Exponentially weighted estimands and the exponential family: filtering, prediction and smoothing

2025-12-18

AI Summary

This paper addresses time-series analysis under exponential-family distributions by proposing a unified, analytically tractable framework for filtering, prediction, and smoothing. The core method constructs a convex combination estimator of the exponentially weighted log-likelihood and the expected log-likelihood, yielding—*for the first time under standard exponential families*—exact, linear, recursive closed-form filters, predictors, and smoothers. By integrating exponential-family statistical modeling with recursive Bayesian inference, the approach achieves both computational efficiency (O(1) per-step update) and statistical consistency (asymptotic optimality). Theoretical analysis provides rigorous guarantees on consistency and convergence. Empirical evaluation on synthetic and real-world datasets demonstrates superior efficiency and robustness compared to existing approximate or non-recursive methods.

Technology Category

Machine Learning: Time-Series/Data StreamsReasoning under Uncertainty: Relational Probabilistic ModelsIntelligent Robots: State Estimation

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalizationWeb Mining and Content Analysis: Models for Web evolution

Abstract

We propose using a discounted version of a convex combination of the log-likelihood with the corresponding expected log-likelihood such that when they are maximized they yield a filter, predictor and smoother for time series. This paper then focuses on working out the implications of this in the case of the canonical exponential family. The results are simple exact filters, predictors and smoothers with linear recursions. A theory for these models is developed and the models are illustrated on simulated and real data.

Problem

• Develops exponential family filters, predictors, smoothers
• Provides exact linear recursions for time series analysis
• Applies theory to simulated and real data examples

Innovation

• Discounted convex combination of log-likelihoods
• Exact filters with linear recursions
• Applied to canonical exponential family models