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I am aware of the general conditional probability rule which says that

$P(ABCD) = P(A|BCD)P(B|CD)P(C|D)P(D)$

But is there any situation where one can write

$P(A|D) = P(A|B)P(B|C)P(C|D)$ where $A,B,C,D$ are random variables.

Thanks

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Try $A\subseteq B\subseteq C\subseteq D$. –  Michael Greinecker Feb 13 '12 at 22:56
    
@suresh Per faq, usage of signature is not recommended, so I removed it. –  Sasha Feb 13 '12 at 22:57
    
@Sasha, I was not aware of this. Thanks for pointing it out –  suresh Feb 13 '12 at 23:02
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Actually, you'd want $A,B,C,D$ to be events rather than random variables. –  Michael Hardy Feb 13 '12 at 23:39
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In general, you run into the problem that you can have both events $A$ and $D$ occur without $B$ and $C$ occuring. If the events are nested, this problem can not occur. –  Michael Greinecker Feb 13 '12 at 23:40
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1 Answer

up vote 2 down vote accepted

This works with Markov chains. It's essentially the definition of a Markov chain.

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