[Paper Review] Online state and parameter estimation in Dynamic Generalised Linear Models

Rui Miguel Vieira, Darren J. Wilkinson|arXiv (Cornell University)|Aug 30, 2016

Gaussian Processes and Bayesian Inference15 references4 citations

TL;DR

This paper proposes online Bayesian inference for Dynamic Generalized Linear Models (DGLMs) using Sequential Monte Carlo (SMC) methods, comparing sufficient statistics-based approaches with Liu and West particle filtering. It demonstrates that sufficient statistics methods delay particle impoverishment, deliver lower MSE in state and parameter estimation, and enable accurate long-horizon forecasts—outperforming Liu and West in accuracy despite higher computational cost, especially on streaming data with complex temporal structures.

ABSTRACT

Inference for streaming time-series is tightly coupled with the problem of Bayesian on-line state and parameter inference. In this paper we will introduce Dynamic Generalised Linear Models, the class of models often chosen to model continuous and discrete time-series data. We will look at three different approaches which allow on-line estimation and analyse the results when applied to different real world datasets related to inference for streaming data. Sufficient statistics based methods delay known problems, such as particle impoverishment, especially when applied to long running time-series, while providing reasonable parameter estimations when compared to exact methods, such as Particle Marginal Metropolis-Hastings. State and observation forecasts will also be analysed as a performance metric. By benchmarking against a "gold standard" (off-line) method, we can better understand the performance of on-line methods in challenging real-world scenarios.

Motivation & Objective

• To address the challenge of real-time Bayesian inference in streaming time-series data with complex, non-linear patterns.
• To evaluate and compare the performance of online SMC methods—specifically sufficient statistics-based filters, Storvik’s method, and Liu and West particle filtering—on real-world datasets.
• To assess trade-offs between computational efficiency, estimation accuracy, and robustness to particle impoverishment in long-running time-series.
• To benchmark online methods against a gold-standard off-line method (Particle Marginal Metropolis-Hastings) for state and parameter estimation.
• To evaluate the predictive performance of state and observation forecasts under different online inference strategies.

Proposed method

• Uses Dynamic Generalized Linear Models (DGLMs) to model time-series data with flexible observation and system distributions, including Poisson and Gaussian families.
• Applies Sequential Monte Carlo (SMC) with resampling to perform online state and parameter estimation in a Bayesian framework.
• Employs sufficient statistics-based filtering to decouple parameter estimation from the state space, reducing particle degeneracy and delaying impoverishment.
• Implements three online inference algorithms: Liu and West (LW) particle filtering, Storvik’s auxiliary particle filter, and a sufficient statistics-based SMC method.
• Utilizes a log-normal prior for variance parameters and inverse-Wishart priors for covariance matrices to enable conjugate updating in the DGLM framework.
• Employs a smoothing parameter (δ = 0.98) in the LW filter to stabilize parameter updates and reduce variance in particle weights.

Experimental results

Research questions

• RQ1 How do sufficient statistics-based SMC methods compare to Liu and West particle filtering in terms of state estimation accuracy and robustness to particle impoverishment in long-running time-series?
• RQ2 To what extent do online SMC methods provide accurate parameter estimates when benchmarked against the gold-standard Particle Marginal Metropolis-Hastings (PMMH) method?
• RQ3 Can sufficient statistics-based methods maintain reliable state and observation forecasts over long horizons (e.g., 6 days) in real-world streaming data?
• RQ4 What is the trade-off between computational cost and estimation accuracy across different online SMC methods in DGLMs?
• RQ5 How do different resampling strategies and particle count (Np) affect the stability and performance of online inference in DGLMs?

Key findings

• Sufficient statistics-based SMC methods achieved significantly lower mean squared error (MSE) in state estimation—e.g., 0.2609 for θ¹—compared to Liu and West (3.283), indicating superior accuracy.
• The Liu and West filter suffered from early particle collapse, leading to poor parameter estimation, with posterior variance estimates deviating markedly from PMMH benchmarks.
• Sufficient statistics methods delayed particle impoverishment, maintaining stable parameter estimates even after 5000 iterations, unlike Liu and West which collapsed after a few hundred iterations.
• State forecasts over a 600-step horizon (≈6 days) were accurate and captured seasonal patterns only with sufficient statistics-based methods, while Liu and West failed to track the trend.
• At Np = 1000, sufficient statistics methods achieved MSE below 0.3 for most states, demonstrating robustness at low particle counts with sub-millisecond computation per iteration.
• Despite higher computational cost, Liu and West outperformed others in speed (4.626 ms/iteration), but its poor estimation quality made it unsuitable for reliable inference.