Simulation and Markov Chain Monte Carlo

Abstract

Simulation is a computer-based exploratory exercise that aids in understanding how the behavior of a random or even a deterministic process changes in response to changes in input or the environment. It is essentially the only option left when exact mathematical calculations are impossible, or require an amount of effort that the user is not willing to invest. Even when the mathematical calculations are quite doable, a preliminary simulation can be very helpful in guiding the researcher to theorems that were not a priori obvious or conjectured, and also to identify the more productive corners of a particular problem. Although simulation in itself is a machine-based exercise, credible simulation must be based on appropriate theory. A simulation algorithm must be theoretically justified before we use it. This chapter gives a fairly broad introduction to the classic theory and techniques of probabilistic simulation, and also to some of the modern advents in simulation, particularly Markov chain Monte Carlo (MCMC) methods based on ergodic Markov chain theory.