改进了分层高斯滤波中的波动率更新，避免精度为负的错误。
Robust volatility updates for Hierarchical Gaussian Filtering
- 通过插值两种能量展开式，重构波动率耦合节点的更新机制。
- 新方法在全参数空间内保持数值稳定，即使预测误差很大也有效。
- 适合需要高鲁棒性贝叶斯推断的研究者，如神经科学建模。
分层高斯滤波（HGF）网络可高效更新代理对环境隐状态的后验分布（信念）。HGF父节点可针对子节点的均值或方差进行调节。当输入节点接收到新信息时，会根据每个节点的一步更新方程，逐级传递信念更新，涉及均值和精度（方差倒数）的计算。然而，原始的方差目标父节点更新公式（波动率耦合）在某些参数区域会导致后验精度为负，这是逻辑上不可能的，导致算法终止报错。本文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，通过在先验预测点和通过拉姆伯特W函数解析求得的第二个极值点之间插值，避免了后验精度为负的问题。所得更新公式在整个参数空间中都具有鲁棒性，即使在大预测误差下也能准确追踪变分后验。
原文摘要 · Abstract (English)
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.
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