First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. 



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> The description of my problem is the following:
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Sample a point $x$ uniformly at random from the solutions satisfying linear **equalities** $Ax = b$.


Specifically, I'm interested in sampling a $n$ x $m$ joint probability distribution of two discrete random variables that satisfies given marginal probabilities.
This is a special case of the more general described above.


Also, this problem is similar in sampling contingency tables with fixed margins, but it is not restricted to integral solutions. 



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> What I have found / tried so far:
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[Uniformly Sampling from Convex Polytopes](https://mathoverflow.net/questions/9854/uniformly-sampling-from-convex-polytopes) (not exactly what I want, as I'm only interested about on the boundaries and not in the polytope)


[Random Sampling a linearly constrained region in n-dimensions...](https://mathoverflow.net/questions/76255/random-sampling-a-linearly-constrained-region-in-n-dimensions) (This is closer to what I want, but can (?) only be applied to the case of $n$ x $2$ dimensional probability distributions)


<http://www.mzlabs.com/JMPubs/Sampling%20Contingency%20Tables-Dyer.pdf> (Paper about sampling contingency tables with fixed margins)


<http://www.mathworks.com/matlabcentral/fileexchange/34208-uniform-distribution-over-a-convex-polytope/content/cprnd.m> (Implementation of a method to sample a point that satisfies a set of linear **inequalities**. I tried enforcing equality constraints but the method does not work)



Sampling point uniformly at random satisfying equality constraints
The equality constraints determine an affine subspace of $\mathbb R^n$. After suitable change of variables, this becomes $\mathbb R^m$ for suitable dimension $m$, and the nonnegativity of the entries of your probability distribution gives you a convex polytope. So it is exactly the case of uniformly sampling from convex polytopes.


