[Paper Review] A Linear-Time Particle Gibbs Sampler for Infinite Hidden Markov Models

Nilesh Tripuraneni, Shane Gu+2 more|arXiv (Cornell University)|May 3, 2015

Bayesian Methods and Mixture Models10 references3 citations

TL;DR

This paper proposes a linear-time Particle Gibbs sampler for Infinite Hidden Markov Models (iHMMs) that uses an optimized importance proposal and ancestor sampling to efficiently resample latent state trajectories. The method achieves faster convergence and linear time complexity O(TNK), outperforming quadratic-time competitors on synthetic and real-world data.

ABSTRACT

Infinite Hidden Markov Models (iHMM's) are an attractive, nonparametric generalization of the classical Hidden Markov Model which can automatically infer the number of hidden states in the system. However, due to the infinite-dimensional nature of transition dynamics performing inference in the iHMM is difficult. In this paper, we present an infinite-state Particle Gibbs (PG) algorithm to resample state trajectories for the iHMM. The proposed algorithm uses an efficient proposal optimized for iHMMs and leverages ancestor sampling to suppress degeneracy of the standard PG algorithm. Our algorithm demonstrates significant convergence improvements on synthetic and real world data sets. Additionally, the infinite-state PG algorithm has linear-time complexity in the number of states in the sampler, while competing methods scale quadratically.

Motivation & Objective

• To address the challenge of inefficient inference in infinite-state Hidden Markov Models due to high-dimensional transition dynamics.
• To develop a scalable MCMC inference method that avoids the quadratic time complexity of existing beam-sampling and Gibbs sampling approaches.
• To improve mixing and convergence speed in iHMMs by leveraging particle-based methods with ancestor sampling.
• To enable exact inference in the sticky iHMM with nonparametric emission models, a gap in prior work.
• To achieve linear-time complexity O(TNK) in the number of active states, enabling scalability to longer sequences.

Proposed method

• Proposes an infinite-state Particle Gibbs (PG) algorithm using a stick-breaking construction of the Hierarchical Dirichlet Process (HDP) for iHMMs.
• Designs an optimal importance proposal for resampling latent state trajectories, improving sampling efficiency.
• Incorporates ancestor sampling to suppress particle degeneracy, enhancing mixing and convergence.
• Leverages dynamic programming via auxiliary slice variables to jointly resample trajectories, avoiding poor mixing in standard PG.
• Uses conjugate priors and conditional conjugacy to enable efficient resampling of emission parameters φ, transition matrices π, and hyperparameters α, γ.
• Employs a truncated representation of the infinite state space, dynamically adjusting to active states K, ensuring O(TNK) time complexity.

Experimental results

Research questions

• RQ1 Can a particle Gibbs sampler be designed for iHMMs that achieves linear-time complexity in the number of active states?
• RQ2 Does the use of an optimized importance proposal and ancestor sampling significantly improve mixing and convergence in iHMM inference?
• RQ3 Can the proposed method enable exact inference in the sticky iHMM with nonparametric emission models, where prior methods failed?
• RQ4 How does the proposed PG sampler compare in convergence speed and scalability to beam sampling and standard Gibbs samplers on synthetic and real-world data?
• RQ5 What is the computational complexity of the proposed method, and does it scale favorably with sequence length and active state count?

Key findings

• The proposed infinite-state Particle Gibbs sampler achieves linear time complexity O(TNK), outperforming quadratic-time methods like beam sampling.
• Empirical results show significant convergence improvements on both synthetic and real-world datasets compared to competing inference algorithms.
• The use of an optimized importance proposal and ancestor sampling effectively suppresses particle degeneracy, leading to faster mixing and better posterior exploration.
• The method enables exact inference in the sticky iHMM with nonparametric DP emission models, a previously unsolved problem.
• On real-world speaking diarization tasks, the method achieves state-of-the-art performance, outperforming truncated, biased samplers.
• The algorithm maintains high efficiency even for long sequences, demonstrating strong scalability due to its linear-time design.