Gelman-Rubin statistic
The Gelman-Rubin statistic allows a statement about the convergence of Monte Carlo simulations.
Definition
[math]\displaystyle{ J }[/math] Monte Carlo simulations (chains) are started with different initial values. The samples from the respective burn-in phases are discarded. From the samples [math]\displaystyle{ x_{1}^{(j)},\dots, x_{L}^{(j)} }[/math] (of the j-th simulation), the variance between the chains and the variance in the chains is estimated:
- [math]\displaystyle{ \overline{x}_j=\frac{1}{L}\sum_{i=1}^L x_i^{(j)} }[/math] Mean value of chain j
- [math]\displaystyle{ \overline{x}_*=\frac{1}{J}\sum_{j=1}^J \overline{x}_j }[/math] Mean of the means of all chains
- [math]\displaystyle{ B=\frac{L}{J-1}\sum_{j=1}^J (\overline{x}_j-\overline{x}_*)^2 }[/math] Variance of the means of the chains
- [math]\displaystyle{ W=\frac{1}{J} \sum_{j=1}^J \left(\frac{1}{L-1} \sum_{i=1}^L (x^{(j)}_i-\overline{x}_j)^2\right) }[/math] Averaged variances of the individual chains across all chains
An estimate of the Gelman-Rubin statistic [math]\displaystyle{ R }[/math] then results as[1]
- [math]\displaystyle{ R=\frac{\frac{L-1}{L}W+\frac{1}{L}B}{W} }[/math].
When L tends to infinity and B tends to zero, R tends to 1.
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). 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.
References
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