Gelman-Rubin statistic

{{Expert needed|1=Mathematics|date=January 2024}}

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

== Definition ==

$J$ Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded.

From the samples $x\_\{1\}^\{(j)\},\backslash dots,\; x\_\{L\}^\{(j)\}$ (of the j-th simulation), the variance between the chains and the variance in the chains is estimated:

:$\backslash overline\{x\}\_j=\backslash frac\{1\}\{L\}\backslash sum\_\{i=1\}^L\; x\_i^\{(j)\}$ Mean value of chain j

:$\backslash overline\{x\}\_*=\backslash frac\{1\}\{J\}\backslash sum\_\{j=1\}^J\; \backslash overline\{x\}\_j$ Mean of the means of all chains

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

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

An estimate of the Gelman-Rubin statistic $R$ then results as{{Cite book|url=https://bookdown.org/rdpeng/advstatcomp/monitoring-convergence.html|title=7.4 Monitoring Convergence | Advanced Statistical Computing|first=Roger D.|last=Peng|via=bookdown.org}}

:$R=\backslash frac\{\backslash frac\{L-1\}\{L\}W+\backslash frac\{1\}\{L\}B\}\{W\}$.

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

A different formula is given by Vats & Knudson.{{cite journal | doi = 10.1214/20-STS812| title = Revisiting the Gelman–Rubin Diagnostic| date = 2021| last1 = Vats| first1 = Dootika| last2 = Knudson| first2 = Christina| journal = Statistical Science| volume = 36| issue = 4| arxiv = 1812.09384}}

== 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.{{citation needed|date= January 2024}}