The Gelman-Rubin statistic allows a statement about the convergence of Monte Carlo simulations.
Definition
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
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
- 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
- ^ Peng, Roger D. 7.4 Monitoring Convergence | Advanced Statistical Computing – via bookdown.org.
- ^ Vats, Dootika; Knudson, Christina (2021). "Revisiting the Gelman–Rubin Diagnostic". Statistical Science. 36 (4). arXiv:1812.09384. doi:10.1214/20-STS812.