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I've been tossing and turning all night as I work out a particular optimality condition in my ongoing work on classifier fusion. Essentially I want to minimize the sum of a multivariate normal distribution function at several points, where the different terms differ in being the probability contained in a open hypercube of the form (-inf, k)x(k, inf)x..., that is each term is the integral of the multivariate normal PDF over the cartesian product of some finite sequence whose elements are either (-inf, k) or (k, inf). These integrals have no analytical expression, they are usually calculated by numerically integrating the normal PDF. These values are available to me through a R library.

In order to find the minima of this, I have to differentiate by k and set it equal to 0. For this I need to numerically estimate the partial derivatives of each integral and add them up. So I have to solve an equation whose terms can only be numerically estimated. How would I go about doing this? Or is there an easier way and I'm making this too brute-force?

Thanks a lot.

Molu

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# Solving an equation with several numerically estimated derivatives

Can you offer guidance or do you also need help?

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