Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects

Markov chain Monte Carlo (MCMC) is a versatile sampling approach that is useful in a wide range of estimation and simulation applications. Fundamentally, MCMC is a powerful general means of generating random samples from probability distributions from which it is otherwise difficult to draw samples. The MCMC method is named for its reliance on the construction of a Markovian (dependent) sequence of random variables. Under modest conditions, the sequence has a limiting probability distribution that corresponds to the distribution of interest, often called the target distribution. The sequence, as will be seen, is easily constructed, and the target distribution may be almost any distribution of interest. These two features of MCMC make it a popular choice for Monte Carlo simulation. The primary forms of MCMC are the Metropolis-Hastings (M-H) algorithm and Gibbs sampling. Although both forms are useful, we focus on M-H, which is more flexible and easier to implement. A general discussion of many aspects of MCMC, including several examples, is given in [1]. As discussed in "Summary," the purpose of this article is to provide some of the theoretical support for M-H that was not given in [1], focusing specifically on the stationarity and convergence of the underlying Markov process.