Estimate a known degrees of freedom student t distribution from data using the EM algorithm

LICENSE.txt

This is free and unencumbered software released into the public domain.

Anyone is free to copy, modify, publish, use, compile, sell, or
distribute this software, either in source code form or as a compiled
binary, for any purpose, commercial or non-commercial, and by any
means.

In jurisdictions that recognize copyright laws, the author or authors
of this software dedicate any and all copyright interest in the
software to the public domain. We make this dedication for the benefit
of the public at large and to the detriment of our heirs and
successors. We intend this dedication to be an overt act of
relinquishment in perpetuity of all present and future rights to this
software under copyright law.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR
OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
OTHER DEALINGS IN THE SOFTWARE.

For more information, please refer to <https://unlicense.org>

t.py

import numpy as np
from scipy import special

def t(X, dof=3.5, iter=200, eps=1e-6):
    '''t
    Estimates the mean and covariance of the dataset
    X (rows are datapoints) assuming they come from a
    student t likelihood with no priors and dof degrees
    of freedom using the EM algorithm.

    Implementation based on the algorithm detailed in Murphy
    Section 11.4.5 (page 362).

    :param X: dataset
    :type  X: np.array[n,d]
    :param dof: degrees of freedom for likelihood
    :type  dof: float > 2
    :param iter: maximum EM iterations
    :type  iter: int
    :param eps: tolerance for EM convergence
    :type  eps: float
    :return: estimated covariance, estimated mean, list of
             objectives at each iteration.
    :rtype: np.array[d,d], np.array[d], list[float]
    '''
    # initialize parameters
    D = X.shape[1]
    N = X.shape[0]
    cov = np.cov(X,rowvar=False)
    mean = X.mean(axis=0)
    mu = X - mean[None,:]
    delta = np.einsum('ij,ij->i', mu, np.linalg.solve(cov,mu.T).T)
    z = (dof + D) / (dof + delta)
    obj = [
        -N*np.linalg.slogdet(cov)[1]/2 - (z*delta).sum()/2 \
        -N*special.gammaln(dof/2) + N*dof*np.log(dof/2)/2 + dof*(np.log(z)-z).sum()/2
    ]

    # iterate
    for i in range(iter):
        # M step
        mean = (X * z[:,None]).sum(axis=0).reshape(-1,1) / z.sum()
        mu = X - mean.squeeze()[None,:]
        cov = np.einsum('ij,ik->jk', mu, mu * z[:,None])/N

        # E step
        delta = (mu * np.linalg.solve(cov,mu.T).T).sum(axis=1)
        delta = np.einsum('ij,ij->i', mu, np.linalg.solve(cov,mu.T).T)
        z = (dof + D) / (dof + delta)

        # store objective
        obj.append(
            -N*np.linalg.slogdet(cov)[1]/2 - (z*delta).sum()/2 \
            -N*special.gammaln(dof/2) + N*dof*np.log(dof/2)/2 + dof*(np.log(z)-z).sum()/2
        )   
        
        if np.abs(obj[-1] - obj[-2]) < eps:
            break
    return cov, mean.squeeze(), obj

Thank you. Cheers.