This is documentation for an old release of SciPy (version 0.18.1). Read this page in the documentation of the latest stable release (version 1.15.1).
scipy.special.multigammaln¶
- scipy.special.multigammaln(a, d)[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 : ndarray
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
Γd(a)=∫A>0e−tr(A)|A|a−(d+1)/2dAwith 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
Γd(a)=πd(d−1)/4d∏i=1Γ(a−(i−1)/2).References
R. J. Muirhead, Aspects of multivariate statistical theory (Wiley Series in probability and mathematical statistics).