A Gibbs sampler constructs a Markov chain by repeatedly drawing coordinates from their full conditional distributions given the current values of the other coordinates. For a two-coordinate target density , a sweep draws and then . Each update preserves the target joint law: retaining one marginal and resampling its exact conditional reconstructs the same joint density. Thus a sweep has the target as a stationary distribution. Convergence from arbitrary starts additionally needs suitable irreducibility and other conditions; disconnected support can trap a chain. Successive output samples are usually dependent.
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