Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 37 / 5 / d / 1 / Solution

The algorithm is a Random-scan Gibbs sampler. Start at a point where the target is positive, with a proper normalizing integral and well-defined full conditional distributions. Its state is the entire coordinate vector. Choosing the initial point fixes the initial law; it need not already be the invariant law. The transition rule below preserves the target irrespective of this choice.