s 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 <span class="math inline">\(\hat{R}\)</span> will
be one. If the chains have not converged to a common distribution,
the <span class="math inline">\(\hat{R}\)</span> statistic will be greater than one.</p>
<p>Gelman and Rubin’s recommendation is that the independent Markov
chains be initialized with diffuse starting values for th