Derivation of Vasicek Entropy Estimator

[thrillhouse86]

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:
$$H(f) = -\int^{\infty}_{-\infty} f(x)log(f(x))dx$$

where f(x) is your probability distribution function

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

Where $$F^{1}$$ 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:

$$\frac{d F^{-1}(p)}{dp} = \frac{1}{f(F^{-1}(p))}$$

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

 

[Stephen Tashi]

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