Edited by Adrian R. Bell, Chris Brooks and Marcel Prokopczuk
Markov Chain Monte Carlo (MCMC) is a method of making Bayesian inference from the posterior distribution of model parameters based on their prior distribution and the likelihood of the observed data. It is especially powerful when dealing with complicated and un-normalized posterior distributions of parameters in a multidimensional or nonlinear model. However, if the likelihood evaluation involves a latent or unobservable process, as in the case of state space models, then the latent process needs to be estimated through a pure Bayesian method or filtering methods such as a Kalman filter or a particle filter. In this chapter, we present the MCMC method for sampling from the posterior distribution where particle filtering is nested in the likelihood evaluation. Firstly, we introduce the general concepts and procedures for Bayesian inference. Next, we establish the general MCMC framework with a few useful algorithms. Finally, we introduce particle filtering and combine it with MCMC for drawing random samples from the posterior distribution. Throughout this chapter, examples based on the Basel II single-factor portfolio credit risk model are used to illustrate the concepts and procedures.
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