GibbsSampling

Gibbssampling, more commonly known as Gibbs sampling, is a Markov chain Monte Carlo method for generating samples from a multivariate probability distribution when direct sampling is difficult. The algorithm iteratively samples each variable from its conditional distribution given the current values of all other variables. The method has strong roots in the 1980s and is widely used in Bayesian computation because the full conditional distributions are often easier to sample from than the joint distribution.

Algorithmically, suppose y = (y1, ..., yN) has target distribution p(y). Initialize y(0). For each iteration t = 1, 2, ..., sequentially update each component yi by drawing from p(yi | y1(t), ..., y_{i-1}(t), y_{i+1}(t-1), ..., yN(t-1)) (i.e., the most recent values for the other components). After a burn-in period, the collected samples approximate the target distribution p(y). The approach relies on the existence and tractability of the full conditional distributions.

Convergence and diagnostics: Under mild regularity conditions, the Gibbs chain has p(y) as its stationary distribution. Convergence is assessed with diagnostics such as trace plots, autocorrelation, and multiple-chain comparisons. Burn-in and thinning are commonly used to reduce autocorrelation and improve approximation quality.

Variants and extensions: Blocked Gibbs sampling updates groups of variables together to improve mixing. Collapsed Gibbs sampling integrates out some parameters to sample latent variables more efficiently. Metropolis-within-Gibbs replaces difficult conditionals with Metropolis steps. Gibbs sampling is extensively used in latent variable models, Bayesian networks, and hierarchical models, including collapsed Gibbs for topic models like latent Dirichlet allocation.

Limitations: Mixing can be slow for highly correlated variables or complex posteriors. The method requires tractable full conditionals or workable approximations, and performance depends on initialization and model structure.