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I'm working with Bayesian Networks and conditional probability. Let $e$, $l$ and $s$ be random variables, the expression $$\sum_l p(e|l)p(l|s)=p(e|s) $$

holds always? If so, how can I derive it using the definition of conditional probability and marginalization?

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If I understand correctly, $s$ is the parent of $l$ and $l$ is the parent of $e$ in the Bayesian network. Therefore, $$ p(e \mid l, s) = p(e \mid l) $$ since the probability of $e$ depends only on $l$. Then \begin{align} \sum_l p(e\mid l)p(l\mid s) = \sum_l p(e\mid l, s)p(l\mid s) = \sum_l p(e, l \mid s) = p(e \mid s) \end{align} where the second equality is due to $$ p(A, B \mid C) = p(A \mid B, C) \cdot p(B \mid C) $$ and the last equality is due to the law of total probability.

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  • $\begingroup$ That is one situation. More generally, the equality, $p(e\mid \ell, s)=p(e\mid \ell)$, means that $e$ and $s$ are conditionally independent when given $\ell$, and vice versa. $$\begin{align}p(e\mid s)~&= \sum_\ell p(e,\ell\mid s) && \text{Law of Total Probability}\\[1ex] &=~\sum_\ell p(e\mid\ell,s)~p(\ell\mid s) &&\text{Definition of Conditional Probability}\\[1ex] &=~\sum_\ell p(e\mid \ell)~p(\ell\mid s) && e\perp s\mid\ell\end{align}$$ $\endgroup$ Mar 21 '17 at 8:08
  • $\begingroup$ @GrahamKemp Good point. $\endgroup$
    – PSPACEhard
    Mar 21 '17 at 10:07
  • $\begingroup$ @GrahamKemp Thanks fo the comment, but I don't see any difference with the answer. The first equation of the answer $p(e|l,s)=p(e|l)$ means that $e$ and $s$ are conditionally independent given $l$. $\endgroup$
    – Ana S. H.
    Mar 21 '17 at 15:00

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