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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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    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.2012-02-13

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