Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.

翻译：预测编码（PC）为人工神经网络训练提供了基于局部和生物机制的替代反向传播方案，但至今仍存在收敛速度慢、且随网络深度增加性能急剧退化的问题。我们将这两个问题归因于一个简化假设：当前PC网络将精度矩阵固定为单位矩阵，摒弃了变分推导中实现快速、局部和贝叶斯特性所必需的精度加权预测误差。通过将预测编码网络表达为深度分层高斯滤波器（HGFs）并恢复精度加权消息传递，我们弥补了这一缺陷，从而在各层获得动态不确定性估计和符合赫布规则的更新范式。所得网络可在单一自由能目标下同时学习激活值、权重和精度，无需全局误差信号，且推理过程无需迭代或自动微分。在FashionMNIST数据集上，我们的方案在epoch级时间开销上接近反向传播的同时收敛更快，并在在线学习、数据效率和概念漂移任务中超越反向传播。由此证明，结合在线精度学习的闭式变分推断为深度预测编码网络提供了可扩展基础，在无需迭代松弛或全局误差信号的前提下，保留了生物可解释性优势。