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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?

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

1 Answer 1

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I'm not clear from your question what type of model you're using, but in Linear Modelling there is a class of problems known as "large p, small n" in the "Variable Selection" literature, the best known solution is probably the Lasso, refer Tibshirani's Lasso page.

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  • $\begingroup$ it appears the OP is doing something along the lines of inverting a function of several variables and is finding that it becomes quite difficult as the number of variables increase. $\endgroup$
    – Macro
    Commented Jun 30, 2012 at 6:36
  • $\begingroup$ ah... if that is the case my answer is probably a delete... $\endgroup$
    – Sean
    Commented Jun 30, 2012 at 6:44

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