Entropy of t-distribution in scipy: How to input degrees of freedom to digamma and beta functions?

Question

The closed-form analytical solution for the entropy of a variable X that follows the t-distribution, derived here, is

Seeing that python has functions scipy.special.digamma and scipy.special.beta, how can the above formula be implemented in code?

What confuses me is that the functions just mentioned do not take the degrees of freedom parameter nu (v) as an input according to the documentation. A running example would help

Answer 1

You can also use the differential entropy of multivariate student t-distribution, where, dim is dimensional, dof is degree of freedom and cmtx is covariance.

import numpy as np
import scipy.special as sc
def compute_true_entropy(dim=1, dof=3, std=False):
    cmtx = np.identity(dim)
    B0 = 0.5*np.log(np.linalg.det(cmtx))
    B1 = sc.gamma((dim+dof)/2)/((sc.gamma(dof/2))*((np.pi*dof)**(dim/2)))
    B2 = ((dof+dim)/2)*(sc.digamma((dof+dim)/2) - sc.digamma((dof)/2))
    entropy = B0 - np.log(B1) + B2
   return entropy

Answer 2

By its definition, the entropy is defined by Shanoon as:

Now if you apply this formula to the Student-t distribution, you will notice that this one already contains the degree of freedom parameter (v):

As a result of the integration you will have in the approximation both Betta and Digmma. If you can the calculation, honestly I couldn't, you will find out that these take v as an input just as a result of the calculation. It is not in their definition.

v varies between 1 (Cauchy distribution) and infinity (the normal distribution).

To simplify the calculations, I used the code below:

import numpy as np
import scipy.special as sc
v = float(input('Degre of freedom '))
v1 = (1+v)/2
v2 = v/2
Entropy_of_Variable_X = v1*(sc.digamma(v1)-sc.digamma(v2))+np.log(np.sqrt(v)*sc.beta(v2,0.5))
print('Entropy of the variable X, of degree of freedom equal to : ', v, 'is ', Entropy_of_Variable_X)

You can pass it a list or something like that to calculate the entropy for multiple distribution.