Gibbs sampling

Remarks

For clause 1, if $y_{-i} = x_{-i}$ and $Z_i(x) > 0$, then $Z_i(y) = Z_i(x)$ and both sides equal $\pi(x) \pi(y) / Z_i(x)$. If $Z_i(x) = 0$ then $\pi(x) = \pi(y) = 0$ for all such $y$, and in all other cases both sides are $0$. Summing detailed balance over $x$ gives $\pi K_i = \pi$, and clause 2 follows because a product or an average of kernels that fix $\pi$ also fixes $\pi$. The same argument works for densities on $\R^d$, with sums replaced by integrals. Stationarity alone does not give convergence of the chain to $\pi$. If $S$ is finite and $\pi(x) > 0$ for every $x$, every entry of $K_1 \cdots K_d$ is positive, so the chain is irreducible and aperiodic and converges to $\pi$ by the convergence theorem. Without positivity this can fail: for $\pi$ uniform on $\set{(0,0), (1,1)} \subset \set{0,1}^2$, each $K_i$ fixes every state of positive mass, so the chain never moves.

Going deeper

For a student

Gibbs sampling is the Markov chain Monte Carlo method of choice when the full conditional distributions are easy to sample. This is the case in many Bayesian hierarchical models with conjugate priors and in lattice models such as the Ising model. Each update is a Metropolis–Hastings step whose proposal is the full conditional, so its acceptance probability is always $1$. The systematic-scan sampler is in general not reversible, though each $K_i$ is; the random-scan sampler is reversible. Strong dependence between coordinates slows mixing, since moves along single coordinates then make little progress; blocking several coordinates together is the usual remedy.