Markov Chain Monte Carlo (MCMC) convergence diagnostics are procedures for assessing whether a Markov chain sampler has reached its stationary distribution and whether a given run length is sufficient for reliable posterior inference. The core problem is that MCMC output is autocorrelated and dependent on starting values; diagnostics aim to detect failure to mix, inadequate burn-in, and convergence to the wrong distribution.
Key Ideas
Burn-in: Initial draws are discarded because the chain may start far from the stationary distribution. Formal burn-in determination: Raftery-Lewis (1992) derives the required burn-in M analytically from a pilot run via a two-state Markov approximation to the binarized chain.
Multi-chain diagnostics (Gelman-Rubin R^): Run m≥2 chains from overdispersed starting values; compute between-chain variance B and within-chain variance W. The potential scale reduction factor (PSRF) R^=V^/W converges to 1 at stationarity. Values R^>1.1 signal incomplete mixing. Requires no knowledge of the target distribution beyond the ability to sample.
Single-chain diagnostics (Geweke spectral): Split a single chain into early (first 10%) and late (last 50%) segments; test equality of means using spectral variance estimates. The Relative Numerical Efficiency (RNE) = σiid2/σMCMC2 quantifies autocorrelation loss.
Run-length planning (Raftery-Lewis 1992): Given target quantile q, tolerance r, and probability s, derives minimum draws Nmin and thinning interval k from pilot chain. More targeted than Gelman-Rubin: tells you how long to run before starting, not whether the current run is adequate.
Wrong-distribution detection (Zellner-Min 1995 Gibbs Sampler Convergence criterion, GSC²): Standard diagnostics can show apparent convergence while the chain is trapped at a local mode. The GSC² criterion exploits the identity p(α∣D)p(β∣α,D)=p(β∣D)p(α∣β,D); violations of symmetry in the Gibbs scan output indicate the chain has converged to the wrong marginals. Applied to detect multimodal failure in Seemingly Unrelated Regression (SUR) posteriors.
Thinning is never beneficial (MacEachern-Berliner 1994): For any stationary chain, the full-chain mean has weakly lower variance than any systematic subsample of the same total run length. The only legitimate reason to thin is storage or downstream processing cost.
Formal ergodicity theory (Tierney 1994): ϕ-irreducibility + aperiodicity + invariance ⇒R^→1 for π-a.e. starting point. Harris recurrence strengthens this to all starting points. These conditions underpin the validity of the above empirical diagnostics.
How It Works
Full details of each method — including the Gelman-Rubin formula, Raftery-Lewis Markov chain derivation, Zellner-Min three criteria (ARC² = Anchored Ratio, DC² = Difference, RC² = Ratio), and spectral variance estimation — are in Gibbs Sampler (§ Convergence Diagnostics) and Markov Chain Monte Carlo (§ Convergence Diagnostics). The entry here is a navigational hub; see those pages for derivations.
Why It Matters
A chain that has not converged produces biased posterior summaries regardless of how many draws are collected after burn-in. The Zellner-Min example — a chain stuck near (0,5) when the true posterior has another mode at (0,−5) — illustrates that Gelman-Rubin and Raftery-Lewis can both report convergence while the sampler is exploring only part of the space. Multiple complementary diagnostics are needed; no single check is sufficient.
Open Questions
All standard diagnostics assume the chain eventually mixes; none detects a sampler that is ϕ-irreducible but mixes on astronomical timescales.
Multivariate generalizations of R^ (Brooks-Gelman 1998 multivariate PSRF) are more powerful but require correlated-chain bookkeeping.
Convergence diagnostics for discontinuous posteriors (point-mass spikes in spike-and-slab, truncated distributions) are not well standardized.