Analytic solutions for Tsallis entropies

Integral over a multivariate gaussian to power q

Let $C in RR^{n xx n}$ be a positive-definite matrix. The probability density function of a multivariate gaussian in $RR^n$ is given by $f : RR^n -> R$ :

$f(x) = A e^(-(1/2)x^T C^-1 x)$

where

$A = 1 / ((2 pi)^(n / 2) det(C)^(1 / 2))$

We assume without loss of generality that the distribution is origin-centered. Let

$I = int_{RR^n} f^q(x) dx = int_{RR^n} A^q e^(-(1/2)(sqrt(q) x)^T C^-1 (sqrt(q) x)) dx$

In the analytic solutions for Renyi entropies this integral is solved as:

$I = A^(q - 1) / q^(n / 2)$

Tsallis entropy of a multivariate gaussian

Using the notation from the previous section, the Tsallis entropy is given for a multivariate gaussian $X$ by:

$H_q(X) = (1 - I) / (q - 1)$

which is fine for evaluation as it is.

Tsallis entropy of a uniform distribution

Let $X$ be random variable with a uniform distribution on $S sub RR^n$ . The corresponding probability density function $f : RR^n -> R$ is given by $f(x) = (chi_S(x)) / (m(S))$ , where $m(S)$ is the Lebesgue measure of $S$ and $chi_S$ is the characteristic function of $S$ .

$I = int_{RR^n} f(x)^q dx = (1 / (m(S)))^q int_{S} 1 dx = 1 / (m^(q - 1)(S))$

Then

$H_q(X) = (1 - I) / (q - 1)$

which is fine for evaluation as it is.

Files

Analytic solutions for Tsallis entropies

Integral over a multivariate gaussian to power q

Tsallis entropy of a multivariate gaussian

Tsallis entropy of a uniform distribution

See also

Files

An aggregate file for analytic Tsallis entropies.

Tsallis entropy of a normal distribution

Tsallis entropy of a uniform distribution