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Let $T$ be a binary tree, where the internal nodes are $Y_1,...,Y_n$ and the leafs are $X_1,...,X_m$. (The $Y$s and the $X$s will eventuall represent a random variable in a Bayesian network.)

We can turn this binary tree into a polytree by directing all the edges downward toward the leafs from the root. What we get is a graphical model or a Bayesian network.

My main question - what other polytrees over the $Y$s and the $X$s can we have such that the marginal distribution $p(X_1,...,X_m)$ is the same like the marginal distribution in the original polytree I just described?

I am guessing we may be able to redirect some edges such that they point to a different node, and still get the same independence assumptions (and hence the same marginal distribution), but I am not sure what would be the permitted inversions.

Thanks!

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You can pick any other node as root and redirect nodes away from it to get Markov equivalent model. Markov equivalent means that two graphical models correspond to identical spaces of distributions.

Let "unshielded collider" be structure of the form A->B<-C

Theorem: Two directed acyclic graphs D1 and D2 are Markov equivalent if and only if D1 and D2 have the same vertex set, same adjacencies, and same unshielded colliders.

See Ch.4 of Meek's "Related Graphical Frameworks: Undirected, Directed Acyclic and Chain Graph Models" http://repository.cmu.edu/cgi/viewcontent.cgi?article=1206&context=philosophy

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  • $\begingroup$ interesting, that helps. $\endgroup$ – gmmodeler Sep 15 '11 at 10:29

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