[Paper Review] Infinite Structured Hidden Semi-Markov Models

Jonathan H. Huggins, Frank Wood|arXiv (Cornell University)|Jun 30, 2014

Bayesian Methods and Mixture Models25 references6 citations

TL;DR

This paper introduces the Infinite Structured Hidden Semi-Markov Model (ISHSMM), a Bayesian nonparametric framework that enables explicit modeling of state dwell times in infinite hidden Markov models with structured transitions, such as left-to-right HMMs. It uses a Chinese Restaurant Process prior over states and a gamma-gamma conjugate prior for duration modeling, enabling automatic inference of unknown change points and persistent state durations without pre-specifying the number of states.

ABSTRACT

This paper reviews recent advances in Bayesian nonparametric techniques for constructing and performing inference in infinite hidden Markov models. We focus on variants of Bayesian nonparametric hidden Markov models that enhance a posteriori state-persistence in particular. This paper also introduces a new Bayesian nonparametric framework for generating left-to-right and other structured, explicit-duration infinite hidden Markov models that we call the infinite structured hidden semi-Markov model.

Motivation & Objective

• Address the challenge of unknown and potentially infinite state cardinality in hidden Markov models.
• Overcome the limitation of standard infinite HMMs in producing rapid state switching by encouraging long state dwell-times.
• Develop a general framework for constructing structured, explicit-duration infinite HMMs, including left-to-right HMMs.
• Enable nonparametric inference over change points in sequence data without assuming a fixed number of segments.
• Provide a scalable inference algorithm that extends existing Gibbs sampling techniques for infinite HMMs.

Proposed method

• Proposes the Infinite Structured HMM (ISHSMM) as a generative model with a Chinese Restaurant Process (CRP) prior over an infinite number of states.
• Introduces explicit duration modeling using a gamma-gamma conjugate prior for state dwell times, allowing direct control over persistence.
• Implements structured transition constraints, such as left-to-right ordering, by restricting state transitions to non-decreasing indices.
• Uses Gibbs sampling for posterior inference, leveraging conjugate priors and conditional distributions derived from the CRP and gamma-gamma hierarchy.
• Applies the model to both synthetic and real-world data, including coal mining disaster counts, with full posterior sampling over states, durations, and change points.
• Employs a nonparametric prior over emission distributions via a Dirichlet process, enabling flexible modeling of observed data.

Experimental results

Research questions

• RQ1 Can a Bayesian nonparametric model be constructed that explicitly models state dwell times while maintaining structured transition constraints?
• RQ2 How can infinite HMMs be extended to support left-to-right or other structured transition patterns without fixing the number of states?
• RQ3 To what extent can the ISHSMM automatically infer the correct number of change points in sequence data without prior assumptions?
• RQ4 How robust is the model to hyperparameter choices, particularly for the duration rate prior?
• RQ5 Can the ISHSMM outperform standard infinite HMMs in segmenting data with long, persistent states?

Key findings

• The ISHSMM successfully infers two dominant change points in the coal mining disaster dataset, located around years 40 and 100, consistent with prior studies.
• Posterior inference on synthetic data shows that the model accurately recovers true state means and duration rates, with high posterior confidence around the first four states.
• The posterior distribution for the final state’s duration rate exhibits high variance, as expected due to the unobserved end-of-sequence duration.
• The model remains robust to hyperparameter choices for the gamma prior on duration rates, producing qualitatively similar results across a wide range of αd values (20 to 200).
• The ISHSMM effectively avoids rapid state switching by encouraging long dwell times through explicit duration modeling, outperforming standard infinite HMMs in segmentation quality.
• The model supports nonparametric inference over both the number of states and the number of change points, enabling automatic discovery of segment structure in sequence data.