12.2 Gibbs Sampling
Gibbs sampling (or Gibbs sampler) of Geman and Geman (1984) and Gelfand and Smith (1990) is perhaps the most popular MCMC method. We introduce the idea of Gibbs sampling by using a simple problem with three parameters. Here the word parameter is used in a very general sense. A missing data point can be regarded as a parameter under the MCMC framework. Similarly, an unobservable variable such as the “true” price of an asset can be regarded as N parameters when there are N transaction prices available. This concept of parameter is related to data augmentation and becomes apparent when we discuss applications of the MCMC methods.
Denote the three parameters by θ1, θ2, and θ3. Let X be the collection of available data and M the entertained model. The goal here is to estimate the parameters so that the fitted model can be used to make inference. Suppose that the likelihood function of the model is hard to obtain, but the three conditional distributions of a single parameter given the others are available. In other words, we assume that the following three conditional distributions are known:
where denotes the conditional distribution of the parameter θi given the data, the model, and the other two parameters. In application, we do not need to know the exact forms ...