Gelman and Rubin's MCMC Convergence Diagnostic

Description

Gelman and Rubin (1992) proposed a general approach to monitoring convergence of MCMC output in which m > 1 parallel chains are updated with initial values that are overdispersed relative to each target distribution, which must be normally distributed. Convergence is diagnosed when the chains have ‘forgotten’ their initial values, and the output from all chains is indistinguishable. The Gelman.Diagnostic function makes a comparison of within-chain and between-chain variances, and is similar to a classical analysis of variance. A large deviation between these two variances indicates non-convergence.

This diagnostic is popular as a stopping rule, though it requires parallel chains. The LaplacesDemon.hpc function is an extension of LaplacesDemon to enable parallel chains. As an alternative, the popular single-chain stopping rule is based on MCSE.

Usage

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Gelman.Diagnostic(x, confidence=0.95, transform=FALSE)

Arguments

x

This required argument accepts an object of class demonoid.hpc, or a list of multiple objects of class demonoid, where the number of components in the list is the number of chains.

confidence

This is the coverage probability of the confidence interval for the potential scale reduction factor (PSRF).

transform

Logical. If TRUE, then marginal posterior distributions in x may be transformed to improve the normality of the distribution, which is assumed. A log-transform is applied to marginal posterior distributions in the interval (0, Inf], or a logit-transform is applied to marginal posterior distributions in the interval (0,1).

Details

To use the Gelman.Diagnostic function, the user must first have multiple MCMC chains for the same model, and three chains is usually sufficient. The easiest way to obtain multiple chains is with the LaplacesDemon.hpc function.

Although the LaplacesDemon function does not simultaneously update multiple MCMC chains, it is easy enough to obtain multiple chains, and if the computer has multiple processors (which is common), then multiple chains may be obtained simultaneously as follows. The model file may be opened in separate, concurrent R sessions, and it is recommended that a maximum number of sessions is equal to the number of processors, minus one. Each session constitutes its own chain, and the code is identical, except the initial values should be randomized with the GIV function so the chains begin in different places. The resulting object of class demonoid for each chain is saved, all objects are read into one session, put into a list, and passed to the Gelman.Diagnostic function.

Initial values must be overdispersed with respect to each target distribution, though these distributions are unknown in the beginning. Since the Gelman.Diagnostic function relies heavily on overdispersion with respect to the target distribution, the user should consider using MCMC twice, first to estimate the target distributions, and secondly to overdisperse initial values with respect to them. This may help identify multimodal target distributions. If multiple modes are found, it remain possible that more modes exist. When multiple modes are found, and if chains are combined with the Combine function, each mode is probably not represented in a proportion correct to the distribution.

The ‘potential scale reduction factor’ (PSRF) is an estimated factor by which the scale of the current distribution for the target distribution might be reduced if the simulations were continued for an infinite number of iterations. Each PSRF declines to 1 as the number of iterations approaches infinity. PSRF is also often represented as R-hat. PSRF is calculated for each marginal posterior distribution in x, together with upper and lower confidence limits. Approximate convergence is diagnosed when the upper limit is close to 1. The recommended proximity of each PSRF to 1 varies with each problem, but a general goal is to achieve PSRF < 1.1. PSRF is an estimate of how much narrower the posterior might become with an infinite number of iterations. When PSRF = 1.1, for example, it may be interpreted as a potential reduction of 10% in posterior interval width, given infinite iterations. The multivariate form 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 target distribution is stationary and normally distributed. The transform argument may be used to improve the normal approximation.

A large PSRF indicates that the between-chain variance is substantially greater than the within-chain variance, so that longer simulation is needed. If a PSRF is close to 1, then the associated chains are likely to have converged to one target distribution. A large PSRF (perhaps generally when a PSRF > 1.2) indicates convergence failure, and can indicate the presence of a multimodal marginal posterior distribution in which different chains may have converged to different local modes (see is.multimodal), or the need to update the associated chains longer, because burn-in (see burnin) has yet to be completed.

The Gelman.Diagnostic is essentially the same as the gelman.diag function in the coda package, but here it is programmed to work with objects of class demonoid.

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

sigma.hat^2 = (n-1)W/n + 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 initial values were chosen to be overdispersed (and this assumes the target distribution is known, see above).

This convergence diagnostic is based on the assumption that each target distribution is normal. A Bayesian probability interval (see p.interval) can be constructed using a t-distribution with mean

mu.hat = Sample mean of all chains combined,

variance

V.hat = sigma.hat2 + B/(mn),

and degrees of freedom estimated by the method of moments

d = 2*V.hat^2 / Var(V.hat)

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((d+3) V.hat /((d+1)W)

Values substantially above 1 indicate lack of convergence. If the chains have not converged, then Bayesian probability intervals based on the t-distribution are too wide, and have the potential to shrink by this factor if the MCMC run is continued.

The multivariate version of Gelman and Rubin's diagnostic was proposed by Brooks and Gelman (1998). Unlike the univariate proportional scale reduction factor, the multivariate version does not include an adjustment for the estimated number of degrees of freedom.

Value

A list is returned with the following components:

PSRF

This is a list containing the point-estimates of the potential scale reduction factor (labelled Point Est.) and the associated upper confidence limits (labelled Upper C.I.).

MPSRF

This is the point-estimate of the multivariate potential scale reduction factor.

References

Brooks, S.P. and Gelman, A. (1998). "General Methods for Monitoring Convergence of Iterative Simulations". Journal of Computational and Graphical Statistics, 7, p. 434–455.

Gelman, A. and Rubin, D.B. (1992). "Inference from Iterative Simulation using Multiple Sequences". Statistical Science, 7, p. 457–511.

See Also

Combine, GIV, is.multimodal, LaplacesDemon, LaplacesDemon.hpc, MCSE, and p.interval.

Examples

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#library(LaplacesDemon)
###After updating multiple chains with LaplacesDemon.hpc, do:
#Gelman.Diagnostic(Fit)

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