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