**Spatio-Temporal Methods in Environmental Epidemiology**

Chapter 5 - The Bayesian Approach in Practice

** Summary **

This chapter describes methods for implementing Bayesian models when their complexity means that simple, analytic solutions

may not be available. From this chapter, the reader will have gained an understanding of the following topics:

- Analytical approximations to the posterior distribution.
- Using samples from a posterior distribution for inference and Monte Carlo integration.
- Methods for direct sampling such as importance and rejection sampling.
- Markov Chain Monte Carlo (MCMC) and methods for obtaining samples from the required
posterior distribution including

Metropolisâ€“Hastings and Gibbs algorithms.
- Using WinBUGS to fit Bayesian models using Gibbs sampling.
- Integrated Nested Laplace Approximations (INLA) as a method for performing efficient
Bayesian inference including the use

of Râ€“INLA to implement a wide variety of latent
process models.

** R CODE **

Example 5.3

Example 5.4