Geweke (1992) proposed a convergence diagnostic for Markov chains. This diagnostic is based on a test for equality of the means of the first and last part of a Markov chain (by default the first 10% and the last 50%). If the samples are drawn from a stationary distribution of the chain, then the two means are equal and Geweke's statistic has an asymptotically standard normal distribution.
The test statistic is a standard Z-score: the difference between the two sample means divided by its estimated standard error. The standard error is estimated from the spectral density at zero, and so takes into account any autocorrelation.
The Z-score is calculated under the assumption that the two parts of the chain are asymptotically independent.
The Geweke.Diagnostic is a univariate diagnostic that is
usually applied to each marginal posterior distribution. A
multivariate form is not included. By chance alone due to multiple
independent tests, 5% of the marginal posterior distributions should
appear non-stationary when stationarity exists. Assessing multivariate
convergence is difficult.
Geweke.Diagnostic(x)This required argument is a vector or matrix of posterior
samples, such as from the output of the LaplacesDemon
function. Each column vector in a matrix is a chain to be assessed.
A minimum of 100 samples are required.
A vector is returned, in which each element is a Z-score for a test of
equality that compares the means of the first and last parts of each
chain supplied as x to Geweke.Diagnostic.
The Geweke.Diagnostic is essentially the same as the
geweke.diag function in the coda package, but
programmed to accept a simple vector or matrix, so it does not require
an mcmc object.
Geweke, J. (1992). "Evaluating the Accuracy of Sampling-Based Approaches to Calculating Posterior Moments". In Bayesian Statistics 4 (ed JM Bernardo, JO Berger, AP Dawid, and AFM Smith). Clarendon Press, Oxford, UK.
burnin,
is.stationary, and
LaplacesDemon
# NOT RUN {
library(LaplacesDemon)
Geweke.Diagnostic(rnorm(100))
Geweke.Diagnostic(matrix(rnorm(100),10,10))
# }
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