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Is it that true?

If yes, how to prove this?

$$ P\left(E_e\right)=\sum _{\eta =1}^{\mathbb{H}} P\left(H_{\eta }\right) P\left(E_e|H_{\eta }\right), $$

where $E_e$ is an generic evidence, $H_\eta$ it's the $\eta$-esim hyphotesis and $\mathbb{H}$ is the cardinality of the hypothesis ($H$) set.

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I don't quite follow the jargon, but the formula looks like the law of total probability. –  Srivatsan Oct 22 '11 at 23:04
    
Looks like very simple yet...but didnt find a way until now. Thx about the link. I will take a look. –  GarouDan Oct 22 '11 at 23:05
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Just be aware that the law of total probability $P(A) = \sum_i P(A\mid B_i)P(B_i)$ assumes $A \subset \cup_i B_i \subset \Omega$ where $\Omega$ is the entire sample space. More simply, people just assume $\cup_i B_i = \Omega$ for simplicity. –  Dilip Sarwate Oct 23 '11 at 0:57
    
@SrivatsanNarayanan, if one fo you answer this question, I will embrace it. Someone else too. I just don't want post my own answer. –  GarouDan Oct 28 '11 at 12:50
    
@DilipSarwate if one fo you answer this question, I will embrace it. Someone else too. I just don't want post my own answer. –  GarouDan Oct 28 '11 at 12:50
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1 Answer

up vote 1 down vote accepted

The formula you quote seems to be just the law of total probability. Assume that the set of events $\{ H_\eta \}_{1 \leq \eta \leq \mathbb H}$ forms a partition of the sample space $\Omega$; i.e., the $H_\eta$'s are pairwise disjoint, and $\bigcup \limits_{\eta = 1}^{\mathbb H} H_\eta = \Omega$.

Now for any event $E_e$, the set of events $\{ E_e \cap H_\eta \}$ forms a partition of $E_e$. Therefore, by additivity, we have $$ P(E_e) = \sum_{\eta = 1}^{\mathbb H} P(E_e \cap H_\eta). \tag{1} $$ Now, by the definition of conditional probability, we have $P(E_e \cap H_\eta) = P(H_\eta) \cdot P(E_e \mid H_\eta)$. Plugging this in $(1)$ we get the claim.*

The following sentence taken from the wikipedia article explains what this theorem means intuitively (notation changed to match ours):

The summation can be interpreted as a weighted average, and consequently the marginal probability, $P(E_e)$, is sometimes called "average probability"; "overall probability" is sometimes used in less formal writings.


*The formula is true even for $\{ H_\eta \}_{\eta \geq 1}$ forms a countably infinite partition of $\Omega$. The proof has to be modified only slightly for this.

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Thx. A good answer! –  GarouDan Oct 29 '11 at 23:42
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