Log likelihood of covariance incorrect?

First of all, I apologize if it’s just me making a mistake here. However, I can’t seem to reconcile my understanding of the gaussian likelihood function with the one in empirical_covariance.py.

In particular:

log_likelihood_ = - np.sum(emp_cov * precision) + fast_logdet(precision)

The term emp_cov * precision should be multiplied by p. In the multivariate gaussian density, it’s the scatter matrix, not the covariance matrix, who’s Frobenius inner product is being computed.

Am I missing something here or is this incorrect?

EDIT: Here’s a source that seems to confirm my interpretation (slide 34).

[agramfort]

for me there is no n (aka n_samples in our formula) as it’s the mean log likelihood.

[agramfort]

@ashaffer I don’t know where the p would come from. You can do a numerical test if you want to confirm. Simulate data that match the model and check that log like tends to 0.