Let $E$ be a bayesian network with 3 nodes : $A$ is the root and $B$ and $C$ are the children of $A$.

$A$ can take 3 values: 0, 1 and 2

$B$ and $C$ can take 2 values: 0 or 1

Imagine that we don't know the conditional probability distributions of $A, B$ and $C$ and we want to find them.

In my situation, I have datas in the following form: I have $S_1, ..., S_k$ such that for any $S_i$, the value of $A$ is constant (for example $S_1$ corresponds to only observation with $A=2$) but is unknown. And I know only one of the values of $B$ and $C$. So, for example:

$S_1$ could be the set $\{[B=1], [C=0], [C=1], [C=1]\}$ and I know that for this 4 points $A$ is constant.

I'd like to find back the cpds of all nodes. I could use the $EM$ algorithm but how do I deal with the fact that some training datas are correlated (ie there are subsets where $A$ is constant) ?



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