Re: Mixture of Gaussian Integral
From: Robert Israel (israel_at_math.ubc.ca)
Date: 06/07/04
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Date: Mon, 7 Jun 2004 11:04:37 -0700 (PDT)
In article <KEWwc.1282$GB4.41814@news.xtra.co.nz>,
Zeke Chan <zekechan@amonics.com> wrote:
>Hi I have a prob here I hope you can help. Let N_a(x) denote Gaussian
>distribution with mean mu_a and cov_a (the subscript indicates the index of
>the distribution). Given a mixture of two gaussians distribution,
>p(b)N_b(x) + p(c)N_c(x), I'd like to the integral
>
>int[ N_a(x) log(p(b) N_b(x) + p(c)N_c(x)) , dx]
>
>Matlab can't solve this. Very grateful if you can find at least an
>approximation to this problem.
Perhaps some clarifications might be helpful. By N_a(x) do you mean the
density of a (one-variable) Gaussian distribution? And are you
integrating from -infinity to infinity?
I doubt that there's a closed-form formula for the answer. If you want
an approximation, useful asymptotic results may be possible. But
you'd need to tell us which parameters are to be considered
small (-> 0) or large (-> infinity).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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