Metropolis-Hastings Reversibility: What is the measure used in the definition of reversibility?

Let $$\pi$$ be a target probability distribution on a measurable space $$(E, \mathcal{E})$$. MCMC obtains dependent samples from $$\pi$$ by using a Markov Chain with transition kernel $$\mathrm{K}:E\times \mathcal{E}\to[0,1]$$ that leaves $$\pi$$ invariant $$\pi = \pi \mathrm{K}$$ In Metropolis-Hastings we check a weaker condition called detailed balance (which implies $$\pi$$-invariance). Detailed balance is a notion of reversibility and it is often defined by saying that the measures $$\pi(d x) \mathrm{K}(x, dx') = \pi(dx')\mathrm{K}(x', dx)$$ are equivalent, which people also write as $$\int_A \pi(dx)\mathrm{K}(x, B) = \int_B \pi(dx)K(x, A)$$

In measure-theoretic terms, how can one define detailed balance?

My guess is that we define a product measure $$\mu$$ on $$(E\times E, \mathcal{E}\otimes \mathcal{E})$$ that is defined as $$\mu(A\times B)= \pi(A) \mathrm{K}(x, B) \qquad A, B\in\mathcal{E} \qquad x\in A$$ and then the statement above essentially means this? $$\mu(A\times B) = \mu(B\times A)$$ I guess this would make sense because we could write $$\int_A \pi(dx) \mathrm{K}(x, B) = \int_{A} \pi(dx) \int_B \mathrm{K}(x, dy) = \int_{A\times B} \pi(dx) \mathrm{K}(x, dy) = \int_{A\times B} \mu(dx, dy) = \mu(A\times B)$$

• I guess my key question is: is the measure they are talking about a product measure? Jun 29 at 13:34
• Yes the measure as defined is symmetric. Jun 29 at 14:04
• @Xi'an Thank you! Is $\mu$ defined correctly here for Metropolis-Hastings then? Jun 29 at 14:09
• Yes, I think so! Jun 29 at 16:05
• @Xi'an thank you! Jun 29 at 16:49