Gelman-Rubin statistic

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

Definition

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  Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded. From the samples   (of the j-th simulation), the variance between the chains and the variance in the chains is estimated:

  Mean value of chain j
  Mean of the means of all chains
  Variance of the means of the chains
  Averaged variances of the individual chains across all chains

An estimate of the Gelman-Rubin statistic   then results as[1]

 .

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

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

Alternatives

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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]

Literature

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  • 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.

References

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