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Help with an integral

  1. May 27, 2014 #1
    This is not a homework question but something that has come up during work and I just need some guidance on what is the best way to solve this.

    I am computing a quantity where one of the terms to compute is a derivative of an integral of the form

    [itex]\nabla_{\mu}log\left(Z(\mu, \Sigma)\right)[/itex] where [itex]Z(\mu, \Sigma)[/itex] is of the form [itex]\int t(w) \, q(w) \,dw[/itex]. My plan was to first get an expression for Z and then take the derivative.

    Now, [itex]q(w)[/itex] is a Gaussian distribution on w and is given by [itex]K_1 \exp\left(-0.5 (w-\mu)^{T} \Lambda (w-\mu)\right)[/itex] where [itex]K_1[/itex] is a constant and [itex]T[/itex] denotes a transpose operation, [itex]\Lambda[/itex] is the inverse of the covariance matrix associated with the Gaussian.

    Similarly, [itex]t(w)[/itex] is also a Gaussian but not on [itex]w[/itex]. t(w) can be written as:

    [itex]t(w) = K_2 \exp\left(-0.5 (y-f(x, w))^{T} \Sigma (y-f(x, w)\right)[/itex]

    where [itex]y[/itex] and [itex]x[/itex] are two observed quantities and can be treated as constants. [itex]f[/itex] is some non-linear function of [itex]w[/itex].

    So, ultimately I need to compute Z as an integral of this complicated expression given by:

    [itex] Z = \int \left[K_2 \exp\left(-0.5 (y-f(x, w))^{T} \Sigma (y-f(x, w)\right)\right] \; \left[K_1 \exp\left(-0.5 (w-\mu)^{T} \Lambda (w-\mu)\right)\right] dw
    [/itex]

    I must admit my upper level calculus skills are not great. I was wondering if someone can comment on how to go about or if there is some other simplifications I can do to solve this.

    Thanks a lot!
     
  2. jcsd
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