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Given the following Bayessian Network:

Graphical representation of a probabilistic model

I wonder when is it reasonable to estimate $p(u\mid c)$ as

$$ p(u\mid c) \approx p(c\mid w=w_1,\ldots,w_t)$$

I want to estimate that because I can't calculate $p(u\mid c)$ because $u$ is not observable. I've been looking for inference and reasoning in bayes networks but I couldn't find any inference like this.

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Are you assuming a particular type of distribution for $u and c$ or is this simply a general exercise? –  jerad Dec 15 '12 at 18:57
    
I don't know what you mean. Could you give an example of such a distribution? –  Kits89 Dec 16 '12 at 10:17
    
I what is the functional form of the conditional distribution $p(u|c)$? Is $u$ a discrete variable? Gaussian? What about c? –  jerad Dec 16 '12 at 21:51
    
Oh, all the variables are discrete. $u$ are users of a search engine, $c$ are categories where the queries that $u$ search ($q$) and webs sites that they visit ($w$) are classified. –  Kits89 Dec 17 '12 at 8:30
    
See this document here under the subsection "Known Structure, Partial Observability" –  jerad Dec 17 '12 at 16:51

1 Answer 1

I'm going to take a Bayesian stab at this:

By Bayes' rule $$ P(u \mid c) = \frac{P(c \mid u) P(u)}{P(c)} $$ $ P(c) $ is just a normalizing constant so what you're really interested in is $$ P(c \mid u) P(u) $$ The first term $ P(c \mid u) $ is the likelihood. This quantity depends on your model of how $ c $ is generated from $ u $. The second term, $ P(u) $ is your prior. This is what you believe the value of $ u $ before observing data.

If you can observe $ c $, then notice that this expression does not depend at all on $ w $ or $ q $. In other words, $ u \perp w,q \mid c $ ($ u $ is conditionally independent of $ w $ and $ q $ given $ c $)

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+1 Thanks for your answer. I see your point. However, I could assume that there is such a dependency, right? I mean, imagine that, in fact, $w,q$ depend on $u$. Then, as I don't have a way of knowing any probability involving $u$, could I use some reasoning pattern to infer $p(u|c)$ from $c$ and $q$?? –  Kits89 Dec 15 '12 at 9:06
    
@Kits89, I think tskuzzy's last statement just meant that according to your graphical model, if $c$ is observed then $w and q$ are independent of $u$. –  jerad Dec 15 '12 at 18:48
    
I know. But in my last comment I was saying to imagine that there is an arrow from $u$ to $w$ in my graphical model. –  Kits89 Dec 16 '12 at 10:18

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