Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence

G. Nuel (1)

1 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques

We focus here on the distribution of the random count N of a regular expression in a multi-state random sequence generated by a heterogenous Markov source. We first briefly recall how classical Markov chain embedding techniques allow reducing the problem to the count of specific transitions in a (heterogenous) order 1 Markov chain over a deterministic finite automaton state space. From this result we derive the expression of both the mgf/pgf of N as well as the factorial and non-factorial moments of N. We then introduce the notion of evidence-based constraints in this context. Following the classical forward/backward algorithm in hidden Markov models, we provide explicit recursions allowing to compute the mgf/pgf of N under the evidence constraint. All the results presented are illustrated with a toy example.

Mots clés: probability generating function; moment generating function; probabilistic graphical model; Bayesian network; sum-product algorithm; forward/backward algorithms

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

G. Nuel. Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence. Methodology and Computing in Applied Probability, 2019, 21 (3), pp.875-887. ⟨10.1007/s11009-019-09700-0⟩. ⟨hal-02350413⟩