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I reviewing some AI books and recently i found the following question that i would like to solve:

In the given Bayes network, decide the conditional independence of the nodes.

Graph is here

Justify and explain your decision of conditional independence (or dependence) for:

Question from 1 to 5 here

based on what i have learned, i think A and B should be indep because they have different incoming arcs. but for question number 3, i think the total answer should be dependent since if we know the E then we can know the A and B.

I will be more than happy if any one can help.

Thanks in advance.

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  • $\begingroup$ What have you tried? Please show your efforts so we might know where the difficulty lies. $\endgroup$ – Graham Kemp Dec 12 '17 at 12:25
  • $\begingroup$ based on what i have learned, i think A and B should be indep because they have different incoming arcs. but for question number 3, i think the total answer should be dependent since if we know the E then we can know the A and B. @Graham Kemp Please correct me if i am wrong and i don't know how to answer the reset of them. Thanks $\endgroup$ – Bilgin Dec 12 '17 at 12:46
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The graph is directed and acyclic, and so is a Bayes network. The way to check for conditional independence in a Bayes network is to moralize the graph. If you have a moral graph and two nodes are not connected then they are conditional independent given the connecting nodes. This is because the adjacent nodes in a moral graph are it's Markov Blanket

Additional Stuff:

One thing to note here is the concept of Markov degeneracy. That is, there are multiple undirected graphs that give the same moral graph. You can see something about this here. This comes up not so much if the graph encodes a model,and is specified a priori, but if you are trying to do inference on learning directed graphs. Some applications are, for example, learning gene pathways.

The common approach to do that is to either learn many different graphs and combine them somehow, or put an ordering such that only acyclic graphs following that ordering can be learned. This is a recent paper on developing a new Bayesian method for graph inference, with the first and second section having a pretty comprehensive description of this problem.

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