[Paper Review] Divide-and-Conquer Bayesian Inference in Hidden Markov Models

Chunlei Wang, Sanvesh Srivastava|arXiv (Cornell University)|May 30, 2021

Markov Chains and Monte Carlo Methods47 references4 citations

TL;DR

This paper proposes a divide-and-conquer Bayesian inference method for hidden Markov models (HMMs) with dependent data, overcoming the limitation of existing methods that assume independent observations. By modifying the likelihood using block-wise prediction filters and scaling by K, the method ensures subset posterior variances match the true posterior asymptotically, enabling accurate, scalable posterior inference via standard combination algorithms with theoretical guarantees in 1-Wasserstein distance.

ABSTRACT

Divide-and-conquer Bayesian methods consist of three steps: dividing the data into smaller computationally manageable subsets, running a sampling algorithm in parallel on all the subsets, and combining parameter draws from all the subsets. The combined parameter draws are used for efficient posterior inference in massive data settings. A major restriction of existing divide-and-conquer methods is that their first two steps assume that the observations are independent. We address this problem by developing a divide-and-conquer method for Bayesian inference in parametric hidden Markov models, where the state space is known and finite. Our main contributions are two-fold. First, after partitioning the data into smaller blocks of consecutive observations, we modify the likelihood for performing posterior computations on the subsets such that the posterior variances of the subset and true posterior distributions have the same asymptotic order. Second, if the number of subsets is chosen appropriately depending on the mixing properties of the hidden Markov chain, then we show that the subset posterior distributions defined using the modified likelihood are asymptotically normal as the subset sample size tends to infinity. The latter result also implies that we can use any existing combination algorithm in the third step. We show that the combined posterior distribution obtained using one such algorithm is close to the true posterior distribution in 1-Wasserstein distance under widely used regularity assumptions. Our numerical results show that the proposed method provides an accurate approximation of the true posterior distribution than its competitors in diverse simulation studies and a real data analysis.

Motivation & Objective

• To address the gap in divide-and-conquer Bayesian methods, which traditionally assume independent data, by extending them to dependent HMMs.
• To develop a likelihood modification that preserves asymptotic variance structure across subsets, ensuring valid posterior approximation.
• To establish theoretical conditions under which subset posteriors are asymptotically normal, enabling use of standard combination algorithms.
• To demonstrate empirically that the proposed block filtered posterior outperforms existing divide-and-conquer competitors in accuracy and efficiency on both simulated and real HMM data.
• To provide a scalable alternative to full-data MCMC and online EM for massive HMM inference, avoiding the computational bottlenecks of data augmentation and tuning challenges of stochastic methods.

Proposed method

• Partition the HMM data sequence into K consecutive blocks of size m = n/K to enable parallel computation.
• Define a modified conditional likelihood pθ(Y[j] | Y[j−1])^K on each subset, where the exponent K compensates for missing data and preserves asymptotic variance structure.
• Use the prediction filter to compute the modified likelihood, ensuring consistency with the true conditional likelihood as m and K grow.
• Perform parallel MCMC sampling on each subset using the modified posterior π(θ | Y[j], Y[j−1]) ∝ π(θ) × [pθ(Y[j] | Y[j−1])]^K.
• Combine posterior draws from all subsets using a standard combination algorithm (e.g., Wasserstein or moment-based methods).
• Theoretical justification relies on the mixing properties of the hidden Markov chain to determine the optimal number of subsets K.

Experimental results

Research questions

• RQ1 Can divide-and-conquer Bayesian inference be extended to HMMs with dependent observations, where standard methods assume independence?
• RQ2 How should the likelihood be modified on each subset to ensure that the subset posterior variance matches the true posterior variance asymptotically?
• RQ3 Under what conditions on the number of subsets K and subset size m is the subset posterior distribution asymptotically normal?
• RQ4 How close is the combined posterior distribution to the true posterior in terms of 1-Wasserstein distance?
• RQ5 Does the proposed block filtered posterior outperform existing divide-and-conquer and full-likelihood methods in accuracy and computational efficiency on HMMs with varying dependence structures?

Key findings

• The block filtered posterior distribution achieves 1-Wasserstein distance to the true posterior that is asymptotically negligible under standard regularity conditions, ensuring theoretical validity.
• For HMMs with weak dependence (S=2,3), the block filtered posterior matches Hamiltonian Monte Carlo (HMC) in accuracy, with median accuracy of 0.97–0.99 across n=10^4 and n=10^5.
• For strongly dependent HMMs (S=5,7), the block filtered posterior maintains median accuracy of 0.51–0.68, while competitors like WASP, DPMC, and PIE drop to near 0.00, demonstrating robustness to dependence.
• The block filtered posterior is more efficient than data augmentation (2.27–0.70 hours vs. 4.53 hours) and faster than HMC for K=n^{1/3} (0.70 hours) on n=10^5 data points.
• As K increases, accuracy decreases due to smaller subset sizes, but the block filtered posterior remains superior to competitors across all K values and HMM configurations.
• The method is robust across diverse simulation settings and real data, with maximum Monte Carlo error of 0.02 in posterior accuracy and 0.01 in real data analysis.