Fast Gibbs Sampling on Bayesian Hidden Markov Model with Missing Observations

TL;DR

This paper introduces a collapsed Gibbs sampler for Bayesian Hidden Markov Models with missing data, offering faster computation and higher sampling efficiency compared to existing methods, especially with many missing entries.

Contribution

The paper proposes a novel collapsed Gibbs sampling method that efficiently handles missing observations in HMMs, improving speed and sampling quality over traditional approaches.

Findings

Achieves comparable estimation accuracy to existing methods.

Produces larger Effective Sample Size (ESS) per iteration.

Significantly reduces computational complexity with many missing entries.

Abstract

The Hidden Markov Model (HMM) is a widely-used statistical model for handling sequential data. However, the presence of missing observations in real-world datasets often complicates the application of the model. The EM algorithm and Gibbs samplers can be used to estimate the model, yet suffering from various problems including non-convexity, high computational complexity and slow mixing. In this paper, we propose a collapsed Gibbs sampler that efficiently samples from HMMs' posterior by integrating out both the missing observations and the corresponding latent states. The proposed sampler is fast due to its three advantages. First, it achieves an estimation accuracy that is comparable to existing methods. Second, it can produce a larger Effective Sample Size (ESS) per iteration, which can be justified theoretically and numerically. Third, when the number of missing entries is large, the sampler has a significant smaller computational complexity per iteration compared to other methods, thus is faster computationally. In summary, the proposed sampling algorithm is fast both computationally and theoretically and is particularly advantageous when there are a lot of missing entries. Finally, empirical evaluations based on numerical simulations and real data analysis demonstrate that the proposed algorithm consistently outperforms existing algorithms in terms of time complexity and sampling efficiency (measured in ESS).