Markov Chain Monte Carlo

Abstract

One of the simplest and most powerful practical uses of the ergodic theory of Markov chains is in Markov chain Monte Carlo (MCMC). Suppose we wish to simulate from a probability density π (which will be called the target density) but that direct simulation is either impossible or practically infeasible (possibly due to the high dimensionality of π). This generic problem occurs in diverse scientific applications, for instance Statistics, Computer Science, and Statistical Physics. Markov chain Monte Carlo offers an indirect solution based on the observation that it is much easier to construct an ergodic Markov chain with π as a stationary probability measure, than to simulate directly from π. This is because of the ingenious MetropolisHastings algorithm which takes an arbitrary Markov chain and adjusts it using a simple accept-reject mechanism to ensure the stationarity of π for the resulting process. The algorithms was introduced by Metropolis et al. (1953) in a Statistical Physics context, and was generalised by Hastings (1970). It was considered in the context of image analysis (Geman and Geman, 1984) data augmentation (Tanner and Wong, 1987). However, its routine use in Statistics (especially for Bayesian inference) did not take place until its popularisation by Gelfand and Smith (1990). For modern discussions of MCMC, see e.g. Tierney (1994), Smith and Roberts (1993), Gilks et al. (1996), and Roberts and Rosenthal (1998b). The number of financial applications of MCMC is rapidly growing (see for example the reviews of Kim et al., 1996 and Johannes and Polson, 2003). In this area, important problems revolve around the need to impute latent (or imperfectly observed) time-series such as stochastic volatility processes. Modern developments have often combined the use of MCMC methods with filtering or particle filtering methodology. In Actuarial Sciences, MCMC appears to have huge potential in hitherto intractabile inference problems, much of this untapped as yet (though see Scollnik, 2001, Ntzoufras and Dellaportas, 2002, and Bladt et al., 2003).