Gelman-Rubin statistic

Template:Expert needed 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 $x_1^{(j)},\dots ,x_L^{(j)}$ (of the j-th simulation), the variance between the chains and the variance in the chains is estimated:

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

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

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

$W=\frac{1}{J}\sum _{j=1}^J(\frac{1}{L-1}\sum _{i=1}^L(x_i^{(j)}-{\bar{x}}_j{)}^2)$ Averaged variances of the individual chains across all chains

An estimate of the Gelman-Rubin statistic $R$ then results as

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

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

A different formula is given by Vats & Knudson.

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.Template:Citation needed

Literature

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