After having read the Wikipedia entry on the degenerate case of the multivariate normal distribution:
https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case
my question is:
What is its entropy ?
In the non-degenerate case the formula is pretty clear, but for the degenerate case the exact formula for the entropy is not given, only the density is given.
I take it that the determinant of the covariance matrix is replaced by its pseudo-determinant. However, my main concern is about the number of parameters in the entropy formula. Because in the degenerate case the covariance matrix is not full-rank, you actually have fewer random variables and the attention is limited to this subspace as the Wikipedia entry nicely explains.
So in the entropy formula which number of parameters do i use for the degenerate case ? The original or the dimension of the subspace of the original variables ? May it also have a different form than the formula of the entropy for the non-degenerate case ?