Monte-Carlo – Applied Probability Notes

Sequential Monte Carlo (SMC)

Sequential Monte-Carlo is a general method of sampling from a sequence of probability distributions $\hat \eta_1,...,\hat \eta_t$ .

Multi-Level Monte Carlo (MLMC)

Multi-Level Monte-Carlo is an Monte-carlo method for calculating numerically accurate estimates when fine grained estimates are expensive, but cheap coarse-grained estimates can be used to supplement this. We considered the simulation of stochastic differential equations, which is the application first proposed, but we note that the approach applies in a variety of other settings.

Markov Chain Monte Carlo (MCMC)

Markov chain Monte Carlo is a variant of the Monte Carlo, where samples are no longer independent but instead are sampled from a Markov chain. This can be useful in Bayesian statistics, or when we sequentially adjust a small number of parameters for a more complex combined distribution.

We cover MCMC, its use in Bayesian statistics, Metropolis-Hastings, Gibbs Sampling, and Simulated Annealing.

Monte-Carlo (MC)

Due to some projects, I’ve decided to start a set of posts on Monte-Carlo and its variants. These include Monte-Carlo (MC), Markov chain Monte-Carlo (MCMC), Sequential Monte-Carlo (SMC) and Multi-Level Monte-Carlo (MLMC). I’ll probably expand these posts further at a later point.

Here we cover “vanilla” Monte-Carlo, importance sampling and self-normalized importance sampling: