Time-coherency of Bayesian priors of transient semi-Markov chains for sequential alignment

This paper proposes a novel insight to the problem of duration modeling for Information Retrieval problems where a discrete sequence of events is estimated from a time-signal using Bayesian models. Since the duration of each event is unknown, a major issue is setting the right Bayesian prior on each of them. Hidden Semi-Markov models (HSMM) allow choosing explicitly any probability distribution for the durations but learning these statistically is a non-parametric problem. In absence of huge training data sets, most algorithms rely on regularization techniques such as choosing parametric classes of distributions but the justifications of such techniques are often heuristics. Among the numerous application domains of HMM-like paradigms, music-to-audio alignment brings two interesting properties. Firstly, a music score informs of the ordering among events. Secondly, it assigns to each event a nominal duration. For alignment tasks the Markov models conveniently model the ordering with transient chains. But the modeling of these nominal durations is a crucial and undermined problematic. This work investigates the relationship of this prior information of duration with the Bayesian priors of a HSMM. Theoretical insights are obtained through the study of the prior state probability of transient semi-Markov chains. Whereas ergodic chain and their convergence to an equilibrium probability are well studied, transient chains constitute an undermined case but of prime importance for real-time inference on HSMM. On the first hand we prove that the non-asymptotical evolution of the state probability has some particular behaviors if the Bayesian priors fulfill several precise conditions, derived from statistical properties like the hazard rate and the tail decay. Then we say that a model is time-coherent if the evolution of the state probability respects the information of ordering and nominal lengths. This leads to several prescriptions on the design of HSMM Bayesian priors. On the other hand we get further prescriptions by comparing the Bayesian priors associated to different nominal lengths. This real-valued parameter comes with a natural ordering; we explain why this ordering among parameters is coherently modeled by some specific stochastic orderings among distributions that are standard in statistics. We conclude by demonstrating the practical consequences of these properties with an experiment of real-time audio-to-score alignment with the HSMM-based software Antescofo developed at Ircam.