The Gelman-Rubin statistic allows a statement about the convergence of Monte Carlo simulations.

${\displaystyle J}$ Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded. From the samples ${\displaystyle x_{1}^{(j)},\dots ,x_{L}^{(j)))$ (of the j-th simulation), the variance between the chains and the variance in the chains is estimated:

${\displaystyle {\overline {x))_{j}={\frac {1}{L))\sum _{i=1}^{L}x_{i}^{(j)))$ Mean value of chain j

${\displaystyle {\overline {x))_{*}={\frac {1}{J))\sum _{j=1}^{J}{\overline {x))_{j))$ Mean of the means of all chains

${\displaystyle B={\frac {L}{J-1))\sum _{j=1}^{J}({\overline {x))_{j}-{\overline {x))_{*})^{2))$ Variance of the means of the chains

${\displaystyle W={\frac {1}{J))\sum _{j=1}^{J}\left({\frac {1}{L-1))\sum _{i=1}^{L}(x_{i}^{(j)}-{\overline {x))_{j})^{2}\right)}$ Averaged variances of the individual chains across all chains

An estimate of the Gelman-Rubin statistic ${\displaystyle R}$ then results as^[1]

${\displaystyle R={\frac ((\frac {L-1}{L))W+{\frac {1}{L))B}{W))}$.

When L tends to infinity and B tends to zero, R tends to 1.

A different formula is given by Vats & Knudson.^[2].

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.^{[citation needed]}

• Vats, Dootika; Knudson, Christina (2021). "Revisiting the Gelman–Rubin Diagnostic". Statistical Science. 36 (4). arXiv:1812.09384. doi:10.1214/20-STS812.
• Gelman, Andrew; Rubin, Donald B. (1992). "Inference from Iterative Simulation Using Multiple Sequences". Statistical Science. 7 (4): 457–472. Bibcode:1992StaSc...7..457G. doi:10.1214/ss/1177011136. JSTOR 2246093.