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Suppose we have the following Bayesian net (or a probabilistic graphical model):

$L \rightarrow X \leftarrow F$, i.e. $P(L,X,F) = P(X|L,F)P(L)P(F)$ and all of these probabilities are known.

Let $\delta(x)$ be a decision rule: given a $x$ it outputs the value of $L$ with the highest posterior. $\delta(x) = \rm{argmax}_\ell P(L=\ell|x)$.

I want to compute the value of $F$ which maximizes the probability of giving a correct decision:

$\rm{argmax}_f \;\;P(\delta(X)=L | F=f) =?$

I'm confused on how to proceed from here. The fact that the decision rule uses the prior probability of L confuses me. So,

$ P(\delta(X)=L | F=f) = \sum_{i=1}^N P(\delta(x)=i | F=f, L=i) P(L=i|F=f)$

In the first term (right handside), the event $L=i$ is given, so in that case, how can $\delta(x)$ function correctly?

I'm either formulating the objective wrongly, or there are some problems with my notation. Any leads, hints will be highly appreciated.

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    Do you know about Dershowitz's conditional confusion during the OJ trial as pointed out by Gigerenzen?2012-08-30
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    @EmreA first according to your formulation at top, $L$ and $F$ are independent. Is that right? Second given $L$ the probability of $\delta$ is of course not $1$ as it is designed previously spanning $L$ into some regions with some associated probabilities.2012-08-31
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    @alancalvitti I've read it from Wikipedia now and no ring belled. Could you please explain?2012-08-31
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    @SeyhmusGüngören, yes, L and F are independent (but only when X is not observed). So what effect does it have? I understand that $\delta$ is designed previously but no $X$ is observed yet, how do you compute the probability of being correct?2012-08-31
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    @EmreA It is just what I saw from your formulation from the second line. You wrote it without mentioning that $F$ and $L$ were independent. Let me have a look at your answer.2012-08-31

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