What it is

Metropolis-Hastings Algorithm is a general framework which includes as special cases the very first and simpler MCMC Markov chain Monte Carlo Algorithm.

How it works

Consider an arbitrary posterior $p(z)$ and its transition matrix $Q(z\to z\ast )$, it cannot always satisfy the detailed balance. If we want to achieve the detailed balance, we should add an acceptance ration $\alpha$, to make the equation correct:

in which one can derive $\alpha$:

This is how Metropolis Hastings work:

1. sample $z^{\ast }\sim Q(z|z_{i-1})$
2. sample $u\sim U(0,1)$
3. $\alpha =min(1,\frac{p(z^{\ast })Q(z|z^{\ast })}{p(z),Q(z^{\ast }|z)})$
4. decision:
   • if $u\le \alpha$, accept $z_i=z^{\ast }$
   • else, reject $z^{\ast }$ and accept $z_i=z_{i-1}$

Gibbs Sampling

• Gibbs sampling is a special type of Metropolis Hastings sampling.
• It is an approach to constructing a Markov chain where the probability of the next sample is calculated as the conditional probability given the prior sample.