# Gibbs Sampling

Gibbs sampling is a special case of the Metropolis-Hastings method, where a sequence of proposal distributions $$q$$ is defined in terms of the conditional distributions of the joint distribution $$p(\mathbf{x})$$, and proposals are always accepted.

In the general case of a system with $$K$$ variables, a single iteration involves sampling one parameter at a time:

$$\begin{array}{l}{x_{1}^{(t+1)} \sim P\left(x_{1} | x_{2}^{(t)}, x_{3}^{(t)}, \ldots, x_{K}^{(t)}\right)} \ {x_{2}^{(t+1)} \sim P\left(x_{2} | x_{1}^{(t+1)}, x_{3}^{(t)}, \ldots, x_{K}^{(t)}\right)} \ {x_{3}^{(t+1)} \sim P\left(x_{3} | x_{1}^{(t+1)}, x_{2}^{(t+1)}, \ldots, x_{K}^{(t)}\right), \text { etc. }}\end{array}$$

## Pros and Cons

1. Suffers the same defects as Metropolis-Hastings methods