rhat

Rhat, short for the potential scale reduction factor, is a diagnostic statistic used to assess convergence of Markov chain Monte Carlo (MCMC) simulations. It compares the variability within each chain to the variability between multiple chains, with values near 1 indicating convergence to the target distribution.

Calculation: Run at least two or more chains from overdispersed starting values. For a scalar parameter, let W be the average within-chain variance and B be the between-chain variance of chain means. The pooled estimate of marginal posterior variance is V_hat = ((n-1)/n) W + (1/n) B, where n is the number of iterations per chain. Then Rhat = sqrt( V_hat / W ). Values substantially greater than 1 indicate potential lack of convergence, while Rhat close to 1 (often < 1.1, or < 1.01 in stricter contexts) suggests convergence. Some implementations report split-Rhat, which splits each chain in two to increase sensitivity.

Usage and interpretation: Software libraries (Stan, PyMC, JAGS) report Rhat for each parameter. Analysts look for Rhat near 1 and examine effective sample size (ESS) to assess precision. Rhat is a diagnostic aid, not a guarantee of convergence.

Limitations: Rhat presumes stationary chains and can be misleading in multimodal or highly skewed posteriors; it may fail to reflect slow convergence or mixing problems. It should be used with other diagnostics such as trace plots, autocorrelations, and ESS.

Extensions: Multivariate Rhat assesses convergence of a vector of parameters. The concept has been extended to more general scale reductions and robust variants.