### Links between probabilistic automata and hidden Markov models: probability distributions, learning models and induction algorithms

P. Dupont, F. Denis, Y. Esposito
2005 Pattern Recognition
This article presents an overview of Probabilistic Automata (PA) and discrete Hidden Markov Models (HMMs), and aims at clarifying the links between them. The first part of this work concentrates on probability distributions generated by these models. Necessary and sufficient conditions for an automaton to define a probabilistic language are detailed. It is proved that probabilistic deterministic automata (PDFA) form a proper subclass of probabilistic non-deterministic automata (PNFA). Two
more » ... es of equivalent models are described next. On one hand, HMMs and PNFA with no final probabilities generate distributions over complete finite prefix-free sets. On the other hand, HMMs with final probabilities and probabilistic automata generate distributions over strings of finite length. The second part of this article presents several learning models, which formalize the problem of PA induction or, equivalently, the problem of HMM topology induction and parameter estimation. These learning models include the PAC and identification with probability 1 frameworks. Links with Bayesian learning are also discussed. The last part of this article presents an overview of induction algorithms for PA or HMMs using state merging, state splitting, parameter pruning and error-correcting techniques. Definition 2. The support L ⊆ * of the semi-distribution is the language L = {u ∈ * | (u) > 0}. Definition 3. A distribution or probabilistic language over * is a semi-distribution such that u∈ * (u) = 1. Semi-probabilistic automata Definition 4. A semi-probabilistic automaton 2 (semi-PA) is a 5-tuple , Q, , , where is a finite alphabet, Q is a finite set of states, : Q × × Q → [0, 1] is a mapping defining the transition probability function, : Q → [0, 1] is a mapping defining the initial probability of each state, and : Q → [0, 1] is a mapping defining the final probability of each state. The following constraints must 2 Such an automaton is called a semi-PA and not a PA as it defines a semi-distribution (see Corollary 1). The supplementary conditions to be satisfied to define a distribution are detailed in Definition 9.