BMK.Diagnostic: BMK Convergence Diagnostic
In LaplacesDemon: Complete Environment for Bayesian Inference

Description Usage Arguments Details Value References See Also Examples

Given a matrix of posterior samples from MCMC, the BMK.Diagnostic function calculates Hellinger distances between consecutive batches for each chain. This is useful for monitoring convergence of MCMC chains.

1	BMK.Diagnostic(X, batches=10)

`X`	This required argument accepts a matrix of posterior samples or an object of class `demonoid`, in which case it uses the posterior samples in `X$Posterior1`.
`batches`	This is the number of batches on which the convergence diagnostic will be calculated. The `batches` argument defaults to 10.

Hellinger distance is used to quantify dissimilarity between two probability distributions. It is based on the Hellinger integral, introduced by Hellinger (1909). Traditionally, Hellinger distance is bound to the interval [0,1], though another popular form occurs in the interval [0,sqrt(2)]. A higher value of Hellinger distance is associated with more dissimilarity between the distributions.

Convergence is assumed when Hellinger distances are below a threshold, indicating that posterior samples are similar between consecutive batches. If all Hellinger distances beyond a given batch of samples is below the threshold, then burnin is suggested to occur immediately before the first batch of satisfactory Hellinger distances.

As an aid to interpretation, consider a matrix of 1,000 posterior samples from three chains: beta[1], beta[2], and beta[3]. With 10 batches, the column names are: 100, 200, ..., 900. A Hellinger distance for the chain beta[1] at 100 is the Hellinger distance between two batches: samples 1-100, and samples 101:200.

A benefit to using BMK.Diagnostic is that the resulting Hellinger distances may easily be plotted with the plotMatrix function, allowing the user to see quickly which consecutive batches of which chains were dissimilar. This makes it easier to find problematic chains.

The BMK.Diagnostic is calculated automatically in the LaplacesDemon function, and is one of the criteria in the Consort function regarding the recommendation of when to stop updating the Markov chain Monte Carlo (MCMC) sampler in LaplacesDemon.

For more information on the related topics of burn-in and stationarity, see the burnin and is.stationary functions, and the accompanying vignettes.

The BMK.Diagnostic function returns an object of class bmk that is a J x B matrix of Hellinger distances between consecutive batches for J parameters of posterior samples. The number of columns, B is equal to the number of batches minus one.

The BMK.Diagnostic function is similar to the bmkconverge function in package BMK.

Boone, E.L., Merrick, J.R. and Krachey, M.J. (2013). "A Hellinger Distance Approach to MCMC Diagnostics". Journal of Statistical Computation and Simulation, in press.

Hellinger, E. (1909). "Neue Begrundung der Theorie quadratischer Formen von unendlichvielen Veranderlichen" (in German). Journal fur die reine und angewandte Mathematik, 136, p. 210–271.

burnin, Consort, is.stationary, and LaplacesDemon.

library(LaplacesDemon)
N <- 1000 #Number of posterior samples
J <- 10 #Number of parameters
Theta <- matrix(runif(N*J),N,J)
colnames(Theta) <- paste("beta[", 1:J, "]", sep="")
for (i in 2:N) {Theta[i,1] <- Theta[i-1,1] + rnorm(1)}
HD <- BMK.Diagnostic(Theta, batches=10)
plot(HD, title="Hellinger distance between batches")