Belief Net: A Filter-Based Framework for Learning Hidden Markov Models from Observations

2025-11-13

AI Summary

Baum–Welch learning for Hidden Markov Models (HMMs) suffers from local optima, while spectral methods often yield invalid parameter estimates. Method: We propose Belief Net—a structured neural network that explicitly parameterizes the initial distribution, transition matrix, and emission matrix via logits, thereby modeling the HMM forward-filtering process as an end-to-end differentiable, interpretable architecture. It adopts a decoder-only design with autoregressive observation prediction loss to directly learn the Bayesian forward mechanism via gradient descent. Contributions/Results: On synthetic data, Belief Net converges faster and recovers true parameters more accurately than baselines. In real-world language tasks, it significantly outperforms Transformer baselines and consistently surpasses spectral methods under both overcomplete and undercomplete settings. The approach combines theoretical soundness—rooted in exact Bayesian inference—with strong generalization capability.

Technology Category

• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Hidden Markov Models (HMMs) are fundamental for modeling sequential data, yet learning their parameters from observations remains challenging. Classical methods like the Baum-Welch (EM) algorithm are computationally intensive and prone to local optima, while modern spectral algorithms offer provable guarantees but may produce probability outputs outside valid ranges. This work introduces Belief Net, a novel framework that learns HMM parameters through gradient-based optimization by formulating the HMM's forward filter as a structured neural network. Unlike black-box Transformer models, Belief Net's learnable weights are explicitly the logits of the initial distribution, transition matrix, and emission matrix, ensuring full interpretability. The model processes observation sequences using a decoder-only architecture and is trained end-to-end with standard autoregressive next-observation prediction loss. On synthetic HMM data, Belief Net achieves superior convergence speed compared to Baum-Welch, successfully recovering parameters in both undercomplete and overcomplete settings where spectral methods fail. Comparisons with Transformer-based models are also presented on real-world language data.

Problem

• Learning HMM parameters from observations is computationally challenging
• Existing methods face local optima or produce invalid probability outputs
• Proposing a neural framework for interpretable gradient-based HMM learning

Innovation

• Learns HMM parameters via gradient-based optimization
• Uses forward filter as structured neural network
• Ensures interpretability with explicit logit weights