### Jasper De Bock, Gert De Cooman

## State sequence prediction in imprecise hidden Markov models

### Abstract

We present an efficient exact algorithm for estimating state sequences from outputs (or observations) in imprecise hidden Markov models (iHMM), where both the uncertainty linking one state to the next, and that linking a state to its output, are represented using coherent lower previsions.
The notion of independence we associate with the credal network representing the iHMM is that of epistemic irrelevance.
We consider as best estimates for state sequences the (Walley--Sen) maximal sequences for the posterior joint state model (conditioned on the observed output sequence), associated with a gain function that is the indicator of the state sequence.
This corresponds to (and generalises) finding the state sequence with the highest posterior probability in HMMs with precise transition and output probabilities (pHMMs).
We argue that the computational complexity is at worst quadratic in the length of the Markov chain, cubic in the number of states, and essentially linear in the number of maximal state sequences.
For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters.

### Keywords

Imprecise hidden Markov model, optimal state sequence, maximality, coherent lower prevision, credal network, epistemic irrelevance.

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### Authors’ addresses

**Jasper De Bock**

Technologiepark- Zwijnaarde 914

9052 Zwijnaarde

**Gert De Cooman**

Technologiepark - Zwijnaarde 914

9052 Zwijnaarde

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