Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 3 / a / Solution

PutThe normalizing identityshows thatis a probability measure. If $x$ and $y$ differ only in coordinate $i$, the transition density from $x$ to $y$ is $p(x,i)\mu (y_i\mid x_{-i})/S(x)$. Thereforewhich is symmetric in $x_i,y_i$. Thus detailed balance holds and the Tempered Gibbs sampler is $\pi$-reversible.