Random-scan Gibbs sampler

A random-scan Gibbs sampler chooses one coordinate, commonly uniformly at random, and redraws it from its complete conditional distribution while leaving all other coordinates fixed. Each coordinate update preserves the target distribution.

A single-coordinate Gibbs sampler update preserves the remaining coordinates and independently redraws the selected coordinate from its full conditional distribution. The joint measure of old and new states factors into a marginal and two identical conditional factors, making it symmetric. A fixed-probability mixture of such kernels therefore satisfies detailed balance.