Method
NumCosmoMathMSetCatalogget_shrink_factor
Declaration [src]
gdouble
ncm_mset_catalog_get_shrink_factor (
NcmMSetCatalog* mcat
)
Description [src]
Gets the current shrink factor which is the multivariate potential scale reduction factor (MPSRF), namely, $$\hat{R}^p = \sqrt{\frac{n - 1}{n} + \left( \frac{m + 1}{m} \right) \lambda_1},$$ where $n$ is the number of points of one chain, $m$ is the number of chains and $\lambda_1$ is the largest eigenvalue of the positive definite matrix $W^{-1}B/n$.
$W$ is the within-chain covariance: $$W = $$ arithmetical mean of the covariance matrices of each chain.
$B$ is the between-chain covariance: $$B = $$ covariance between the means of each chain.
Refined version: $$\hat{R}^p = \sqrt{\frac{\hat{d} + 3}{\hat{d} + 1} \left(\frac{n - 1}{n} + \left( \frac{m + 1}{m} \right) \lambda_1\right)},$$ where $\hat{d} = 2 \hat{V}^2 / \widehat{Var}(\hat{V})$, $$\hat{V} = \frac{n -1}{n}W + \frac{m + 1}{m} \frac{B}{n}.$$
Some references for this MCMC convergence diagnostic: [Brooks and Gelman (1998)][XBrooks1998], [Gelman and Rubin (1992)][XGelman1992], SAS/STAT.