maxframe.tensor.special.multigammaln#

maxframe.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

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

Array API Standard Support

multigammaln has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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).