Inference on a probabilistic graphical model with observed continuous variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T17:50:47Z http://mathoverflow.net/feeds/question/97999 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97999/inference-on-a-probabilistic-graphical-model-with-observed-continuous-variable Inference on a probabilistic graphical model with observed continuous variable eakbas 2012-05-25T22:25:26Z 2012-05-28T04:59:17Z <p>Suppose we have the following probabilistic graphical model: </p> <p>$L \rightarrow X, L \rightarrow Y, F \rightarrow Y, X \rightarrow D, Y \rightarrow D$. </p> <p>That is, the joint is: $P(D,X,Y,L,F) = P(D|X,Y)P(X|L)P(Y|L,F)P(L)P(F)$. </p> <p>Here $L,F,D$ are discrete, $X,Y$ are continuous random variables (having proper pdf's). </p> <p>I want to compute $P(D=d,L=l | X=x, F=f)$. Note that we have already observed ($x$) for one of the continuous variables.</p> <p>Here is my solution attempt so far: </p> <p>$P(d,l | x, f) = \frac{P(d,l,x, f)}{P(x,f)}$ Let's ignore the denominator for now. The numerator: </p> <p>$P(d,l,x,f) = \int_y P(d|x,y) P(x|l) P(y|l,f)P(l)P(f)$</p> <p>Now, $P(x|l)$ and $P(y|l,f)$ should be zero because they are continuous variables. But we have already observed the value of X. So, I am not sure how to proceed from now on. My guess is that we should use the pdf's instead of probabilities: </p> <p>$P(d,l,x,f) = \int_y P(d|x,y)\;f_{X|L}(x|l)\;f_{Y|L,F}(y|l,f)\;P(l)\;P(f)dy$</p> <p>Is this correct? </p> <p>If this is correct, then another question is how to compute this integral. Would the following work?</p> <p>Sample a $y$ from $f_{Y|L,F}(Y|l,f)$, compute $P(d|x,y)$ and then compute $g(y) = P(d|x,y)\;f_{X|L}(x|l)\;P(l)\;P(f)$. Do this a large number of times and take the average of $g(y)$'s. ??? </p> <p>Any ideas, hints, directions would be highly appreciated. </p>