# Estimating the probabilities of many events given only intersections [closed]

We are interested in approximating the probability of $n$ (possibly dependent) events $\{e_1,\dots, e_n\},$ but we can only estimate the probability of an intersection of any $q$ of them:

$$P(e_{k_1}\cap e_{k_2}\cap \dots \cap e_{k_q})$$

The problem's model gives us the following identity. We have an algebraic function $f:\underset{q \text{ times}}{\underbrace{[0,1]\times \dots \times [0,1]}}\rightarrow [0,1]$ where:

$$f\left(P(e_{k_1}),P(e_{k_2}),\dots, P(e_{k_n})\right) = P(e_{k_1}\cap e_{k_2}\cap \dots \cap e_{k_q}).$$

$q$ is fairly small (it's 9), but $n$ can be on the order of dozens to several thousand.

If we took enough samples we would have a chance of just using algebra to solve for each $P(e_i)$ individually, but the amount of samples and computation time needed to do this seems huge.

Is there a good way to approach such a problem?

• Several thousand is small computationally maybe you need to profile and analyze your code or consider parallel computing Commented Sep 28, 2012 at 17:13