Adaptive Rejection Metropolis Sampling within Gibbs Sampling

TLDR

Gibbs sampling is a powerful inference method that requires sampling from full conditional distributions, which are often complex and computationally expensive, and while adaptive rejection sampling efficiently handles log‑concave conditionals, many applied fields such as population pharmacokinetics produce non‑log‑concave conditionals and noisy data that necessitate robust error models. The paper extends adaptive rejection sampling with a Hastings‑Metropolis step to handle non‑log‑concave full conditionals and proposes a robust nonlinear full probability model for population pharmacokinetic data. The method incorporates a Hastings‑Metropolis step into adaptive rejection sampling and employs a robust t‑distributed error structure within a nonlinear mixed‑effects pharmacokinetic model. We demonstrate that our method enables Bayesian inference for this model, through an analysis of antibiotic administration in new‑born babies.

Abstract

Gibbs sampling is a powerful technique for statistical inference. It involves little more than sampling from full conditional distributions, which can be both complex and computationally expensive to evaluate. Gilks and Wild have shown that in practice full conditionals are often log‐concave, and they proposed a method of adaptive rejection sampling for efficiently sampling from univariate log‐concave distributions. In this paper, to deal with non‐log‐concave full conditional distributions, we generalize adaptive rejection sampling to include a Hastings‐Metropolis algorithm step. One important field of application in which statistical models may lead to non‐log‐concave full conditionals is population pharmacokinetics. Here, the relationship between drug dose and blood or plasma concentration in a group of patients typically is modelled by using nonlinear mixed effects 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 is appropriate to account for outlying observations and/or patients. We propose a robust nonlinear full probability model for population pharmacokinetic data. We demonstrate that our method enables Bayesian inference for this model, through an analysis of antibiotic administration in new‐born babies.