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Short question

If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked out before? (Please include references.)

[Update: I am looking for definitions that are either (a) axiomatic or (b)a limit/Radon-Nykodym derivative/other construction; in both cases similar to the definition of the standard conditional expectation $\mathbf{E}[Y \mid X=x]$ for a random variable $X$. (Possibly one may also be able to use known facts about conditional expectation as well.) I am really hoping that the definition obeys

  1. $x \mapsto \mathbf{E}[Y \mid X\ni x]$ is $T_X$-capacitable, and
  2. $\int \mathbf{E}[Y\mid X\ni x]\, d\mathbf{T}_X(x)= \mathbf{E}[Y]$ (where the LHS is a Choquet integral).

I don't expect anyone to reinvent this, just to point me in the right direction in the literature.]

Long question

Let $X$ be a random closed set (that is $X$ is a random variable taking values in the space $\mathcal{F}(\mathbb{X})$ of closed sets in a (locally compact if needed) Polish space $\mathbb{X}$.). Let $\mathbf{P}_X$ be the distribution of $X$, that is the associated probability measure on $\mathcal{F}(\mathbb{X})$. Let $\mathbf{T}_X$ be the capacity of $X$, that is $\mathbf{T}_X(A)=\mathbf{P}(A \cap X \neq \varnothing)$ for $A \subseteq \mathbb{X}$. Let $\int f\, d\mathbf{T}_X$ denote the Choquet integral, that is $$\int f\, d\mathbf{T}_X = \int_0^\infty \mathbf{T}_X\{x : f(x) > t\}\, dt,$$ and in the case of random closed sets, we have $$\int f\, d\mathbf{T}_X = \int \sup_{x\in F} f(x)\,d\mathbf{P}_X(F).$$

Given a random variable $Y$, I would like to define $\mathbf{E}[Y \mid X \ni x]$. Clearly, when $\mathbb{X}$ is discrete this is easy to define. However, in general I suspect we need to think of the map $$x \mapsto \mathbf{E}[Y \mid X\ni x]$$ as a $\mathbf{T}_X$-capacitable map (defined quasi-everywhere), and that $$\int \mathbf{E}[Y\mid X\ni x]\, d\mathbf{T}_X(x)= \mathbf{E}[Y].$$

While I could take the time to work all this out, I imagine this has been studied and written about in papers and books (which I can't seem to find off hand).

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  • $\begingroup$ Try checking out Palm measure. I think this is exactly what you want. $\endgroup$
    – Anthony Quas
    Commented Mar 13, 2014 at 23:09
  • $\begingroup$ @AnthonyQuas, I will look more into Palm measures, but at a first glance, they don't seem to be what I want. They are for point processes/random measures. Whereas here I am working with random sets. I don't see the connection... $\endgroup$
    – Jason Rute
    Commented Mar 13, 2014 at 23:31
  • $\begingroup$ Nice question... We can ask question about the function $g(x)=E(Y\mid X=x)$, defined by $g(X)=E(Y\mid X)$, as long as they don't distinguish $g$ from another function $h$ where $g(x)=h(x)$ for almost all $x$. But in this case if $g(C)=E(Y\mid X=C)$ then for each particular $x$, $\{C\in x\}$ is a particular measure zero sets of $C$'s and therefore forbidden... $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 14, 2014 at 1:37
  • $\begingroup$ Perhaps "Graf, S. (1980). A Radon-Nikodym theorem for capacities. J. Reine Angew. Math. 320, 192–214." might be a good launching off point. $\endgroup$
    – Daniel Roy
    Commented Mar 14, 2014 at 9:19
  • $\begingroup$ Or "Hess, C. (1999). Conditional expectation and martingales of random sets. Pattern Recognition 32, 1543–1567." $\endgroup$
    – Daniel Roy
    Commented Mar 14, 2014 at 9:20

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