Abstract
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 Metropolis–Hastings 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.
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