Gibbs Sampling: A Powerful Markov Chain Monte Carlo Technique

Introduction

Gibbs sampling, also known as the Gibbs sampler, is a powerful Markov Chain Monte Carlo (MCMC) algorithm for sampling from specified multivariate probability distributions. It is widely used in statistics and machine learning for various applications.

How Gibbs Sampling Works

Suppose we have a random variable of interest that can be expressed as components x = (x1, x2, ..., xd). We aim to simulate the distribution of x from its multivariate probability distribution. Gibbs sampling involves generating a sequence of pairs of random variables (X0, Y0), (X1, Y1), ..., (Xn, Yn), where: - X0 = x - Each Yi is generated from the conditional distribution of Yi given Xi - Each Xi+1 is generated from the conditional distribution of Xi+1 given Yi, X0, X1, ..., Xi