Derivation of Vasicek Entropy Estimator

  • #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:

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
7,713
1,519
Maybe it's "integration by sustitution" with [itex] x = F^{-1}(p) [/itex]
 

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