Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 2 / a / Solution

Given the current state $x=(x_1,\dots ,x_d)$, a Random-scan Gibbs sampler chooses $I$ uniformly from $\{1,\dots ,d\}$, leaves $x_{-I}$ unchanged, and samples the new coordinate from the complete conditional distributionIts transition kernel isand it leaves $\pi$ invariant.