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Given three events $E$, $A$ and $B$, is it possbile to decompose the joint-conditional probability $P(E \vert A, B)$, as a expression in terms of non-joint-conditional probabilities, and marginal probabilities, as following?

$$ P(E \vert A, B) \equiv Expression-Only-Using \\ \{P(E), P(A), P(B), P(E \vert A), P(A \vert E), P(E \vert B), P(B \vert E), P(A \vert B), P(B \vert A) \} $$

It seems to me that, this is impossible unless some conditional independence is assumed. For example,

$P(E \vert A, B) \propto P(A, B \vert E) P(E)$

if assume $A$ and $B$ are independent in the condition of $E$, we further get

$P(A, B \vert E) P(E) \propto P(A \vert E)P(B \vert E) P(E)$

Phil
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  • I assume your difficulty is at a more basic level, so I give an elaborate answer below. –  Jul 23 '24 at 09:04
  • The motivation of this question is for the data analysis. Given some observational data, it is usually relatively easy to estimate the probabilities given the non-joint condition. But sometimes we want to further estimate the probability given the joint conditions. – Phil Jul 26 '24 at 05:07
  • If you have a particular data analysis problem, maybe posting that problem separately with more details would be more helpful. I can hardly read from this post any applied component. These are mostly theoretical discussions. – Zack Fisher Jul 26 '24 at 05:57
  • @ZackFisher My apologies if it caused the confusion. But, no, this is not for a particular data analysis problem. It's just motivated by some thoughts about a class of data analysis problems. It's meant for theoretical discussions. – Phil Jul 26 '24 at 15:11

1 Answers1

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In general, it is not always possible. We can remove / standardize the margins, and what is left after that is the Copula, which is no longer decomposable.

Ryan Shen
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  • Thanks for the reply. I'm not sure yet if this answers my original questions or not. I'm wondering why the copula is more useful than the original joint distribution function. I mean, even if one could estimate each marginal distribution, it still remains difficult to estimate the complete (joint) copula function. – Phil Jul 26 '24 at 05:22
  • That's valid comment and many statisticians don't use copulas at all. However, there are also useful copulas being used in applied areas. The idea is that once we remove or standardize the marginals, we can work on the pure dependence component, if the latter is of primary interest. It's more useful to study the inter-relationship among variables. – Ryan Shen Jul 26 '24 at 05:48
  • Could you please elaborate a bit about the statement that, copula is "more useful to study the inter-relationship among variables"? – Phil Jul 26 '24 at 15:23