Purpose and Organization of This Book

The development and implementation of computational methods for drawing random samples from the incompletely known posterior has revolutionized Bayesian statistics. Computational Bayesian statistics breaks free from the limited class of models where the posterior can be found analytically. Statisticians can use observation models, and choose prior distributions that are more realistic, and calculate estimates of the parameters from the Monte Carlo samples from the posterior. Computational Bayesian methods can easily deal with complicated models that have many parameters. This makes the advantages that the Bayesian approach offers accessible to a much wider class of models. 

This book aims to introduce the ideas of computational Bayesian statistics to advanced undergraduate and first-year graduate students. Students should enter the course with some knowledge in Bayesian statistics at the level of Bolstad (2007). This book builds on that background. It aims to give the reader a big-picture overview of the methods of computational Bayesian statistics, and to demonstrate them for some common statistical applications. 

In Chapter 2, we look at methods which allow us to draw a random sample directly from an incompletely known posterior distribution. We do this by drawing a random sample from an easily sampled distribution, and only accepting some draws into the final sample. This reshapes the accepted sample so it becomes a random sample from the incompletely known posterior. These methods work very well for a single parameter. However, they can be seen to become very inefficient as the number of parameters increases. The main use of the.se methods is as a small step that samples a single parameter as part of a Gibbs sampler. 

Chapter 3 compares Bayesian inferences drawn from a numerical posterior with Bayesian inferences from the posterior random sample. 

Chapter 4 reviews Bayesian statistics using conjugate priors. These are the classical models that have analytic solutions to the posterior. In computational Bayesian statics, these are useful tools for drawing an individual parameter as steps of the Markov chain Monte Carlo algorithms. 

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