Parameter redundancy and identifiability in hidden Markov models

Abstract

Hidden Markov models are a flexible class of models that can be used to describe time series data which depends on an unobservable Markov process. As with any complex model, it is not always obvious whether all the parameters are identifiable, or if the model is parameter redundant; that is, the model can be reparameterised in terms of a smaller number of parameters. This paper considers different methods for detecting parameter redundancy and identifiability in hidden Markov models. We examine both numerical methods and methods that involve symbolic algebra. These symbolic methods require a unique representation of a model, known as an exhaustive summary. We provide an exhaustive summary for hidden Markov models and show how it can be used to investigate identifiability.

Uncovering ecological state dynamics with hidden Markov models

“…For maximum likelihood estimation, the risk of false convergence to a local rather than the global maximum of the likelihood must not be underestimated. In addition to the general advice to avoid overly complex models (Lavine, 2010; Cole, 2019), the main strategy to reduce this risk is to try many initial parameter vectors within the maximisation.…”

A guide to state–space modeling of ecological time series

“…44) for the first observation (y 1 ), the second element is the marginal likelihood for the first two observations (y 1:2 ), etc. This straightforward exhaustive summary works well for HMMs (Cole, 2019), but can be impractical for SSMs with continuous states as it involves integration. Suitable, but more complex to derive, exhaustive summaries for SSMs are given in Cole and McCrea (2016).…”

Guest editor’s introduction to the special issue on “Hidden Markov Models: Theory and Applications”

“…Cole [7] describes three different approaches for investigating identifiability and detecting parameter redundancy in the context of HMMs. Given the fairly complex structure of HMMs in many applied settings, with not only two stochastic processes but also different dependence structures that can be assumed, the issue of estimability in general is clearly of much interest when working with HMMs.…”

“…Given the fairly complex structure of HMMs in many applied settings, with not only two stochastic processes but also different dependence structures that can be assumed, the issue of estimability in general is clearly of much interest when working with HMMs. Two of the procedures discussed in Cole [7], the Hessian method and the log-likelihood profile method, are based on numerical techniques, while the third uses symbolic algebra. The paper provides guidance on the practical feasibility of these approaches.…”