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In Graphical Models and Bayesian Networks, how do you implement XOR problem?

I read bayesian network vs bayes classifier here:

A Naive Bayes classifier is a simple model that describes particular class of Bayesian network - where all of the features are class-conditionally independent. Because of this, there are certain problems that Naive Bayes cannot solve (example below). However, its simplicity also makes it easier to apply, and it requires less data to get a good result in many cases.

Example: XOR You have a learning problem with binary features x_1, x_2 and a target variable y = x_1 XOR x_2.

In a Naive Bayes classifier, x_1 and x_2 must be treated independently - so you would compute things like "The probability that y = 1 given that x_1 = 1" - hopefully you can see that this isn't helpful, because x_1 = 1 doesn't make y = 1 any more or less likely. Since a Bayesian network does not assume independence, it would be able to solve such a problem.

I googled, but could not figure out how. Can someone give me a hint or good references? Thanks!

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1 Answers1

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This is actually fairly simple.

The DAG of the model would look like

 x1 -> XOR <- x2

The probability distribution for the XOR node can then be written

x1  x2  | P(XOR=1|x1,x2)
0   0   |    0
0   1   |    1
1   0   |    1
1   1   |    0
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