A Gentle Introduction to Markov Chain Monte Carlo (MCMC)

The Clever Machine

Applying probabilistic models to data usually involves integrating a complex, multi-dimensional probability distribution. For example, calculating the expectation/mean of a model distribution involves such an integration. Many (most) times, these integrals are not calculable due to the high dimensionality of the distribution or because there is no closed-form expression for the integral available using calculus. Markov Chain Monte Carlo (MCMC) is a method that allows one to approximate complex integrals using stochastic sampling routines. As MCMC’s name indicates, the method is composed of two components, the Markov chain and Monte Carlo integration.

Monte Carlo integration is a powerful technique that exploits stochastic sampling of the distribution in question in order to approximate the difficult integration. However, in order to use Monte Carlo integration it is necessary to be able to sample from the probability distribution in question, which may be difficult or impossible to do directly. This…

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