Hierarchical Bayesian models of perception and learning feature prominently in contemporary cognitive neuroscience where, for example, they inform computational concepts of mental disorders. This includes predictive coding and hierarchical Gaussian filtering (HGF), which differ in the nature of hierarchical representations. Predictive coding assumes that higher levels in a given hierarchy influence the state (value) of lower levels. In HGF, however, higher levels determine the rate of change at lower levels. Here, we extend the space of generative models underlying HGF to include a form of nonlinear hierarchical coupling between state values akin to predictive coding and artificial neural networks in general. We derive the update equations corresponding to this generalization of HGF and conceptualize them as connecting a network of (belief) nodes where parent nodes either predict the state of child nodes or their rate of change. This enables us to (1) create modular architectures with generic computational steps in each node of the network, and (2) disclose the hierarchical message passing implied by generalized HGF models and to compare this to comparable schemes under predictive coding. We find that the algorithmic architecture instantiated by the generalized HGF is largely compatible with that of predictive coding but extends it with some unique predictions which arise from precision and volatility related computations. Our developments enable highly flexible implementations of hierarchical Bayesian models for empirical data analysis and are available as open source software.

翻译：感知与学习的分层贝叶斯模型在当代认知神经科学中占据重要地位，例如用于构建精神疾病的计算概念。这包括预测编码和分层高斯滤波（HGF），两者在分层表征的性质上有所不同。预测编码假设给定层级结构中较高层级会影响较低层级的状态（值）。但在HGF中，较高层级决定较低层级的变化速率。本文扩展了HGF所依赖的生成模型空间，纳入了类似于预测编码和通用人工神经网络的状态值之间的非线性分层耦合形式。我们推导了对应于这种HGF泛化的更新方程，并将其概念化为连接信念节点网络，其中父节点要么预测子节点的状态，要么预测其变化速率。这使得我们能够：（1）构建模块化架构，在网络每个节点中实现通用计算步骤；（2）揭示广义HGF模型隐含的分层消息传递机制，并将其与预测编码下的可比方案进行比较。我们发现广义HGF实例化的算法架构与预测编码基本兼容，但通过精度和波动性相关计算产生的独特预测对其进行了扩展。我们的进展使得针对实证数据分析的分层贝叶斯模型能够实现高度灵活的部署，并以开源软件形式提供。