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The rare event approximation for event $A$ and $B$ means the upper-bound approximation $P(A\cup B)=P(A)+P(B)-P(A\cap B)\leq P(A)+P(B)$. Now by inclusion-exclusion-principle $$P(A\cup B\cup \neg C)= P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap \neg C) -P(B\cap\neg C) +P(A\cap B\cap \neg C) \leq P(A)+P(B)+P(C)+P(A\cap B\cap \neg C)$$

and now by Wikipedia, this is the general form

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so does the rare-event-approximation means removal of minus terms in the general form of inclusion-exclusion principle aka the below?

$$\mathbb P\left(\cup_{i=1}^{n}A_i\right)\leq \sum_{I\subset\{1,3,...,2h-1\}; |I|=k}\mathbb P(A_I)$$

where $2h-1$ is the last odd term in $\{1,2,3,...,n\}$.

Example

For example in the case of three events, is the below true rare-event approximation?

$$P(A\cup B\cup \neg C) \leq P(A)+P(B)+P(\neg C)+P(A\cap B\cap \neg C)$$

P.s. I am studying probability-risk-assessment course, Mat-2.3117.

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1 Answer 1

up vote 1 down vote accepted

Removing all the negatives certainly gives an upper bound. But if one looks at the logic of the inclusion-exclusion argument, whenever we have just added, we have added too much (except possibly, at the very end). So at any stage just before we start subtracting again, our truncated expression gives an upper bound.

Thus one obtains upper bounds by truncating after the first sum, or the third, or the fifth, and so on.

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Do you mean the rare-event approximation for $P(A\cup B\cup C)$ with the three events is not unique i.e. I can approximate the original expressions in multiple ways by removing between one and all minus terms? –  hhh Apr 4 '13 at 0:17
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There is $\Pr(A)+\Pr(B)+\Pr(C)$, and of course the full absolutely correct three-term expression, which is not really an approximation. Things get more interesting at, say, $5$ terms, where we can stop after $\sum \Pr(A_i)$, or $\sum \Pr(A_i\cap A_j\cap A_k)$ (or at the end, but then we are not really approximating). Note that your approximation around the third line is unduly pessimistic, you don't want to add the $\Pr(A\cap B\cap C)$ term. –  André Nicolas Apr 4 '13 at 0:22
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The least upper bound is the exact answer. We want an easy to compute upper bound. In surprisingly many cases, $\sum \Pr(A_i)$ is good enough. When it isn't, $\sum \Pr(A_i)-\sum\Pr(A_i\cap A_j)+\sum\Pr(A_i\cap A_j\cap A_k)$ is often good enough. One may even have to go two steps further. –  André Nicolas Apr 4 '13 at 0:40
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Yes, by two steps further I mean what you wrote. The approximation is useful when individual probabilities are smallish, so that probabilities of long intersections are very small. –  André Nicolas Apr 4 '13 at 0:53
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That's another kind of rare event approximation. It comes up when one is doing the theory of the relationship between the exponential, the Poisson, and the binomial when $np$ is "small." We end up assuming that in a small amount of time $\Delta t$, the probability of two or more events is negligible. –  André Nicolas Apr 4 '13 at 1:09

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