Английская Википедия:Gelman-Rubin statistic

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

Definition

<math>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>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>\overline{x}_j=\frac{1}{L}\sum_{i=1}^L x_i^{(j)}</math> Mean value of chain j

<math>\overline{x}_*=\frac{1}{J}\sum_{j=1}^J \overline{x}_j</math> Mean of the means of all chains

<math>B=\frac{L}{J-1}\sum_{j=1}^J (\overline{x}_j-\overline{x}_*)^2</math> Variance of the means of the chains

<math>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>R</math> then results as

<math>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.Шаблон:Citation needed

Literature

• Шаблон:Cite journal
• Шаблон:Cite journal