multivariate student t-distribution with python

Question

To generate samples with multivariate t-distribution I use this function:

def multivariatet(mu,Sigma,N,M):
    '''
    Output:
    Produce M samples of d-dimensional multivariate t distribution
    Input:
    mu = mean (d dimensional numpy array or scalar)
    Sigma = scale matrix (dxd numpy array)
    N = degrees of freedom
    M = # of samples to produce
    '''
    d = len(Sigma)
    g = np.tile(np.random.gamma(N/2.,2./N,M),(d,1)).T
    Z = np.random.multivariate_normal(np.zeros(d),Sigma,M)
    return mu + Z/np.sqrt(g)

but what I am looking for now is the multivariate student t-distribution it self so I can calculate the density of elements where dimension > 1.

That will be something like stats.t.pdf(x, df, loc, scale) of the package scipy but in multi-dimensional space.

Answer 1

I coded the density by myself:

import numpy as np
from math import *

def multivariate_t_distribution(x,mu,Sigma,df,d):
    '''
    Multivariate t-student density:
    output:
        the density of the given element
    input:
        x = parameter (d dimensional numpy array or scalar)
        mu = mean (d dimensional numpy array or scalar)
        Sigma = scale matrix (dxd numpy array)
        df = degrees of freedom
        d: dimension
    '''
    Num = gamma(1. * (d+df)/2)
    Denom = ( gamma(1.*df/2) * pow(df*pi,1.*d/2) * pow(np.linalg.det(Sigma),1./2) * pow(1 + (1./df)*np.dot(np.dot((x - mu),np.linalg.inv(Sigma)), (x - mu)),1.* (d+df)/2))
    d = 1. * Num / Denom 
    return d

Answer 2

This evaluates the log pdf of the multivariate student-T distribution for n by d data matrix X:

from scipy.special import gamma
from numpy.linalg import slogdet

def multivariate_student_t(X, mu, Sigma, df):    
    #multivariate student T distribution

    [n,d] = X.shape
    Xm = X-mu
    V = df * Sigma
    V_inv = np.linalg.inv(V)
    (sign, logdet) = slogdet(np.pi * V)

    logz = -gamma(df/2.0 + d/2.0) + gamma(df/2.0) + 0.5*logdet
    logp = -0.5*(df+d)*np.log(1+ np.sum(np.dot(Xm,V_inv)*Xm,axis=1))

    logp = logp - logz            

    return logp