Question

When implementing Gibbs sampling for a multivariate distribution, what must be true about the full conditional distributions used in each step for the chain to converge to the joint posterior?

Accepted Answer

For a Gibbs sampler to converge to the correct joint posterior distribution, the full conditional distributions must be mathematically derived from the joint distribution itself. A full conditional distribution is the probability distribution of a single variable conditioned on all other variables currently in the model. Specifically, if you have a joint distribution for variables X and Y, the full conditional for X is the joint distribution evaluated while treating Y as a fixed constant. The Markov chain generated by Gibbs sampling converges to the target joint distribution because the sampling process satisfies the condition of detailed balance, which ensures the chain is reversible and has the joint distribution as its stationary distribution. Each step in the process must update variables such that the new state is sampled directly from these exact full conditional distributions. If the full conditionals are incorrect or do not represent the exact conditional structure of the joint distribution, the chain will converge to an unknown or incorrect distribution rather than the desired joint posterior.

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When implementing Gibbs sampling for a multivariate distribution, what must be true about the full conditional distributions used in each step for the chain to converge to the joint posterior?