[Paper Review] Tree-based Particle Smoothing Algorithms in a Hidden Markov Model

Dong Ding, Axel Gandy|arXiv (Cornell University)|Aug 25, 2018

Target Tracking and Data Fusion in Sensor Networks21 references3 citations

TL;DR

This paper proposes a tree-based particle smoothing algorithm (TPS) that uses divide-and-conquer sequential Monte Carlo to estimate joint smoothing distributions in hidden Markov models. By constructing intermediate target distributions based on filtering or smoothing estimates, TPS achieves stable, low-complexity sampling with improved accuracy over conventional methods, especially in non-linear models with high transition variance.

ABSTRACT

We provide a new strategy built on the divide-and-conquer approach by Lindsten et al. (2017) to investigate the smoothing problem in a hidden Markov model. We employ this approach to decompose a hidden Markov model into sub-models with intermediate target distributions based on an auxiliary tree structure and produce independent samples from the sub-models at the leaf nodes towards the original model of interest at the root. We review the target distribution in the sub-models suggested by Lindsten et al. and propose two new classes of target distributions, which are the estimates of the (joint) filtering distributions and the (joint) smoothing distributions. The first proposed type is straightforwardly constructible by running a filtering algorithm in advance. The algorithm using the second type of target distributions has an advantage of roughly retaining the marginals of all random variables invariant at all levels of the tree at the cost of approximating the marginal smoothing distributions in advance. We further propose the constructions of these target distributions using pre-generated Monte Carlo samples. We show empirically the algorithms with the proposed intermediate target distributions give stable and comparable results as the conventional smoothing methods in a linear Gaussian model and a non-linear model.

Motivation & Objective

• Address the challenge of efficient and accurate smoothing in hidden Markov models where exact solutions are intractable.
• Overcome path degeneracy and high computational complexity in standard particle smoothing methods, which often scale quadratically with sample size.
• Develop a flexible, non-sequential approach to smoothing using a binary tree decomposition of the state space and intermediate target distributions.
• Improve proposal quality in importance sampling by incorporating filtering or smoothing estimates as intermediate targets.
• Enable scalable, stable, and accurate smoothing with reduced computational burden compared to conventional particle smoothers.

Proposed method

• Decompose the joint smoothing distribution into sub-models using an auxiliary binary tree structure, with leaf nodes representing individual states and internal nodes representing state blocks.
• Define intermediate target distributions at non-leaf nodes based on either unnormalized likelihoods (TPS-L), filtering distributions (TPS-EF), or smoothing distributions (TPS-ES).
• Use sequential resampling and merging via importance sampling to propagate samples from leaf nodes to the root, gradually approximating the full smoothing distribution.
• Construct intermediate targets using parametric (e.g., normal approximation) or non-parametric (e.g., piecewise constant density estimation) methods from pre-generated Monte Carlo samples.
• Adjust computational cost by tuning the number of particles at each level, enabling linear complexity relative to sample size.
• Implement the algorithm using a backward propagation strategy that maintains marginal distribution consistency across tree levels, especially in TPS-ES.

Experimental results

Research questions

• RQ1 Can a divide-and-conquer approach with tree-structured intermediate targets improve the stability and efficiency of particle smoothing in HMMs?
• RQ2 How do different intermediate target distributions—based on filtering, smoothing, or likelihood—impact the accuracy and computational cost of smoothing?
• RQ3 To what extent can parametric or non-parametric estimation of intermediate targets enhance proposal quality in importance sampling steps?
• RQ4 Does the proposed tree-based method outperform conventional particle smoothers in terms of mean square error and Kolmogorov–Smirnov statistics, especially in non-linear models?
• RQ5 Can the method achieve linear computational complexity while maintaining accuracy comparable to quadratic-cost smoothers?

Key findings

• TPS-EF (estimating filtering distributions) produced the smallest mean square errors in the linear Gaussian model and consistently lowest average Kolmogorov–Smirnov statistics in non-linear models.
• TPS-ES (estimating smoothing distributions) provided the best approximation of the true smoothing distribution, with significantly lower Kolmogorov–Smirnov statistics than TPS-EF, though at higher computational cost.
• TPS-L (likelihood-based targets) showed the smallest error in the linear model but exhibited highly unstable performance in non-linear settings due to poor initial proposal quality.
• In non-linear models with high transition variance relative to emission noise, TPS-EF outperformed other algorithms substantially, demonstrating robustness to model misspecification.
• The parametric and non-parametric constructions of intermediate targets enabled stable and accurate smoothing, with non-parametric methods showing strong performance in non-linear scenarios.
• The overall computational complexity of TPS can be reduced to linear in sample size, offering a significant advantage over standard particle smoothers with quadratic complexity.