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[P] Approximating product of a discrete and continuos distribution in a mixture model

[P] Approximating product of a discrete and continuos distribution in a mixture model

I am working in a mixture model with expectation propagation. When I am trying to compute the normalizing constant, z_i, I have an integral that is not closed form.

My integral looks as follows:

Integral to solve

B_j is a continuous value that is usually << 1. The mean, and variance are known values. The EDCM distribution is a bounded Dirichlet-multinomial distribution. This will result in a discrete discrete distribution since EDCM is discrete.

  1. Can Laplace approximation be used since the distributions to be approximated are not completely continuous?
  2. Can I use BBVI? Posterior p(B|x) propto p(x,B) with an approximation q(B) that is gaussian. The approximation will be continuous, so can I use only the normalization factor of the approximate distribution to approximate p(x) (ie 1/(2pi)^(V/2)|sigma|^1/2). Will this value be considered a proper discrete distribution?
  3. Any other suggestions? Maybe just sample (replace the integral with a summation). I have to iterate many times this approximation.

submitted by /u/cuenta4384
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