Robust volatility updates for Hierarchical Gaussian Filtering

2026-05-01

AI Summary

This work addresses a critical instability in hierarchical Gaussian filtering, where variance coupling with parent nodes can yield negative posterior precisions and cause algorithmic failure in certain parameter regimes. To resolve this issue, the authors propose an improved variational energy approximation based on quadratic expansions at two key locations: the prior predictive mode and a second mode analytically determined via the Lambert W function. By interpolating between these expansions, they construct a robust node update mechanism that integrates variational inference, quadratic approximation, and interpolation strategies. This approach guarantees non-negative posterior precision across the entire parameter space. Experimental results demonstrate that the proposed algorithm remains stable under all parameter conditions and accurately tracks the variational posterior distribution even in the presence of large prediction errors.

Abstract

Hierarchical Gaussian Filtering (HGF) networks allow for efficient updating of posterior distributions (beliefs) about hidden states of an agent's environment. HGF parent nodes can target the mean or variance of their children. New information entering at input nodes leads to a cascade of belief updates across the network according to one-step update equations for each node's mean and precision (inverse variance). However, the original form of the update equations for variance-targeting parents(volatility coupling) can in some regions of parameter space lead to negative posterior precision, a logical impossibility which causes the updating algorithm to terminate with an error. In this report, we introduce a modified quadratic approximation to the variational energy of volatility-coupled nodes that avoids negative posterior precision. The key idea is to interpolate between two quadratic expansions of the variational energy: one at the prior prediction and one at a second mode whose location is obtained in closed form via the Lambert W function. The resulting update equations are robust across the entire parameter space and faithfully track the variational posterior even for large prediction errors.

Problem

• Hierarchical Gaussian Filtering
• volatility coupling
• posterior precision
• variational energy
• negative precision

Innovation

• Hierarchical Gaussian Filtering
• volatility coupling
• variational energy
• Lambert W function
• robust update equations