Maximum Likelihood Estimation for Linear Gaussian Covariance Models
Relatore: Prof. Piotr W. Zwiernik, attualmente visiting presso il Dip di Matematica dell'Università di Genova,
Introduce: Antonio Forcina
mercoledì 15 luglio 2015, ore 16, Aula 101
Riassunto:
We study parameter estimation in linear Gaussian covariance models,
which are p-dimensional Gaussian models with linear constraints on the
covariance matrix. Maximum likelihood estimation for this class of
models leads to a non-convex optimization problem which typically has
many local optima. We prove that the log-likelihood function is
concave over a large region of the cone of positive definite matrices.
Using recent results on the asymptotic distribution of extreme
eigenvalues of the Wishart distribution, we provide sufficient
conditions for any hill climbing method to converge to the global
optimum. Remarkably, our numerical simulations indicate that our
results remain valid for p as small as 2. An important consequence of
this analysis is that for sample sizes n>14p, maximum likelihood
estimation for linear Gaussian covariance models behaves as if it were
a convex optimization problem.
Joint work: Caroline Uhler, Donald Richards