scipy.special.multigammaln

scipy.special.multigammaln(a, d)

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

Parameters

andarray

The multivariate gamma is computed for each item of a.

dint

The dimension of the space of integration.

Returns

resndarray

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

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

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).

Examples

>>> from scipy.special import multigammaln, gammaln
>>> a = 23.5
>>> d = 10
>>> multigammaln(a, d)
454.1488605074416

Verify that the result agrees with the logarithm of the equation shown above:

>>> d*(d-1)/4*np.log(np.pi) + gammaln(a - 0.5*np.arange(0, d)).sum()
454.1488605074416