Bruce E. Trumbo

Using Gibbs Samplers to Compute Bayesian Posterior Distributions

In Chapter 8, we introduced the fundamental ideas of Bayesian inference, in which prior distributions on parameters are used together with data to obtain posterior distributions and thus interval estimates of parameters. However, in practice, Bayesian posterior distributions are often difficult to compute.

Appendix: Getting Started with R

This appendix focuses on some specific features and commands of R that you will need in the first few chapters of this book. If you have never used R before—or you need a review of the basics—we recommend starting here. Throughout the book, we show the R code required for each new concept, briefly explaining the new R commands involved. If you want more detailed information, you can refer to the introductory and general reference manuals available on the R website.

Introduction to Bayesian Estimation

The rest of this book deals with Bayesian estimation. This chapter uses examples to illustrate the fundamental concepts of Bayesian point and interval estimation. It also provides an introduction to Chapters 9 and 10, where more advanced examples require computationally intensive methods.

Generating Random Numbers

Much of this book deals with simulation methods for probability models, also called Monte Carlo methods. We have seen a few introductory examples in Chapter 1. Even for some models that are easy to specify in a theoretical form, it may be difficult or impossible to “do the math” necessary to obtain the numerical results required in practice. Because of recent advances in computer hardware and software, simulation methods now offer feasible solutions to some of these troublesome computational problems.

Examples of Markov Chains with Larger State Spaces

In Chapter 6, we took advantage of the simplicity of 2-state chains to intro- duce fundamental ideas of Markov dependence and long-run behavior using only elementary mathematics. Markov chains taking more than two values are needed in many simulations of practical importance. These chains with larger state spaces can behave in very intricate ways, and a rigorous mathematical treatment of them is beyond the scope of this book. Our approach in this chapter is to provide examples that illustrate some of the important behaviors of more general Markov chains.

Sampling from Applied Probability Models

In Chapter 3, we used the sampling method to find probabilities and expectations involving a random variable with a distribution that is easy to describe but with a density function that is not explicitly known. In this chapter, we explore additional applications of the sampling method. The examples are chosen because of their practical importance or theoretical interest. Some- times, analytic methods can be used to get exact results for special cases, thus providing some confidence in the validity of more general simulation results. Also, in an elementary way, some of the examples and problems show how simulation can be useful in research. At least they have illustrated this to us personally because we have gained insights from simulation in these settings that we might never have gained by analytic means.

Monte Carlo Integration and Limit Theorems

In Chapter 1, we did a few simulations by sampling from finite populations. In Chapter 2, we discussed (pseudo)random numbers and the simulation of some familiar discrete and continuous distributions. In this chapter, we investigate how simulation is used to approximate integrals and what some fundamental limit theorems of probability theory have to say about the accuracy of these approximations. Section 3.1 sets the stage with elementary examples that illustrate some methods of integration.

Using WinBUGS for Bayesian Estimation

Historically, an important roadblock to using Bayesian inference has been the difficulty of computing posterior distributions of parameters. Thus, a major focus of this book is to show how such computations can be done using modern hardware and software.

Introductory Examples: Simulation, Estimation, and Graphics

Because simulation is a major topic of this book, it seems appropriate to start with a few simple simulations. For now, we limit ourselves to simulations based on the R-language function sample.

Markov Chains with Two States

A stochastic process is a collection of random variables, usually considered to be indexed by time. In this book we consider sequences of random variables X 1,X 2…, viewing the subscripts 1,2,… as successive steps in time. The values assumed by the random variables X n are called states of the process, and the set of all its states is called the state space. Sometimes it is convenient to think of the process as describing the movement of a particle over time. If X 1 = i and X2 = j, then we say that the process (or particle) has made a transition from state i at step 1 to state j at step 2. Often we are interested in the behavior of the process over the long run—after many transitions.