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Entropy calculation (was Re: help with an integral)

• To: mathgroup at smc.vnet.net
• Subject: [mg5379] Entropy calculation (was Re: help with an integral)
• From: lewin at ipgp.jussieu.fr (Eric LEWIN)
• Date: Thu, 5 Dec 1996 14:50:19 -0500
• Organization: IPGP Geochimie
• Sender: owner-wri-mathgroup at wolfram.com

[courtesy copy to the author]
[cross-post and follow-up to "sci.stat.math"]

Dans l'article <57649c$28t at dragonfly.wolfram.com>, "Dimitris Agrafiotis"
<dimitris at 3dp.com> =E9crivait:

> I have a rather complicated integral to compute and I was wondering if
> Mathematica can help me do it. Unfortunately, I don't have access to
> this package. The integral is as follows:
>=20
>          +inf
> S =3D -integral(p(x) ln(p(x)))
>          -inf

This is the Shannon entropy of your distribution, isn't it ?

Did i read somewhere that the entropy of the sum of independant random
variables is egal to the sum of their individual entropies ?

> where
>                 n
> p(x) =3D (1/a) * SUM(exp^-((x - c(i))^2 / b)
>                i=3D1
>=20
> and a, b, and c(i) are constants. p(x) is essentially a sum of
> n Gaussians, each centered at its own c(i). x and c(i) are d-
> dimensional vectors. Can the integral be computed/approximated,
> and if so, how many dimensions can we handle? How about the
> sample size, n (i.e. the number of Gaussians)? Can that be
> arbitrarily large?

Individual entropies can here be calculated, since these are all normal
random variables; as far as i can remind, S(i) =3D cst + ln(std(i)). So, =
if
the theorem of addition of entropies can be applied, the answer should be
pretty simple. But i am not sure for the theorem... Can some confirm ?

--=C9ric Lewin

Tout avantage a ses inconv=E9nients, et r=E9ciproquement.
------------------------------- J.Rouxel [Les Shadoks] La vengeance du Ma=
rin