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I have a graphical model with binary variables Y, X1, X2, and observed data D.

D depends on Y X1 and X2 depend on Y

When Y is false, X1 and X2 are independent:

(D) <-- (Y) --> (X1)
         |
         V
        (X2)

However, when Y is true, X1 and X2 are not independent and must be modeled in a joint manner:

(D) <-- (Y) --> (X1,X2)

I've tried putting in an "edge" variable E between X1 and X2 that is influenced by Y; I can get the correct behavior when Y is false, but not when Y is true:

(D) <-- (Y) --> (X1)
         |       ^
         V       E
       (X2) <- /

(that is Y --> E and E --> X1 and E --> X2)

If I reverse the directed edges so that they point from X1 and X2 to E, then the moral graph once again always treats X1 and X2 as joint.

So I can't seem to encode state-dependent independence...

Is there established theory about such networks (if so what is this relationship called)? Is there a way to encode or transform the graph with temporary variables or something so that, if you know given D, Y must be false, that you can use the first graph (where X1 and X2 are independent given Y), and have faster inference?

Of course, I should point out that this is a toy example; the real one is huge.

Thank you very much for your help. And sorry for the ascii graphs, I was raised by wolves.

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You cannot express state-dependent independencies with standard directed (Bayesian networks) or undirected graphical models (Markov networks). But you can use an extension called gates which are described

      T. Minka & J. Winn, Gates: A graphical notation for mixture models, 2008

and were designed specifically for this purpose.

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  • $\begingroup$ This is exactly what I was looking for. "Thank you, kind raccoon!" $\endgroup$ – user Aug 9 '13 at 11:14

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