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.

CmdStanPy Diagnostics

After fitting with CmdStanPy, run:

• fit.summary(): Returns DataFrame with 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 visual checks via ArviZ.

ArviZ Workflow

Convert to InferenceData:

idata = az.from_cmdstanpy(
    fit,
    log_likelihood="log_lik",
    observed_data={"y": y_obs}
)

Run numerical diagnostics:

• 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

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)
• az.plot_parallel(): Parallel coordinates showing divergent vs non-divergent draws

Common Issues

• Divergences + low BFMI: Geometry problems (funnels, stiff regions). Reparameterize or increase adapt_delta.
• High R̂, good visuals: Chains haven't run long enough. Extend iterations.
• Low ESS, good R̂: High autocorrelation. Reparameterize or run longer.
• Max treedepth warnings: Strong correlations. Reparameterize or simplify model.
• Multimodality in plot_posterior: Identification problem or multiple modes.

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