Convergence Diagnostics

Convergence Diagnostics

Use this skill when checking MCMC convergence after fitting Stan models. Convergence means chains mixed and explored the same target, and you have enough effective draws.

Running diagnostics

The canonical workflow computes convergence from az.summary — see summarize() in fit-pipeline > references/posterior_fit.py, which applies the thresholds below and writes them to summary.json. It avoids the OOM that raw fit.diagnose() can hit on large datasets (N > 10K).

For raw access: fit.summary() returns R_hat / ESS_bulk / ESS_tail / MCSE per parameter; fit.diagnose() checks for divergences, max treedepth, low E-BFMI, low ESS, high R-hat. If diagnose reports no problems, you still need the visual checks below.

For ArviZ-side numerical diagnostics on an InferenceData object:

• az.rhat(idata) — rank-normalized split R-hat
• az.ess(idata) — bulk and tail effective sample size
• az.bfmi(idata) — Bayesian fraction of missing information
• az.mcse(idata) — Monte Carlo standard error
• az.summary(idata) — all diagnostics in one table

See inferencedata-handling for converting fit → InferenceData.

Thresholds

Must achieve:

• R̂ < 1.01 (all parameters). Measures chain agreement.
• ESS bulk and tail ≥ 400 per parameter. Enough effective draws.
• BFMI ≥ 0.3 per chain. Adequate energy exploration.
• MCSE << posterior SD. Monte Carlo error small relative to uncertainty.
• No divergent transitions after warmup.

Visual Diagnostics

Chain mixing and stationarity:

• az.plot_trace() — should show "fat fuzzy caterpillars", no trends or stuck chains. Divergences shown as vertical lines.
• az.plot_rank() — rank histograms should be uniform and similar across chains. U-shapes or skew indicate poor mixing.

Autocorrelation and ESS:

• az.plot_autocorr() — slow decay indicates high correlation and low ESS
• az.plot_ess(kind="evolution") — ESS growth over draws; should keep climbing
• az.plot_ess(kind="local") — ESS in local windows/quantiles; checks tail exploration

HMC-specific pathologies:

• az.plot_energy() — overlays energy transitions vs marginal energy. Low BFMI shows mismatch.
• az.plot_pair(divergences=True) — localizes divergences in parameter space (funnels, tight correlations). For hierarchical models: plot each group-level parameter vs log(population_scale) with divergent transitions highlighted. Divergences near small scale = centered funnel (use non-centered). Divergences near large scale = inverted funnel (use centered).
• az.plot_parallel() — parallel coordinates showing divergent vs non-divergent draws

Common Issues

For tooling pitfalls (ArviZ lowercase vs CmdStanPy uppercase column names; KeyErrors on conversion), see stan > Known Issues and inferencedata-handling > Common failures.

Convergence Without Identifiability

A model can have R-hat < 1.01, zero divergences, and adequate ESS while being structurally non-identified. Convergence diagnostics only verify the sampler explored its target — they do not verify the target is well-posed.

Signs of non-identifiability despite clean convergence:

• Pairs plots show ridge, surface, or circular structures (parameters trade off against each other)
• Posterior SDs do not shrink proportional to 1/√N when data size increases
• Parameter pairs with |correlation| > 0.95
• Mixture-model components show R-hat > 10 without divergences (label-switching); fix with ordering constraints, not longer chains.

If suspected: classify the degeneracy (additive redundancy a+b=const, multiplicative a×b=const, label symmetry) and reparameterize to identified quantities (anchor one reference level, use contrasts instead of absolute effects, apply ordering constraints for mixture components).

You never prove convergence, only build a strong circumstantial case. The sampler tells you about your model — listen to it.