Monte Carlo methods broadly describe techniques that use random sampling to solve problems. These techniques are especially powerful when the underlying problem involves some kind of uncertainty. The general method involves performing large numbers of simulations, each sampling different inputs according to a given probability distribution, and then aggregating the results to give a better approximation of the true solution than any individual sample solution.
Markov Chain Monte Carlo (MCMC) is a specific kind of Monte Carlo simulation in which we construct a Markov chain of successively better approximations of the true distribution that we seek. This works by accepting or rejecting a proposed state, sampled at random, based on carefully selected acceptance probabilities at each stage, with the aim of constructing a Markov chain whose unique stationary distribution is precisely the unknown distribution that we wish to...
