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.

翻译：分层高斯滤波（HGF）网络能够有效更新关于智能体环境中隐藏状态的后验分布（信念）。HGF父节点可针对子节点的均值或方差进行调控。输入节点接收新信息时，会根据各节点均值与精度（逆方差）的单步更新方程引发网络中信念更新的级联效应。然而，在参数空间的某些区域，原始形式的方差调控父节点（波动率耦合）更新方程可能导致后验精度为负值——这种逻辑矛盾会使更新算法因错误终止。本报告提出一种修正的二次近似方法，用于波动率耦合节点的变分能量，从而避免负后验精度。核心思想是在两种变分能量二次展开之间进行插值：一种基于先验预测，另一种基于通过朗伯W函数闭式求解的第二模态位置。由此所得更新方程在整个参数空间内保持鲁棒性，且即使面对较大预测误差也能准确追踪变分后验。