I am reading a paper in which the author expresses the log-likelihood function for a gaussian as

$L(\Theta^{(k)}) = -N$ log det $R^{(k)}$ - tr $\left[{R^{(k)}}^{-1} \hat{R}\right]$

where $N$ is the number of samples, $R^{(k)}$ is the low-rank approximation (of rank $k$) of the true covariance matrix $R$, $\hat{R} = \frac{1}{N-1} \sum_{i=1}^N \textbf{x}_i\textbf{x}_i^H$ is the sample covariance (each vector $\textbf{x}_i \in \mathbb{R}^p$ is a sample). The vector $\Theta$ is not relevant for my question (it has the parameters on which the matrix $R^{(k)}$ depends).

I agree almost 100% with this expression except for the fact that I think that the sample covariance should appear multiplied by $N-1$ in the log-likelihood function, but it doesn't matter right now.

Then, the author says that it is straightforward to obtain this expression for the log-likelihood estimate:

$L(\hat{\Theta}) = \textrm{log} \left(\frac{\prod_{i=k+1}^p l_i^\frac{1}{p-k}}{\frac{1}{p-k} \sum_{k+1}^p l_i} \right)^{N(p-k)}$

where ${l_i}$ are the eigenvalues of the sample covariance $\hat{R}$. Well, it has certainly not been straightforward to me. I would appreciate if someone could show me how this step is made.

Just in case, the paper is "Detection of Signals by Information Theoretic Criteria" by Wax and Kailath.


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