classical, well-tested, and require no special hardware. For simple models (\(k \leq 2\)), they remain the best choice for finding the MLE.

Balance Wheel extends their reach: faster evaluations for complex models, Bayesian posteriors via HMC, continuous demography, and multi-population scaling. It trades the certainty of exact numerical computation for the speed of neural approximation.

phlash and SMC++ operate on sequence-level data (not the SFS) and use the SMC likelihood. They capture LD information that the SFS discards. For single-population or two-population inference where LD is informative, they may be more powerful than any SFS-based method.

The right choice depends on the question, the data, and the resources available.