  We consider penalized estimation in hidden Markov models (HMMs) with
multivariate Normal observations. In the moderate-to-large dimensional setting,
estimation for HMMs remains challenging in practice, due to several concerns
arising from the hidden nature of the states. We address these concerns by
$\ell_1$-penalization of state-specific inverse covariance matrices. Penalized
estimation leads to sparse inverse covariance matrices which can be interpreted
as state-specific conditional independence graphs. Penalization is nontrivial
in this latent variable setting; we propose a penalty that automatically adapts
to the number of states $K$ and the state-specific sample sizes and can cope
with scaling issues arising from the unknown states. The methodology is
adaptive and very general, applying in particular to both low- and
high-dimensional settings without requiring hand tuning. Furthermore, our
approach facilitates exploration of the number of states $K$ by coupling
estimation for successive candidate values $K$. Empirical results on simulated
examples demonstrate the effectiveness of the proposed approach. In a
challenging real data example from genome biology, we demonstrate the ability
of our approach to yield gains in predictive power and to deliver richer
estimates than existing methods.
