Joint distribution with specified marginals

Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal distributions $r(x), r(y)>0,$ for each $x,y$ respectively. Can we always find functions $f(x), g(y)$ such that

$\sum_y p(x,y)f(x)g(y) = r(x)$

$\sum_x p(x,y)f(x)g(y) = r(y)$?

It appears that we should always be able to do this, but I would like an explicit expression for a solution $f,g$ in terms of $r(x), r(y).$ Thanks.

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2 Answers

This is called the generalized matrix scaling problem and several other names. Both the theory and associated algorithmic problems have been studied. I suggest you start with this paper and the papers it cites.

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There are some notational issues. The two different marginals and the joint distribution for X and Y need to be distinguished by different choice of symbols. My suggestions are: $p(x,y), p_X(x), p_Y(y)$, and similarly $r_X(x), r_Y(y)$. Now your question is existence of $f(x), g(y)$ such that $\sum_y p(x,y) p_X(x) g(y) = r_X(x),$ and similarly the other one. As the LHS is a summation over $y$, it must be a function purely one $x$.

Now using your hypothesis that you are in finite discrete space the existence of $f(x), g(y)$ can be interpreted as a question of existence of solutions to two linear system of equations.

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The question appears to generalize the problem of scaling the rows and columns of a nonnegative matrix in order to make the matrix doubly-stochastic. That can always be done under some weak conditions, but it certainly cannot be done by solving linear equations. It is much harder than that. –  Brendan McKay Jul 27 '13 at 1:00
Looks like I have not given the problem much thought. –  P Vanchinathan Jul 28 '13 at 5:07