  In various applications involving hidden Markov models (HMMs), some of the
hidden states are aliased, having identical output distributions. The
minimality, identifiability and learnability of such aliased HMMs have been
long standing problems, with only partial solutions provided thus far. In this
paper we focus on parametric-output HMMs, whose output distributions come from
a parametric family, and that have exactly two aliased states. For this class,
we present a complete characterization of their minimality and identifiability.
Furthermore, for a large family of parametric output distributions, we derive
computationally efficient and statistically consistent algorithms to detect the
presence of aliasing and learn the aliased HMM transition and emission
parameters. We illustrate our theoretical analysis by several simulations.
