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Fast Gibbs Sampling on Bayesian Hidden Markov Model with Missing Observations

Dongrong Li, Tianwei Yu, Xiaodan Fan

TL;DR

This paper tackles hidden Markov models with missing observations, where EM and standard Gibbs methods struggle with non-convexity and slow mixing. It introduces a collapsed Gibbs sampler that analytically marginalizes missing observations and their associated latent states, reducing the exploration space and per-iteration cost. The authors establish convergence (via spectral-gap arguments) and complexity benefits, showing a per-iteration cost of $O((1-p)nT)$ and improved ESS, particularly at high missingness, supported by simulations and real-data analyses. The approach yields comparable estimation accuracy while delivering substantial speedups and sampling efficiency, making it well-suited for large-scale sequential data with substantial missingness, such as electronic health records.

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).

Fast Gibbs Sampling on Bayesian Hidden Markov Model with Missing Observations

TL;DR

This paper tackles hidden Markov models with missing observations, where EM and standard Gibbs methods struggle with non-convexity and slow mixing. It introduces a collapsed Gibbs sampler that analytically marginalizes missing observations and their associated latent states, reducing the exploration space and per-iteration cost. The authors establish convergence (via spectral-gap arguments) and complexity benefits, showing a per-iteration cost of and improved ESS, particularly at high missingness, supported by simulations and real-data analyses. The approach yields comparable estimation accuracy while delivering substantial speedups and sampling efficiency, making it well-suited for large-scale sequential data with substantial missingness, such as electronic health records.

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).
Paper Structure (23 sections, 1 theorem, 29 equations, 3 figures, 8 tables, 2 algorithms)

This paper contains 23 sections, 1 theorem, 29 equations, 3 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Hidden Markov Models
  3. Notations
  4. Method
  5. Problem Formulation
  6. Description of the Collapsed Model
  7. Posterior Sampling
  8. Sampling from the Conditional of $\mathbf{z}_o$
  9. Sampling From the Conditional of the Parameters
  10. Predictive Distribution
  11. Forecasting
  12. Predictive Distribution from a New Sample
  13. Missing Observations Imputation
  14. Complexity Analysis
  15. Convergence Rate Analysis
  16. ...and 8 more sections

Key Result

Theorem 3.2

The spectral gaps of the three Gibbs samplers are ordered as:

Figures (3)

  • Figure 1: Time consumed per 1000 iterations of different samplers under different missing probabilities with the random missing mechanism. We only present the averaged time consumption because the computational complexity is deterministic (In fact, the standard deviation of computational time across different experiments is also very small, mainly caused by the jitter of the system). The line with circle markers represents the averaged consumed time of the partially collapsed Gibbs sampler and the line with diamond markers represents the consumed time of the vanilla Gibbs sampler. The average consumed time of the proposed collapsed sampler under different missing probabilities is specified by the line with a square markers. As the missing rate goes high, the proposed method becomes significantly faster than the competitive methods in terms of absolute time consumed per iteration under the random missing mechanism. Moreover, a linear decay in our method's computational time can be concluded from the figure, which verifies the results presented in complexity analysis.
  • Figure 2: Sequences with the blockwise missing pattern. White blocks represent missing entries in the observed sequences
  • Figure 3: Comparison of average time consumed per 1000 iterations over different samplers with the block missing mechanism. Since the computational complexity is deterministic and there is little uncertainty, we only present the mean (In fact, the true standard deviation is also rather small, mainly caused by system's jitter). The line with circle markers represents the averaged consumed time of the partially collapsed Gibbs sampler and the line with diamond markers represents the consumed time of the vanilla Gibbs sampler. The average consumed time of the proposed collapsed sampler under different missing probabilities is specified by the line with a square markers. The proposed method is significantly faster than any other existing methods when the missing probability is high with the block missing mechanism. Moreover, a linear decay in our method's computational time can be concluded from the figure, which verifies the results presented in complexity analysis.

Theorems & Definitions (3)

  • Definition 3.1
  • Theorem 3.2
  • Proof 1: Proof of Theorem \ref{['thm:rate']}