L'échantillonnage de Gibbs est une chaîne de Markov Monte Carlo (MCMC) algorithm used for obtaining a sequence of samples from a distribution de probabilité conjointe when direct sampling is difficult. This technique is particularly useful in high-dimensional spaces where traditional sampling methods may fail.
The core idea behind Gibbs Sampling is to iteratively sample from the conditional distributions of each variable, given the current values of all the other variables. For instance, in a scenario with two variables, X and Y, the algorithm would first sample a value for X from its conditional distribution P(X|Y), then update Y by sampling from P(Y|X), and repeat this process. This results in a sequence of samples that converge to the target distribution conjointe.
One of the key advantages of Gibbs Sampling is its simplicity and ease of implementation, especially when the conditional distributions are easy to sample from. It is widely used in various fields such as Statistiques bayésiennes, machine learning, and image processing. However, Gibbs Sampling can have slow convergence rates, particularly if the variables are highly correlated.
En pratique, les premiers échantillons générés par l'échantillonnage de Gibbs sont souvent rejetés (un processus connu sous le nom de burn-in) pour permettre à la chaîne de converger vers la distribution cible. Les échantillons restants peuvent ensuite être utilisés pour estimer les propriétés de la distribution, telles que les moyennes, les variances et d'autres statistiques.