Profile likelihood for a Gaussian precision matrix ($-\log \det \Omega +\mathrm{tr}(S\Omega )$)

For independent multivariate normal observations with unknown mean and nonsingular covariance, profiling the mean at its sample value gives negative log-likelihood proportional to $-\log \det \Omega +\mathrm{tr}(S\Omega )$ over $\Omega \succ 0$. The sample covariance matrix uses divisor $n$. Its minimizer exists exactly when $S\succ 0$, and is then $S^{-1}$.