# Gibbs sampling

Gibbs sampling is a family of Markov Chain Monte Carlo algorithm where we conditionally sample from each parameter while holding the others fixed, successively. It is useful when the joint distribution is difficult to sample from (no closed-form expression, for instance) but where the conditional distribution of each variable is known.

## General principle

Assume that we want to generate \(N\) samples \(\left\{\tilde{\Theta}^{(n)}\right\}_{n=1 \dots N}\) from the joint distribution \(P\left(\theta_1, \dots, \theta_D\right)\). We start with an initial position \(\tilde{\Theta}^{(0)}\), and to generate \(\tilde{\Theta}}^{i+1}\) we successively draw from the following conditional distributions:

- \(\tilde{\theta}^{(i+1)}_1 \sim P(\theta_1\; |\; \theta_2 = \tilde{\theta_2}^{(i)}, \dots, \theta_D = \tilde{\theta_D}^{(i)})\)
- \(\tilde{\theta}^{(i+1)}_2 \sim P(\theta_2\; |\; \theta_1 = \tilde{\theta_1}^{(i+1)}, \theta_3 = \tilde{\theta_3}^{(i)} \dots, \theta_D = \tilde{\theta_D}^{(i)})\)
- \(\tilde{\theta}^{(i+1)}_j \sim P(\theta_j\; |\; \theta_1 = \tilde{\theta_1}^{(i+1)}, \dots, \theta_{j-1} = \tilde{\theta}_{j-1}^{(i+1)}, \theta_{j+1} = \tilde{\theta}_{j+1}^{(i)}, \dots, \theta_D = \tilde{\theta_D}^{(i)})\)

***Gibbs sampling is a particular case of MCMC where the proposal distribution is the same as the acceptance distribution so we always keep the sample with probability 1**

## Block Gibbs sampling

Accelerate.

## Using auxiliary variables

It is not uncommon to introduce auxiliary variables to the model to make computations easier. Let us consider the *augmented likelihood* \(P(y, \omega | \theta)\), so that the original likelihood:

\[ P(Y, \omega|\theta) = P(Y|\omega, \theta) P(\omega) \]

This works/is useful iff:

- Marginalizing the augmented likelihood returns the original likleihood \(\int P(Y, \omega | \theta) \mathrm{d} \omega = \int P(Y|\omega, \theta) P(\omega) \mathrm{d}\omega = P(Y|\theta)\)
- The prior \(P(\theta)\) is conjugate to \(P(Y|\omega, \theta)\) so \(P(\theta|Y) \propto P(Y | \theta) P(\theta) = \int P(Y|\omega, \theta) P(\omega) P(\theta) \mathrm{d}\omega\)

This trick is used to build Gibbs samplers for:

- The Bernoulli logit regression (Polya-Gamma augmentation)
- The Negative Binomial logit regression (Idem)
- The Horsehoe prior (using the inverse-gamma expansion of the Half-Cauchy distribution)