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How is entropy of a general probability measure defined?

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I can suggest en.wikipedia.org/wiki/Differential_entropy –  falagar Jul 23 '10 at 13:10
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Did you try Google and/or Wikipedia? –  Qiaochu Yuan Jul 23 '10 at 15:14
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I am casting the final vote to close, because even though the original question has generated some interesting details, it wasn't asked with enough background context., nor with indication of what the questioner already knows or has read. –  Yemon Choi Jul 23 '10 at 20:31
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closed as off topic by Gjergji Zaimi, Steve Huntsman, Andrey Rekalo, Qiaochu Yuan, Yemon Choi Jul 23 '10 at 20:31

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3 Answers

up vote 6 down vote accepted

The Entropy of a function $f$ with respect to a measure $\mu$ is

$$ Ent_{\mu}(f)=\int f \log f d\mu - \int f d\mu \log(\int f d\mu ) $$

The entropy of a probability distribution $P$ with respect to $\mu$ is given by $ Ent(\frac{dP }{d\mu })$. I a not aware of a general definition that would not implie a reference (here it is $\mu$) measure ...

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Entropy is not defined for a single probability measure!

Entropy is a relative thing, you define between two measure $\mu$ and $\nu$. Then the entropy is defined by $$ \mathcal{E}(\mu,\nu) = \begin{cases} - \int w(x) \log(w(x)) d\nu(x),& d \mu = w d\nu; \\\ -\infty, & otherwise.\end{cases}, $$ I might have messed up the signs. $w$ is the Radon-Nikodym derivate of $\mu$ with respect to $\nu$.

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It is not. If a probability measure on $\mathbb{R}$ is absolutely continuous and has density $f$, then "entropy" usually refers to the differential entropy, defined in the Wikipedia page falagar linked to. If the probability measure has discrete support, entropy is defined by an analogous formula, given in this Wikipedia page. In the most classical treatments, these are the only situations covered at all.

However, both of these are special cases of the more general notion of relative entropy that Helge and robin girard pointed out: in the continuous case the reference measure ($\nu$ in Helge's notation, $\mu$ in robin's notation) is Lebesgue measure, and in the discrete case the reference measure is counting measure on the support.

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