Gelman-Rubin statistic

Short description: Statement about the convergence of Monte Carlo simulations

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.

Literature

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