### Rhat (\(\hat{R}\))

Quick definition One way to monitor whether a chain has converged to the equilibrium distribution is to compare its behavior to other randomly initialized chains. This is the motivation for the Gelman and Rubin potential scale reduction statistic \(\hat{R}\). The \(\hat{R}\) statistic measures the ratio of the average variance of samples within each chain to the variance of the pooled samples across chains; if all chains are at equilibrium, these will be the same and \(\hat{R}\) will be one. If the chains have not converged to a common distribution, the \(\hat{R}\) statistic will be greater than one.

#### More details

Gelman and Rubin’s recommendation is that the independent Markov chains be initialized with diffuse starting values for the parameters and sampled until all values for \(\hat{R}\) are below 1.1. Stan allows users to specify initial values for parameters and it is also able to draw diffuse random initializations itself.

Details on the computatation of \(\hat{R}\) and some of its limitations can be found in the 'Markov Chain Monte Carlo Sampling' chapter of the Stan Modeling Language User's Guide and Reference Manual.