Gibbs Sampling Explained | MCMC & Bayesian Workflow

How can we generate samples from a complicated joint probability distribution when direct sampling is difficult? In this video, we explore Gibbs sampling, one of the most important Markov Chain Monte Carlo (MCMC) methods used in Bayesian statistics. Gibbs sampling works by repeatedly sampling each parameter from its full conditional distribution, while keeping the remaining parameters fixed. By cycling through these conditional distributions, we construct a Markov chain whose long-run distribution corresponds to the target joint distribution. Topics include: • What Gibbs sampling is • Why we need MCMC methods • Joint vs conditional distributions • Full conditional distributions • The basic Gibbs sampling algorithm • Iterative parameter updating • Markov chains and stationary distributions • Burn-in and retained samples • Convergence intuition • Using Gibbs samples to approximate posterior quantities • Why Gibbs sampling is useful in Bayesian inference • Advantages and limitations of Gibbs sampling The basic idea is: Sample θ₁ | θ₂ → Sample θ₂ | θ₁ → Repeat Rather than sampling directly from a complicated joint posterior distribution, Gibbs sampling breaks the problem into a sequence of simpler conditional sampling steps. After the Markov chain has converged, the generated draws can be used to estimate posterior means, probabilities, credible intervals, predictions, and other quantities of interest. Gibbs sampling is an important bridge between theoretical Bayesian inference and practical Bayesian computation, especially when posterior distributions cannot be handled analytically. This video is part of my Bayesian Statistics Explained series, where we build Bayesian concepts and computational methods step by step. Subscribe for clear and intuitive explanations of statistics, probability, Bayesian modelling, and data science. #GibbsSampling #MCMC #BayesianStatistics