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My question concerns conditional probability and chaining those distributions.

I have not been able to find a rule that allows me to derive $P(A|C)$, when $P(A|B)$ and $P(B|C)$ are given for three dependent events A, B, and C.

I believe it is not true that $P(A|C) = P(A|B) * P(B|C)$. Is it possible to calculate $P(A|C)$, if more information is known, such as $P(B)$ or $P(C)$

Any pointers or references are welcome.

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  • $\begingroup$ The events are independent? Then $P(A\,|\,B)=P(A)$ and so on. $\endgroup$
    – lulu
    Commented Apr 19, 2019 at 14:22
  • $\begingroup$ and if they are dependent? $\endgroup$
    – Amir
    Commented Apr 19, 2019 at 14:23
  • $\begingroup$ Please edit your post to ask a clear question. $\endgroup$
    – lulu
    Commented Apr 19, 2019 at 14:23
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    $\begingroup$ You can say $P(A \mid C)= P(A \mid B, C)P(B \mid C)+ P(A \mid B^c, C)P(B^c \mid C)$ $\endgroup$
    – Henry
    Commented Apr 19, 2019 at 14:23
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    $\begingroup$ If $A$'s relationship with $C$ just depended on $B$, i.e. $P(A \mid B, C)=P(A \mid B)$ and $P(A \mid B^c, C)=P(A \mid B^c)$ then you could also say $P(A \mid C)= P(A \mid B)P(B \mid C)+ P(A \mid B^c)P(B^c \mid C)$; if in addition $A \subset B$ so $P(A \mid B^c)=0$ then you could say $P(A \mid C)= P(A \mid B)P(B \mid C)$. But this requires some strong assumptions which generally are not correct $\endgroup$
    – Henry
    Commented Apr 19, 2019 at 14:29

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