Adaptive Rejection Metropolis Samplingwithin Gibbs Sampling

By W. R. GILKSt and N. G. BEST

Medical Research Council Biostatistics Unit, Cambridge, UK

K. K. C. TANAddenbrooke's Hospital, Cambridge, UK

[Received October 1993. Final revision October 1994]

SUMMARY
Gibbs sampling is a powerful technique for statistical inference. It involves little more thansampling from full conditional distributions, which can be both complex and computationallyexpensive to evaluate. Gilks and Wild have shown that in practice full conditionals are often logconcave,and they proposed a method of adaptive rejection sampling for efficiently sampling fromunivariate log-concave distributions. In this paper, to deal with non-log-concave full conditionaldistributions, we generalize adaptive rejection sampling to include a Hastings-Metropolis algorithmstep. One important field of application in which statistical models may lead to non-log-concavefull conditionals is population pharmacokinetics. Here, the relationship between drug dose andblood or plasma concentration in a group of patients typically is modelled by using non-linear mixedeffects models. Often, the data used for analysis are routinely collected hospital measurements,which tend to be noisy and irregular. Consequently, a robust (t-distributed) error structure isappropriate to account for outlying observations and/or patients. We propose a robust non-linearfull probability model for population pharmacokinetic data. We demonstrate that our methodenables Bayesian inference for this model, through an analysis of antibiotic administration innew-born babies.Keywords: Bayesian computation; Gibbs sampling; Markov chain Monte Carlo method;Metropolis algorithm; Pharmacokinetic model; Random variate generation