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Derivation of Vasicek Entropy Estimator

  1. Oct 14, 2011 #1
    Hey All - I am trying to solve a problem that should be really easy (at least every paper I read says the step is!)

    I'm trying to understand where the Vasicek entropy estimator comes from:

    I can write the differential entropy of a system as:
    [tex]
    H(f) = -\int^{\infty}_{-\infty} f(x)log(f(x))dx
    [/tex]

    where f(x) is your probability distribution function

    Apparently it is an easy step that you can re-write this in the form:
    [tex]
    H(f) = \int^{1}_{0} log(\frac{d}{dp}F^{-1}(p)) dp
    [/tex]

    Where [tex] F^{1} [/tex] is the inverse of the culmative distribution function.

    I've tried using the derivative of an inverse function that I learned when trying to find the derivative of inverse trig functions but all I got was:

    [tex]
    \frac{d F^{-1}(p)}{dp} = \frac{1}{f(F^{-1}(p))}
    [/tex]

    By the way just in case I need to point it out - this isn't homework - its a stupid problem which is making me feel incredibly stupid not being able to solve

    Thanks,
    Thrillhouse
     
    Last edited: Oct 14, 2011
  2. jcsd
  3. Oct 14, 2011 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    Maybe it's "integration by sustitution" with [itex] x = F^{-1}(p) [/itex]
     
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