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In the MC and MCMC literature one commonly finds statements of the following form (see e.g. Roberts & Rosenthal, 2004):

$$ \int_{x \in \mathcal X} \pi(dx)P(x, dy) = \pi(dy). $$

What is the motivation for this type of notation?

In particular I find the following points interesting:

  • Why does the measure $\pi$ appear to the left of the kernel $P(x, dy)$? That is, why not follow the convention that the integrating measure appears to the right, viz. $\int f(x) \mu(dx)$?
  • What's the precise meaning of the statement? In the text referenced, the statement is proceeded by the clause that it holds for all $x,y\in\mathcal X$. I can't make sense of this. The statement is not a function of $x$ as I read it, and I don't see how the $y$ in $dy$ matters. I'm not sure I even understand what $dy$ even means in this setting.

My interpretation of the statement is the following:

$$ \int_\mathcal{X}P(x, A)\pi(dx) = \pi(A), $$

for every $A$ in the relevant $\sigma-$field on $\mathcal X$.

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The order doesn't matter, because scalar multiplication is commutative. People just write it left to right because usually $x$ happens before $y$ in time. They're not just breaking from convention, though. You can integrate over $x$, OR over $y$, or both. They will all mean something different.

In different notation this is basically saying something like $f(y) = \int f(y|x)f(x)dx$, although this is less general because it assumes the existence of densities.

For more information on kernels, see https://en.wikipedia.org/wiki/Markov_kernel

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  • $\begingroup$ (+1) Thanks for your interest. I don't quite understand what you're saying. The link you sent follows the convention that I'm used to from measure theory, namely that when integrating a kernel w.r.t a measure the measure appears to the right. In the MCMC literature the measure always acts from the left on the kernel. Same thing in your density example, you don't write $\int dx f(y\mid x)f(x)$. $\endgroup$
    – KOE
    Jan 28, 2016 at 15:07
  • $\begingroup$ You're looking at the semidirect product section on the wiki link? Yeah it's the same thing, you can change the order. The image of these functions is in the reals, so they commute. It's just convention. And actually yeah I have seen $\int dx ...$ a few times before hah $\endgroup$
    – Taylor
    Jan 28, 2016 at 15:10
  • $\begingroup$ Sure, you can do anything as long as it's clear what it means mathematically, it's just notation. I think there may be a good reason though why in the MCMC literature they almost always use this notation. Could you expand your point about $x$ happens before $y$ to clarify this? $\endgroup$
    – KOE
    Jan 28, 2016 at 15:18
  • $\begingroup$ Well, the first MC in MCMC stands for markov chain. Kernels take old random variables ($x$), and then give you probabilities for your future random variable ($y$) via a kernel. Idk about you but I generally think of time going from left to right $\endgroup$
    – Taylor
    Jan 28, 2016 at 15:39

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