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We consider penalized estimation in hidden Markov models (HMMs) withmultivariate Normal observations. In the moderate-to-large dimensional setting,estimation for HMMs remains challenging in practice, due to several concernsarising from the hidden nature of the states. We address these concerns by$\ell_1$-penalization of state-specific inverse covariance matrices. Penalizedestimation leads to sparse inverse covariance matrices which can be interpretedas state-specific conditional independence graphs. Penalization is nontrivialin this latent variable setting; we propose a penalty that automatically adaptsto the number of states $K$ and the state-specific sample sizes and can copewith scaling issues arising from the unknown states. The methodology isadaptive and very general, applying in particular to both low- andhigh-dimensional settings without requiring hand tuning. Furthermore, ourapproach facilitates exploration of the number of states $K$ by couplingestimation for successive candidate values $K$. Empirical results on simulatedexamples demonstrate the effectiveness of the proposed approach. In achallenging real data example from genome biology, we demonstrate the abilityof our approach to yield gains in predictive power and to deliver richerestimates than existing methods.

Penalized estimation in high-dimensional hidden Markov models with state-specific graphical models