x
, together with upper and lower confidence limits. Approximate
convergence is diagnosed when the upper limit is close to 1. For
multivariate chains, a multivariate value is calculated that bounds
above the potential scale reduction factor for any linear combination
of the (possibly transformed) variables.The confidence limits are based on the assumption that the stationary distribution of the variable under examination is normal. Hence the `transform' parameter may be used to improve the normal approximation.
gelman.diag(x, confidence = 0.95, transform=FALSE, autoburnin=TRUE,
multivariate=TRUE)
mcmc.list
object with more than one chain,
and with starting values that are overdispersed with respect to the
posterior distribution.x
should be transformed to improve the normality of the
distribution. If set to TRUE, a log transform or logit transform, as
appropriate, will be applied.start(x)
is less than end(x)/2
then start
of series will be adjusted so that only second gelman.diag
. This is a list with the
following elements:Point est.
) and their upper
confidence limits (labelled Upper C.I.
).x
.gelman.diag
class has its own print
method.gelman.diag
diagnostic is applied to a
single variable from the chain. It is based a comparison of within-chain
and between-chain variances, and is similar to a classical analysis of
variance.There are two ways to estimate the variance of the stationary distribution: the mean of the empirical variance within each chain, $W$, and the empirical variance from all chains combined, which can be expressed as $$\widehat{\sigma}^2 = \frac{(n-1) W }{n} + \frac{B}{n}$$ where $n$ is the number of iterations and $B/n$ is the empirical between-chain variance.
If the chains have converged, then both estimates are unbiased. Otherwise the first method will underestimate the variance, since the individual chains have not had time to range all over the stationary distribution, and the second method will overestimate the variance, since the starting points were chosen to be overdispersed.
The convergence diagnostic is based on the assumption that the target distribution is normal. A Bayesian credible interval can be constructed using a t-distribution with mean $$\widehat{\mu}=\mbox{Sample mean of all chains combined}$$ and variance $$\widehat{V}=\widehat{\sigma}^2 + \frac{B}{mn}$$ and degrees of freedom estimated by the method of moments $$d = \frac{2*\widehat{V}^2}{\mbox{Var}(\widehat{V})}$$ Use of the t-distribution accounts for the fact that the mean and variance of the posterior distribution are estimated.
The convergence diagnostic itself is $$R=\sqrt{\frac{(d+3) \widehat{V}}{(d+1)W}}$$ Values substantially above 1 indicate lack of convergence. If the chains have not converged, Bayesian credible intervals based on the t-distribution are too wide, and have the potential to shrink by this factor if the MCMC run is continued.
Brooks, SP. and Gelman, A. (1998) General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7, 434-455.
gelman.plot
.