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SciPy

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scipy.special.multigammaln

scipy.special.multigammaln(a, d)[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 : 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

Γd(a)=A>0etr(A)|A|a(d+1)/2dA

with the condition a>(d1)/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

Γd(a)=πd(d1)/4di=1Γ(a(i1)/2).

References

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