I have an equation of the form: $$ Pr(A, B, C|D) = Pr(A, B|C,D)Pr(C|D)$$ but I'm having difficulty justifying why it is correct. How can you get the right hand side from the left?
2026-03-29 02:44:40.1774752280
Factoring a joint conditional probability into a specific form.
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By definition, $$Pr(A, B, C|D)=\frac{Pr(A, B, C, D)}{Pr(D)}=\frac{Pr(A, B, C, D)}{Pr(C, D)}\frac{Pr(C, D)}{Pr(D)}=Pr(A, B|C, D)Pr(C|D).$$