Evidence and uncertainty

For Bayesian inference, the evidence is the integral of the likelihood against the normalized prior. It is the normalizing constant of the unnormalized posterior density. Interpreting this value as Bayesian evidence requires retaining the likelihood’s normalization constants.

For tempered SMC, the full posterior is reached at beta=1. Evidence at an earlier temperature is the normalizing constant of that intermediate target. For nested-sampling SMC, use the shell-evidence interface described in Working with results; the ordinary state normalizing constant tracks constrained-prior mass instead. The uncertainty estimators below apply to ordinary SMC weighting.

SMCResult provides three estimates of log-evidence uncertainty from a single SMC run:

Method	Use	Main limitation
"incremental"	Default; uses each incremental weight update	Ignores dependence between SMC iterations
"lee-whiteley"	Uses the complete particle genealogy	Requires multinomial resampling at every iteration
"fixed-lag-lee-whiteley"	Uses only recent particle ancestry	Trades genealogical stability for truncation bias

Usage

log_evidence_variance estimates \(\operatorname{Var}(\log \widehat Z)\). log_evidence_error returns the corresponding standard error:

log_evidence = result.log_evidence()
variance = result.log_evidence_variance()
error = result.log_evidence_error()

lw_variance = result.log_evidence_variance(method="lee-whiteley")

fixed_lag_variance = result.log_evidence_variance(
    method="fixed-lag-lee-whiteley",
    lag=10,
)

These values describe Monte Carlo error in the SMC evidence estimate, not uncertainty in the statistical model or posterior.

Assumptions and limitations

incremental

This estimator is inexpensive, requires no ancestry, and supports adaptive temperatures and adaptive resampling. It ignores dependence between successive SMC iterations and may therefore underestimate uncertainty.

It requires log_evidence_increment_variance in every particle state’s metadata. This is recorded by FixedTemperedWeighting and AdaptiveTemperedWeighting; missing values produce nan. The correct particle-count scaling is already present in the squared normalised weights, so no extra \(N\) or \(N-1\) factor is needed.

lee-whiteley

This is the full-genealogy estimator described by Lee and Whiteley (2018). It requires at least two particles, a constant particle count, exactly one resampling step per iteration, resampling at every iteration, and multinomial resampling.

sequince records the scheme used by its built-in resampling steps. The estimator warns when it finds systematic, stratified, missing, or unknown scheme information. A custom resampling step should set its resampling_scheme class attribute. The calculation still returns a value after a warning, but it does not then have the stated theoretical justification. The estimator also becomes unreliable as the time-zero ancestor families collapse.

fixed-lag-lee-whiteley

This estimator uses only the most recent resampling events. Short lags retain more ancestor families but omit older variance contributions; long lags approach the full-genealogy estimator and its degeneracy.

lag counts resampling events, not temperature updates or SMC iterations. Skipped adaptive-resampling events do not consume lag. The constant particle count and multinomial-resampling assumptions still apply, and each iteration must contain exactly one resampling-step output. Mutation should not mix particle paths.

Recent-ancestor variance estimation is discussed by Olsson and Douc (2019). Their estimator targets normalised particle expectations rather than evidence. The method provided here is an evidence-specific fixed-lag Lee–Whiteley estimator.

Diagnostics

Inspect the selected genealogy before relying on a fixed-lag estimate:

diagnostics = result.fixed_lag_lee_whiteley_diagnostics(lag=10)
print(diagnostics.traced_resampling_events)
print(diagnostics.distinct_ancestors)
print(diagnostics.family_ess)

The diagnostics report the requested and traced lag, total resampling events, distinct selected-lag ancestors, and family ESS. A family ESS near one indicates severe genealogical depletion. Comparing several plausible lags is generally more informative than reporting one arbitrary choice.

Finite-sample Lee–Whiteley estimates can be negative. The variance method preserves such values because they signal an unreliable estimate; log_evidence_error raises ValueError because no real-valued standard error can then be computed.

Repeated independent SMC runs remain the most direct uncertainty check when they are affordable.