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I am having some trouble discerning joint distributions. The joint distribution below is given for $p(x_1,x_2,x_3,x_4)$ however by following the algebra, applying Bayes rule etc., I cannot reach the joint distribution.

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Or the Markov chain here

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In such cases (or all) do we always need to know what the graph looks like, otherwise how can we construct the joint distribution? Or can we build the graph from individual probabilities, and looking at that, we can apply the algebra (factoring in the conditional independence), allowing us to reach the joint distribution this way?

Thanks,

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If you use the Bayes rule, you marginalize your joint density to some conditional densities. This is the most general picture that you can have. In case you have additional information, about the dependence/independence structure of random variables you can make further simplifications.

At this point a graphical model dictates such dependencies and makes the problem specific.

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I see.. so the algebra by itself might not be enough, and the probability graph can contain more information that'll help constructing a joint distribution. –  BB_ML Sep 18 '12 at 14:54
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Yes for an exact model, the probability graph is necessary. In terms of theoretical model one might be looking for $P(a|b,c)$ however the graph migh be saying that $a$ is independent of $(b,c)$. This will simplify the problem especially when these probabilites are calculated from the data. –  Seyhmus Güngören Sep 18 '12 at 15:10

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