Gelman-Rubin statistic explained

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 ^{(j)} _{1} , ... , x ^{(j)} _{L} (of the j-th simulation), the variance between the chains and the variance in the chains is estimated: \overline{x} _{j = (1)/(L)} \sum ^{L} _{i =1} x ^{(j)} _{i} Mean value of chain j \overline{x} _{* = (1)/(J)} \sum ^{J} _{j =1} \overline{x} _{j} Mean of the means of all chains B = (L)/(J -1) \sum ^{J} _{j =1} (\overline{x} _{j-\overline{x}} ^{2} ^{*)} Variance of the means of the chains W = (1)/(J) \sum ^{J} _{j =1} \left( (1)/(L -1) \sum ^{L} _{i =1} (x ^{(j)} _{i - \overline{x}} ^{2 \right)} ^{j)} Averaged variances of the individual chains across all chains

An estimate of the Gelman-Rubin statistic

R then results as^[1] R = ((L -1)/(W + (1)/(L) B L))/(W) .

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

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

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.

Literature

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

References

1. Book: Peng, Roger D.. 7.4 Monitoring Convergence | Advanced Statistical Computing. bookdown.org.
2. 10.1214/20-STS812. Revisiting the Gelman–Rubin Diagnostic. 2021. Vats. Dootika. Knudson. Christina. Statistical Science. 36. 4. 1812.09384.