Introduction to Bayesian Estimation and Copula Models of Dependence

4 Markov Chain Monte Carlo Methods

In our introduction to Markov chain Monte Carlo (further often abbreviated as MCMC) methods we will try to avoid all details which are not immediately necessary to understand the main ideas of the algorithms. Thus we sacrifice mathematical rigor and computational convenience. We also avoid detailed descriptions of more modern and more complicated MCMC methods. For a more systematic exposure to MCMC we can recommend excellent texts of Gamerman and Lopes [9] and Robert and Casella [26]. Computational details are treated with attention by Bolstad [3] and Kruschke [17], and also with special reference to R in [25].

4.1 Markov Chain Simulations for Sun City and Ten Coins

In this section we will ask our readers to pretend being even more ignorant about the Sun City weather situation than in Sections 3.5 and 3.6. Here we assume that not only we are unable to analytically derive the stationary distribution for the Markov chain of daily weather changes from its transition matrix, but that this matrix itself is not known to us.

Therefore, in order to determine the long-term proportion of rainy days (which can be also treated as the mean value of the binomial variable taking two values: 1 if it rains, and 0 if it is sunny), we can use neither the analytic tools based on solving linear equations developed in Section 3.5 nor the direct Markov chain simulation described in Section 3.6.

For such a weird situation we may introduce the following two-stage ...