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(I believe this is meta-question...)

I find myself able to pose various questions related to "constrained random variables," but I am unsure what techniques might be useful to go about answering them. I'm looking for suggestions of specific techniques in analysis or other branches of math that might assist in answering the following sorts of questions:

  • Let $X$ and $Y$ be two real valued random variables with a joint distribution $g(x,y)$. Suppose the marginals of $X$ and $Y$ are both gaussian. Also let $c(x,y)$ be a function from $\mathbb{R}^2 \rightarrow \mathbb{R}$ such that $$ Pr[c(X,Y) \leq 0] \leq p $$ What are the necessary conditions on $c(x,y)$ if $g(x,y)$ is known to be multivariate normal (i.e. I am looking to the characterize the class of constraint functions that still allow the possibility of $g(x,y)$ to be multivariate normal)?

  • Maximum Entropy Sampling Let $X_1, \ldots, X_n$ be real valued random variables with given marginal distributions $m_1(x), \dots, m_n(x)$. Furthermore it is also given that the random variables jointly satisfy the following (fully-specified) contraints: $$ \begin{array}{rcl} Pr[c_1(X_1, \ldots, X_n) \leq 0] & \leq & p_1 \\ & \vdots & \\ Pr[c_k(X_1, \ldots, X_n) \leq 0] & \leq & p_k\end{array} $$ Let $\mathcal{G}$ be the set of joint distributions $g(x_1, \ldots, x_n)$ which satisfy the given constraints and yield the given marginals. Clearly, $\mathcal{G}$ is potentially infinite. However, let $g_{maxent}(x_1, \ldots, x_n)$ be the element of $\mathcal{G}$ which has maximum entropy (let's assume it possible to work out reasonable conditions on $X_i$, $m_i$, $c_i$, etc to ensure that $g_{maxent}$ is unique). How would one go about drawing random samples from $g_{maxent}$? In particular, is there some sort of algorithm that can implicitly draw samples without having to actually calculate $g_{maxent}$. For instance, what if we sample the mariginals, and then for each contraint, we flip a (biased) coin to see if it is "active", and accept the sample if it satisfies all "active" constraints -- will this lead to samples from $g_{maxent}$? The proceeding algorithm leads to potentially a large number of rejected samples if the contraints are tight -- is there a way to design more efficient algorithms?

Hopefully these two examples give some gist of the sorts of problems I'm interested in, but I can add a few more if these are either unclear or not compelling...

So my specific questions are:

  • Is there a specific branch of probability that deals with these sorts of issues? If so, what is it called (I've tried googling and occasionally come up with very specific references to narrow issues surrounding constrained random variables, but I've yet to find anything with a broad scope)? In particular, are there any texts that discuss this?

  • If I were interested in trying to answer the questions on my own, what sorts of techniques from real analysis, functional analysis, etc might be helpful? (I'm hoping for answers of the following sort "You should have a strong grasp of Theorem X, Y and Z" or "You should understand Functional Analysis at the level of such and such textbook") Also, if these sorts of problems have been posed forever and are completely intractable that's good to know as well.

Also, while these questions come up from actual modeling scenarios, I'm also just curious and looking for an excuse to delve deeper into Analysis in a somewhat applied context -- I find it difficult to maintain perspective and cultivate intuition when I slog through a subject like functional analysis in the abstract, but I'm hoping that if I focus on the specific task of constrained random variables, I'll be able to stay motivated and also be able to better retain what I learn.


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migrated from Aug 10 '12 at 17:34

This question came from our discussion, support, and feature requests site for people studying math at any level and professionals in related fields.

meta means about the site, not about mathematics from a meta perspective. – joriki Aug 10 '12 at 17:46

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