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jax.scipy.special.multigammaln

jax.scipy.special.multigammaln(a, d)

Returns the log of multivariate gamma, also sometimes called the

generalized gamma.

LAX-backend implementation of multigammaln(). Original docstring below.

andarray

The multivariate gamma is computed for each item of a.

dint

The dimension of the space of integration.

resndarray

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

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

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

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

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

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