Notation question for probability measures on product spaces

[economicsnerd]

I asked this in the logic&probability subforum, but I thought I'd try my luck here.

...

Let $(A,\mathcal A), (B,\mathcal B)$ be measurable spaces. Let $p$ be a probability measure on $(A,\mathcal A)$, and let $q:A\to\mathcal P(B,\mathcal B)$ be a measurable function which takes each $a\in A$ to some probability measure $q(\cdot|a)$ on $(B,\mathcal B).$ Then there is a unique probability measure $\mu$ on $(A\times B, \mathcal A\otimes\mathcal B)$ which has $$\mu(\hat A\times \hat B) = \int_{\hat A} q(\hat B|\cdot)\text{ d}p$$ for every $\hat A\in\mathcal A, \hat B\in\mathcal B.$

The question: Is there a typical thing to call $\mu$? Does it have a name, in terms of $p$ and $q$? How about notation? $pq$? $p\otimes q$ (which would be misleading)? $q\circ p$? $q^p$? I looked around and couldn't find anything consistent.

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[economicsnerd]

1.5 years later, I'm still curious about this!

 

[S.G. Janssens]

Let me start by saying that I do not know the answer to your question, I have not seen this construction myself. (Not surprising, as this is not really my field.) However, it made me curious. Perhaps I can ask you some questions in return?

1. At first you demand that $q$ should be measurable as a map to $\mathcal{P}(B,\mathcal{B})$. I assume you put the Borel $\sigma$-algebra on $\mathcal{P}(B,\mathcal{B})$, but assuming which topology? Do we use total variation or weak$^{\ast}$ convergence? Is it even necessary to demand that $q$ be measurable, since in the integral appears only $q(\hat{B}|\cdot)$? Isn't it enough to ask that the maps
$$
q(\hat{B}|\cdot) : A \to [0,1] \qquad (*)
$$
are measurable for every $\hat{B} \in \mathcal{B}$?

2. Suppose $q$ is continuous when $\mathcal{P}(B,\mathcal{B})$ is equipped with the topology of weak$^{\ast}$ convergence, which seems natural to me. Then I believe that it is generally not true that the map in (*) is continuous, as I seem to remember that weak$^{\ast}$ convergence does not, in general, imply setwise convergence. Probably (*) can still be shown to be measurable, but isn't this phenomenon annoying?

3. You might find more information about notation and terminology in the beautiful book by Bogachev on measure theory. If not, I propose $p \odot q$.

4. Is it possible to explain the origin of this construction to someone not very familiar with probability nor economics?

 

[economicsnerd]

Hi Krylov,

1) As it turns out, what you've provided is the definition I'm aware of for measurability of a map whose codomain is the space of probability measures on $(B, \mathcal{B})$. In fact, I'm used to seeing the "standard" $\sigma$-algebra on $\mathcal P(B, \mathcal{B})$ as the one generated by your maps.
[In the case where the latter is a (say) compact metrizable space equipped with its Borel $\sigma$-algebra, I don't remember whether the two definitions are equivalent, but I think they might be. I'm a bit rusty on this.]

2) You're correct here. I haven't found it annoying yet, but there are always reasons to want continuity when one doesn't have it.

3) I'll check it out! If I don't find something, though, I do like your notation.

4) FYI...
The setting I'm looking at is one in which a decision maker cares about some state of the world $a\in A$ distributed according to $p$, where $(A, \mathcal{A}, p)$ is some probability space. Our decision maker isn't going to know $a$, but he'll have some partial information about it. The way we model this is by saying he'll hear a message $b\in B$, where $(B, \mathcal{B})$ is another space (think of $B$ as a language). In each state $a\in A$, he hears a message $b\in B$ distributed according to $q(\cdot| a) \in \mathcal P(B,\mathcal{B})$. This is, in some sense, a fully specified model.
A useful piece of language for the modeler to have here is the decision maker's "beliefs" about $a$ given a message $b$. The most convenient way to do this is to have some underlying probability space $(\Omega, \Sigma, r)$ on which $a, b$ can be viewed as random variables and then look at random variables of the form $\mathbb E [\mathbf1_{a\in \hat A} | b ]$ for $\hat A \in \mathcal A$. As it turns out, even though I don't have a name for it, $(A\times B, \mathcal A \otimes \mathcal B, p \odot q)$ is a particularly convenient such probability space.

 

[WWGD]

I am not sure I understood well, but maybe $\mu$ may be (related to) some sort of modal operator? Or an operator in some type of non-traditional logic ?