I am trying to construct large joint distributions through smaller joint distributions and I'm not sure how to approach the literature.

I am curious if there exists a function which can take n subjoint distributions and approximate a larger joint distribution.

f(P(X,Y), P(X,Z)) = P(X,Y,Z) + e

I am also curious whether overlapping variables (variables in both sub joint distributions) lower the error in comparison to disjoint sub joint distributions.

P(X,Y,Z,A) - f(P(X,Y,A), P(X,Z)) < P(X,Y,Z,A) - f(P(X,Y), P(Z,A))?

I have a feeling that this has something to do with inclusion-exclusion and mutual information.

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    $\begingroup$ bivariate margins - even three of them - do not determine trivariate distributions; typically there would be an infinity of possible solutions. You would have to specify the manner in which you want to combine them. $\endgroup$ – Glen_b Mar 5 '19 at 23:48
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    $\begingroup$ For my use-case, its fine if they dont determine as long as I can bound the error. Could you elaborate more and/or point to some literature on this? Similarly, I'm fine exploring (n-k)variate margins (where n is the full join distribution and k < n) if that simplifies the problem space. $\endgroup$ – badbayesian Mar 6 '19 at 22:12
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    $\begingroup$ I can't tell from your question what topics you've already researched. Are you familiar with copulas, Sklar's Theorem, and the Fréchet–Hoeffding bounds and are seeking to generalize those? Are you familiar with graphical models and RBMs and seeking a more general model? You're likely to get more answers if you can supply more context. $\endgroup$ – olooney Mar 18 '19 at 13:57

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