The Gelman-Rubin statistic allows a statement about the convergence of Monte Carlo simulations.
Definition Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded. From the samples (of the j-th simulation), the variance between the chains and the variance in the chains is estimated: Mean value of chain j Mean of the means of all chains Variance of the means of the chains Averaged variances of the individual chains across all chains
An estimate of the Gelman-Rubin statistic then results as .
When L tends to infinity and B tends to zero, R tends to 1.
A different formula is given by Vats & Knudson.
Alternatives The Geweke Diagnostic compares whether the mean of the first x percent of a chain and the mean of the last y percent of a chain match.
Literature
References