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I was revisiting some basic concepts on graphical models and factorization of distributions and noticed that all the examples I see only have factors that include, at most, one conditioned random variable (i.e. P(A | B, C, D)). Is there are reason that I shouldn't (or can't) decompose P(A, B, C, D) to include a factor like P(A, B | C, D)? If I can't/shouldn't, why? If I can, how should the visualization look like? Should it look like this?

factorization of P(A, B, C, D) that includes P(A, B | C, D)

If yes, isn't this the same visualization for a decomposition like the following: P(A, B, C, D) = P(A | C, D) * P(B | C, D) * P(C) * P(D)? If yes, does that mean two different factorizations can have the same visualization?

Thanks in advance!

echo66
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    That could make sense if $A$ and $B$ depend on each other, but if they do not then it may introduce unnecessary complication for the graphical model. See for example https://en.wikipedia.org/wiki/Graphical_model#Cyclic_Directed_Graphical_Models – Henry Jun 28 '23 at 17:16
  • Thanks for the response, Henry.

    Does that mean that the visualization I post would be only applicable to P(A, B, C, D) = P(A | C, D) * P(B | C, D) * P(C) * P(D) but not to P(A, B, C, D) = P(A, B | C, D) * P(C) * P(D) ?

     – echo66 Jun 28 '23 at 18:06
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    Moving away from graphical models, if $A$ and $B$ are conditionally independent given $C$ and $D$ then this mean $P(A,B \mid C,D) = P(A \mid C,D) , P(B \mid C,D)$. But in graphical models your visualisation would suggest using the right hand side [and multiplying it by $P(C)P(D)$ where $C$ and $D$ are presumed to be unconditionally independent of each other] – Henry Jun 28 '23 at 22:07

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