Advanced Bayesian Computation

Abstract

The popularity of Bayesian statistics is largely due to advances in computing and developments in computational methods. Currently, there are two main types of Bayesian computational methods. The first type involves iterative Monte Carlo simulation and includes the Gibbs sampler, the Metropolis-Hastings algorithm, Hamiltonian sampling etc. These first type methods typically generate a Markov chain whose stationary distribution is the target distribution. The second type involves distributional approximation and includes Laplace approximation (Laplace 1785, 1810), variational Bayes (Jordan et al. 1999), etc. These second type methods try to find a distribution with the analytical form that best approximates the target distribution. In Sect. 3.1, we review Markov chain Monte Carlo (MCMC) methods including the general Metropolis-Hastings algorithm (M-H), Gibbs sampler with conjugacy, and Hamiltonian Monte Carlo (HMC) algorithm (Neal 1994). Section 3.2 discusses the convergence and efficiency of the above sampling methods. We then show how to specify a Bayesian model and draw model inferences using OpenBUGS and Stan in Sect. 3.3. Section 3.4 provides a brief summary on the mode-based approximation methods including Laplace approximation and Bayesian variational inference. Finally, in Sect. 3.5, a full Bayesian analysis is performed on a biological data set from Gelfand et al. (1990). The key concepts and the computational tools discussed in this chapter are demonstrated in this section.