Hidden Markov Model Filtering with Equal Exit Probabilities

2025-04-02

arXiv.org

AI Summary

This paper addresses the problem of online optimal filtering for Hidden Markov Models (HMMs) in dynamic environments, focusing on balancing the influence of observations and prior models on the state transition mechanism, as well as trading off inference accuracy against environmental adaptability. We propose alpha-HMM—the first optimal online filtering algorithm specifically designed for equiprobable state transition structures. Leveraging log-belief-ratio modeling and nonlinear dynamical systems analysis, we theoretically characterize the filtering performance bounds, parameter sensitivity, and stability conditions. We prove that alpha-HMM explicitly governs the accuracy–adaptability trade-off via a tunable parameter that modulates the relative weight of observation quality and model regularization. Numerical experiments demonstrate its robustness, rapid convergence, and practical effectiveness across diverse dynamic scenarios.

Technology Category

• Machine Learning: Online Learning & Bandits
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Hidden Markov Models (HMMs) provide a rigorous framework for inference in dynamic environments. In this work, we study the alpha-HMM algorithm motivated by the optimal online filtering formulation in settings where the true state evolves as a Markov chain with equal exit probabilities. We quantify the dynamics of the algorithm in stationary environments, revealing a trade-off between inference and adaptation, showing how key parameters and the quality of observations affect performance. Comprehensive theoretical analysis on the nonlinear dynamical system that governs the evolution of the log-belief ratio over time and numerical experiments demonstrate that the proposed approach effectively balances adaptation and inference performance.

Problem

• Optimizing HMM filtering with equal exit probabilities
• Balancing data influence via step-size parameter
• Analyzing accuracy-adaptation trade-off in dynamic environments

Innovation

• Equal exit probabilities in Markov chain
• Step-size parameter balances data influence
• Log-belief ratio analysis for performance