scipy.special.

multigammaln#

scipy.special.multigammaln(a, d)[source]#

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

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

Array API Standard Support

multigammaln has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

See Support for the array API standard for more information.

References

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

Examples

>>> import numpy as np
>>> 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