multigammaln(a, d)

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

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

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.

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

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

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.special._spfun_stats.multigammaln

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