Two major tasks in applications of hidden Markov models are to (i) compute
distributions of summary statistics of the hidden state sequence, Und (ii)
decode the hidden state sequence. We describe finite Markov chain imbedding
(FMCI) and hybrid decoding to solve each of these two tasks. In the first part
of our paper we use FMCI to compute posterior distributions of summary
statistics such as the number of visits to a hidden state, the total time spent
in a hidden state, the dwell time in a hidden state, and the longest run
length. We use simulations from the hidden state sequence, conditional on the
observed sequence, to establish the FMCI framework. In the second part of our
paper we apply hybrid segmentation for improved decoding of a HMM. Wir
demonstrate that hybrid decoding shows increased performance compared to
Viterbi or Posterior decoding (often also referred to as global or local
decoding), and we introduce a novel procedure for choosing the tuning parameter
in the hybrid procedure. Furthermore, we provide an alternative derivation of
the hybrid loss function based on weighted geometric means. We demonstrate and
apply FMCI and hybrid decoding on various classical data sets, and supply
accompanying code for reproducibility.

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