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

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.

Because $f$ is a continuous bijection $\mathbb{R}\to \mathbb{R}$, it is a Borel isomorphism. Conditional on $X_{-i}=x_{-i}$, the $i$th coordinate of $F(X)$ has the pushforward of $\pi (d{x}_i\mid x_{-i})$ under $f$. Therefore applying one $K$-update and then $F$ has exactly the same law as applying one $Q$-update to $F(x)$, using the same random coordinate.

Formally, for every Borel set $A$,Composition preserves this conjugacy, so induction on $t$ givesHence

Total variation is invariant under a measurable bijection with measurable inverse. Using part (b),Every $z\in {\mathbb{R}}^d$ equals $F(x)$ for a unique $x$, so