multigammaln

→Tensor

multigammaln(a: Tensor, p: int)

source

Logarithm of the multivariate gamma function $\Gamma_p(a)$ .

Computes $\log \Gamma_p(a)$ , the natural generalisation of lgamma that appears in the normalising constants of the Wishart and inverse-Wishart distributions over positive-definite matrices, and in matrix-Bayes log-evidence formulas.

Parameters

aTensor

Real argument; any floating-point dtype. Each entry must satisfy

a > (p - 1)/2

for the function to be finite (otherwise the underlying lgamma returns inf).

pint

Dimensionality of the matrix argument; must be a positive integer.

Returns

Tensor

$\log \Gamma_p(a)$ element-wise over a, same shape and dtype as a.

Notes

Definition:

\log \Gamma_p(a) = \frac{p(p-1)}{4}\,\log \pi + \sum_{i=1}^{p} \log \Gamma\!\left(a + \frac{1 - i}{2} \right).

For p = 1 this reduces to $\log \Gamma(a)$ . Raises ValueError if p < 1.

Examples

>>> import lucid
>>> from lucid.special import multigammaln
>>> multigammaln(lucid.tensor([3.0, 5.0]), p=2)
Tensor([1.7918, 5.4538])