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Suppose we have a joint distribution $P(D,X,L) = P(D|X)P(X|L)P(L)$. Here, D and L are discrete but X is a continuous random variable. I want to compute $P(D=d)$. How do I do this numerically? The fact that $X$ is continuous confuses me.

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A little bit more complicated version of this problem is at…. Comments/answers needed. – eakbas May 26 '12 at 2:15
up vote 1 down vote accepted

I'm assuming that your "continuous" is actually "absolutely continuous", i.e. $X$ has a density.

$$P(D=d) = E[P(D=d|X)] = \int dx \ P(D=d|X=x) f_X(x) = \int dx \sum_\ell \ P(D=d|X=x) f_{X|L}(x|\ell) P(L=\ell)$$

where the sum is over the possible values of $L$, and $f_{X|L}(x|\ell)$ is the conditional density of $X$ given $L=\ell$.

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