Algorithms for state-estimation and model calibration with finite-emission Hidden semi-Markov Models (HsMMs)

The next statistics seminar will be presented by Dr. Paul Malcolm from the Defence Science and Technology Group (DSTG).

Title: Algorithms for state-estimation and model calibration with finite-emission Hidden semi-Markov Models (HsMMs)
Speaker: Dr. W. P. Malcolm
Time and location : 2-3pm in F11.01.145. Chemistry Lecture Theatre 1 or Zoom
Abstract :

A semi-Markov process evolving in discrete-time is a finite-state stochastic process in which state-sojourns need not be geometrically distributed, as is necessarily the case with Markov models. It’s well known that any first-order time-homogenous Markov chain has geometric sojourns for all its states. This feature creates a convenient simplicity, however, it also excludes modelling scenarios that do not have geometric sojourns.

To start our seminar we first recall the standard HMM and how estimators for it may be derived by recasting to a probabilistic graphical model (PGM), and thereafter exploiting conditional independences. This method can also be applied to HsMM, but with added complexities. Instead we take a different approach by first identifying a state-space in which any HsMM may be recast as a first-order time-homogenous HMM on a lattice. It is shown that an ideal state-space to achieve such a recasting consists of states and natural numbers, where our natural numbers correspond to how long, at time k, the processes has been in the state it’s in at time k.

To compute our estimation schemes we use what is loosely called an abstract form of Bayes rule and an auxiliary probability measure, under which our observation sequence is independent and identically distributed and is statistically independent of the latent state.

Filters, smoothers and an EM algorithm are presented as recursions of un-normalised probabilities. This simplifies coding and facilitates numerical stability. Finally, we discuss the issue of parameter complexity by considering low-parameter sojourn models, such as a shifted binomial model, a shifted Poisson model and convex combinations of shifted binomial models.