Inference and assessing convergence#

Part 3 · Stage 9 · ⛓️ MCMC: Gibbs, Metropolis & HMC · Lesson 072 of 144 · intermediate

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Important

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The two questions#

MCMC draws are neither independent nor, at first, from the posterior. So two questions must be answered before any summary is trusted. Has the chain converged — is it now sampling the target rather than still travelling toward it? And how much information do its correlated draws actually carry?

Warm-up, and many chains#

Discard an initial warm-up phase (the sampler is also adapting its step size there). Then run several chains from dispersed starting points: a single chain stuck in one mode looks perfectly healthy from the inside. Convergence is diagnosed by disagreement between chains, which requires more than one.

R-hat#

The classical diagnostic compares between-chain to within-chain variance: if the chains have forgotten where they started and are exploring the same distribution, the pooled variance is no larger than the average within-chain variance, and

\[\hat{R} = \sqrt{\frac{\widehat{\mathrm{var}}^{+}(\theta \mid y)}{W}} \;\longrightarrow\; 1 .\]

Vehtari, Gelman, Simpson, Carpenter and Bürkner showed the original \(\hat{R}\) has serious flaws — it fails to detect trouble when a chain has a heavy tail or when the variance differs across chains. Their replacement rank-normalises the draws (robust to heavy tails), folds them about the median to catch differing scales, and splits each chain in half to catch drift within a chain. Modern practice: require \(\hat{R} < 1.01\), not the old 1.1.

Effective sample size#

Autocorrelation means \(S\) draws carry the information of \(S_{\text{eff}} < S\) independent ones. Report bulk-ESS (governing the mean and sd) and tail-ESS (governing the 5% and 95% quantiles) separately, and require both above about 400 — a chain can mix well in the middle and crawl in the tails.

import arviz as az
az.summary(idata)                 # r_hat, ess_bulk, ess_tail, mcse_mean, mcse_sd
az.plot_rank(idata)               # rank plots: uniform bars = chains agree
idata.sample_stats["diverging"].sum()    # HMC: must be zero

Prefer rank plots to trace plots: with many chains the “fat hairy caterpillar” is unreadable, whereas rank histograms should be uniform across chains if all are sampling the same distribution.

What convergence is not#

These diagnostics are necessary, not sufficient. They can only detect the failures the chains happened to reveal: a mode never visited by any chain leaves no trace in \(\hat{R}\). And a converged sampler faithfully reproduces the posterior of a model that may be wrong — computation and model adequacy are separate questions, checked by separate tools. Convergence buys you the right posterior for the model you actually wrote.

Hint