# mars.tensor.special.multigammaln#

mars.tensor.special.multigammaln(a, b, **kwargs)[source]#

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

Return type

ndarray

Notes

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

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

References

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