scipy.special.multigammaln

scipy.special.multigammaln(a, d)

Returns the log of multivariate gamma, also sometimes called the generalized gamma.

Parameters:

a : ndarray

The multivariate gamma is computed for each item of a.

d : int

The dimension of the space of integration.

Returns:

res : ndarray

The values of the log multivariate gamma at the given points a.

Notes

The formal definition of the multivariate gamma of dimension d for a real a is

\[\begin{split}\Gamma_d(a) = \int_{A>0} e^{-tr(A)} |A|^{a - (d+1)/2} dA\end{split}\]

with the condition \(a > (d-1)/2\), and \(A > 0\) being the set of all the positive definite matrices of dimension d. Note that a is a scalar: the integrand only is multivariate, the argument is not (the function is defined over a subset of the real set).

This can be proven to be equal to the much friendlier equation

\[\Gamma_d(a) = \pi^{d(d-1)/4} \prod_{i=1}^{d} \Gamma(a - (i-1)/2).\]

References

R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics).