Trying to find a solution to the maximum-entropy probability distribution Q(x,y,z) constrained to reproduce the marginal distributions P(x,y), P(y,z), and P(z,x) from some other distribution P(x,y,z).<p>It is known that Q takes the form Q=a(x,y)<i>b(y,z)</i>c(z,x) for some functions a,b,c to be determined by solving the system of equations:<p>P(x,y) = sum_z Q(x,y,z)<p>P(y,z) = sum_x Q(x,y,z)<p>P(z,x) = sum_y Q(x,y,z)<p>It's not clear there exists a general closed-form solution. Iterative algorithms are known. This type of problem comes up in a number of interesting contexts. For instance, testing for non-trivial multi-variable interactions in dynamical systems such as neural networks or spin networks, performing joins on probabilistic databases, constructing reduced models of probability distributions, and in some cooperative game theory problems.<p>Examples:
<a href="https://www.princeton.edu/~wbialek/our_papers/schneidman+al_03b.pdf" rel="nofollow">https://www.princeton.edu/~wbialek/our_papers/schneidman+al_...</a><p><a href="http://vldb.org/conf/1987/P071.PDF" rel="nofollow">http://vldb.org/conf/1987/P071.PDF</a><p><a href="https://doi.org/10.6028/jres.072b.019" rel="nofollow">https://doi.org/10.6028/jres.072b.019</a><p><a href="https://www.mdpi.com/1099-4300/16/4/2161" rel="nofollow">https://www.mdpi.com/1099-4300/16/4/2161</a>
Edit: formatting