03: BSL Inference — Mutation-Selection Model

What is BSL?

Bayesian Synthetic Likelihood is a "likelihood-free" inference method — like ABC-SMC (notebook 02), it does not require a mathematical formula for the probability of the data given the parameters. Instead, it uses simulation to approximate that probability.

The biological analogy: imagine you want to know whether a particular combination of mutation rate, DFE shape, and other parameters could have produced the fitness trajectory you observed in a population. You cannot solve this analytically — the mutation-selection process is too complex. So instead, you run 50 simulations at those parameter values and look at the results. If the observed data looks "typical" compared to those 50 simulations, the parameters are plausible. If the observed data is an extreme outlier, those parameters are unlikely.

BSL formalizes this intuition: it fits a multivariate normal (bell curve) distribution to the summary statistics from the 50 simulations, then evaluates how probable the observed statistics are under that bell curve. This gives a number — the synthetic likelihood — that can be plugged into standard MCMC machinery to explore parameter space and build a posterior distribution.

How does BSL differ from ABC-SMC?

Feature	ABC-SMC (notebook 02)	BSL (this notebook)
Core question	"Are these simulations close enough to the data?"	"What is the probability of the data under a normal model fitted to these simulations?"
Epsilon threshold	Required — defines "close enough" (arbitrary choice)	Not needed — uses a proper likelihood instead
Convergence diagnostics	Heuristic (particle diversity, ESS of weights)	Formal MCMC diagnostics (R-hat, ESS, trace plots)
Sampling mechanism	Sequential importance sampling with particles	MCMC random walk (BlackJAX adaptive Metropolis)
Key assumption	Good distance metric between simulations and data	Summary statistics are approximately multivariate normal
Computational cost per step	One simulation per particle	50 simulations per MCMC step (more expensive, but fewer steps needed)

Why use both? If ABC-SMC and BSL agree on the inferred parameter values, we have strong evidence that the results are robust — they are not artifacts of either method's assumptions. This is especially important when the conclusions have biological significance, such as determining whether a population sits above or below the mutational meltdown threshold in the Basener & Sanford framework.

1. BSL Posterior Distributions

Figure 1: Marginal posterior distributions from adaptive BSL. Each panel shows the posterior distribution for one of the five inferred parameters. Red dashed lines mark the true parameter values used to generate the synthetic data.

Reading these plots biologically:

mu (mutation rate): The posterior for mu tells us how confidently we can estimate the per-offspring mutation rate from population fitness data alone. A tight peak near the true value (0.1) means the fitness trajectory is highly informative about mutation rate — which makes biological sense, since mutation rate directly controls the rate of fitness change in d(m̄)/dt ≈ Var(m) + μ · E_g[s] · b̄.
gamma_shape (DFE shape): Values below 1.0 indicate an L-shaped distribution of fitness effects, where most mutations have small effects but a heavy tail of large-effect mutations exists. The posterior width here reflects how well the data distinguishes between different DFE shapes.
gamma_scale (DFE scale): Together with gamma_shape, this determines the mean deleterious effect (shape × scale). The posterior may show correlation with gamma_shape, since different shape/scale combinations can produce similar mean effects.
p_beneficial: The fraction of mutations that are beneficial. This is typically the hardest parameter to pin down because beneficial mutations are rare (0.5%) and their signal is small relative to the dominant deleterious mutation load.
sigma_env_ind: Environmental noise on fitness expression. This parameter captures non-genetic variation in fitness and can partially trade off with genetic variance in the summary statistics.

Computational details:

n_sims=50 — 50 simulations per likelihood evaluation, controlling the variance of the synthetic likelihood estimate
chunk_size=10 — simulations are processed in batches of 10 on the GPU to manage memory usage
n_warmup=200, n_samples=500 — 200 burn-in steps discarded, 500 posterior samples retained
n_chains=2 — two independent chains for convergence assessment
4 adaptation phases — the proposal distribution is tuned in 4 phases of 40 steps each

MCMC diagnostics: why they matter

A key advantage of BSL over ABC-SMC is that it produces standard MCMC chains with formal convergence diagnostics. These tell us whether we can trust the posterior:

R-hat (Gelman-Rubin statistic): Compares variance within each chain to variance between the two chains. Values near 1.0 (below 1.05) mean both chains have converged to the same distribution — they agree on the answer. In plain language: the sampler started from two different places in parameter space and arrived at the same conclusions. Values above 1.1 indicate the chains have not yet converged and the results are unreliable.
ESS (Effective Sample Size): Because MCMC samples are correlated (each step depends on the previous one), 500 raw samples may contain only ~100–200 effectively independent draws. ESS above 100 is generally adequate for estimating posterior means and credible intervals. Low ESS suggests the chain needs to run longer or the proposal needs better tuning.
Acceptance rate: The fraction of proposed MCMC moves that were accepted. Rates of 20–40% are typical for well-tuned random-walk Metropolis in multiple dimensions. Very low rates (<10%) mean the proposals are too large; very high rates (>90%) mean the proposals are too small and the chain is barely moving.

In plain language: convergence means the sampler has explored enough of the five-dimensional parameter space to give reliable answers. It has visited all the plausible parameter combinations, not just a small corner. Good diagnostics (R-hat near 1.0, ESS > 100) are our assurance that running the sampler longer would not substantially change the posterior distributions shown above.

03: Bayesian Synthetic Likelihood (BSL)

1. BSL Posterior Distributions