What's the difference between MCMC and particle MCMC?

Question

Markov chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its equilibrium distribution.

So how does the 'particle' bit augment the 'MCMC' bit?

Answer 1

Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters.

Answer 2

In order to perform the usual Metropolis-Hastings step in a MCMC algorithm, one needs to know the closed form of the likelihood function for the observations. When this function is not available (most non-linear and non-Gaussian hidden Markov models), one can substitute it by an unbiased estimator. This unbiased estimator can be obtained through one iteration of a particle filter over the latent process. Basically:

1 - run a particle filter over the latent states to build an unbiased estimate of the likelihood;

2 - perform a Metropolis-Hastings step over the parameters;

3 - iterate until some convergence criterion is satisfied.