Analysis of Financial Time Series, Third Edition

12.3 Bayesian Inference

Conditional distributions play a key role in Gibbs sampling. In the statistical literature, these conditional distributions are referred to as conditional posterior distributions because they are distributions of parameters given the data, other parameter values, and the entertained model. In this section, we review some well-known posterior distributions that are useful in using MCMC methods.

12.3.1 Posterior Distributions

There are two approaches to statistical inference. The first approach is the classical approach based on the maximum-likelihood principle. Here a model is estimated by maximizing the likelihood function of the data, and the fitted model is used to make inference. The other approach is Bayesian inference that combines prior belief with data to obtain posterior distributions on which statistical inference is based. Historically, there were heated debates between the two schools of statistical inference. Yet both approaches have proved to be useful and are now widely accepted. The methods discussed so far in this book belong to the classical approach. However, Bayesian solutions exist for all of the problems considered. This is particularly so in recent years with the advances in MCMC methods, which greatly improve the feasibility of Bayesian analysis. Readers can revisit the previous chapters and derive MCMC solutions for the problems considered. In most cases, the Bayesian solutions are similar to what we had before. In some cases, the ...