Metropolis–Hastings kernel

Assume AC. The Metropolis–Hastings procedure takes as input a standard-Borel space $(\mathsf{X},\mathcal{X})$ (Standard Borel spaces), a target probability measure $\pi$ (Probability measures and probability spaces), and a probability kernel $Q$ from $(\mathsf{X},\mathcal{X})$ to itself (Measure kernel and probability kernel).

1. On the product sigma-algebra (The product sigma-algebra and its finite iterates), form the proposal-flow measure $\mu$ (Measures on sigma-algebras) on $\mathsf{X}\times \mathsf{X}$ by $\mu (C)={\int }_{\mathsf{X}}Q(x,C_x)\pi (dx),C_x=\{y:(x,y)\in C\}\mathrm{.}$ This is a probability measure by A prior and model kernel define a joint law. Let $\tau (x,y)=(y,x)$ and set ${\mu }^{\mathsf{T}}(C)=\mu ({\tau }^{-1}C)$.

2. Set $\eta =\mu \wedge {\mu }^{\mathsf{T}}$, the greatest measure dominated by both flows. Put ${\lambda }_0=\mu +{\mu }^{\mathsf{T}}$, which is a finite measure by Nonnegative scalar multiples and countable weighted sums of measures are measures, and choose nonnegative measurable representatives $f=d\mu \mathrm{/}d{\lambda }_0$ and $g=d{\mu }^{\mathsf{T}}\mathrm{/}d{\lambda }_0$ with $f+g=1$ everywhere after modification on a ${\lambda }_0$-null set. Define $\alpha (x,y)=\{\begin{array}{ll}\min \{1,g(x,y)\mathrm{/}f(x,y)\},& f(x,y)>0,\\ 1,& f(x,y)=0.\end{array}$ This is a jointly measurable version of $d\eta \mathrm{/}d\mu$ taking values in $[0,1]$.

3. For $x\in \mathsf{X}$, compute the integrated acceptance probability $a(x)={\int }_{\mathsf{X}}\alpha (x,y)Q(x,dy)$ and rejection probability $r(x)=1-a(x)$.

4. Return the transition rule $P(x,A)={\int }_{\mathsf{X}}{\mathbf{1}}_A(y)\alpha (x,y)Q(x,dy)+{\mathbf{1}}_A(x)r(x),A\in \mathcal{X}\mathrm{.}$ The second term retains every rejected proposal as a self-transition.

The item The common submeasure and its bounded density proves the common-submeasure construction and density-version claims in step 2.

An RN derivative is an almost-everywhere equivalence class, not a distinguished pointwise function (The Radon-Nikodym derivative as an almost-everywhere equivalence class). The explicit convention in step 2 selects a measurable version. Changing that version on a $\mu$-null set leaves the accepted joint flow unchanged, though it can change $P(x,\cdot )$ for a $\pi$-null set of starting states. The theorem Metropolis–Hastings flow is reversible and invariant ↗ proves that $P$ is a probability kernel and that $\pi$ is invariant; the procedure alone does not assert irreducibility, uniqueness, convergence, a rate, or a CLT.