Extra hidden states under emission misspecification in hidden Markov models

Additional states in a hidden Markov model (HMM) can compensate for restrictive emission distributions without representing additional scientific regimes. We study this distinction using the established representation of mixture components as refined states. Under compactness, mixture identifiability and positive aggregate transitions, we prove that a simple‐emission HMM with too few components has an observed‐law Kullback–Leibler loss that grows at least linearly with sequence length. An exact refinement, or an aggregate HMM with mixture emissions, removes this approximation gap. Simulations illustrate the resulting incentive to add states. An analysis of tumor allele fractions on chromosome 1 then compares five Gaussian states with three regional profiles containing five Gaussian components. The fitted emissions and regional assignments are nearly the same after grouping the Gaussian states, while the mixture model uses 14 fewer parameters. This example illustrates how additional states can describe variation within a regional profile rather than additional regional regimes. The number of fitted states alone therefore does not determine the number of scientifically distinct regimes.