We present a categorical formulation of the cognitive frameworks of
Predictive Processing and Active Inference, expressed in terms of string
diagrams interpreted in a monoidal category with copying and discarding. This
includes diagrammatic accounts of generative models, Bayesian updating,
perception, planning, active inference, and free energy. In particular we
present a diagrammatic derivation of the formula for active inference via free
energy minimisation, and establish a compositionality property for free energy,
allowing free energy to be applied at all levels of an agent's generative
model. Aside from aiming to provide a helpful graphical language for those
familiar with active inference, we conversely hope that this article may
provide a concise formulation and introduction to the framework.

我们提出了一种基于范畴的认知框架，使用具有复制和丢弃功能的范畴内解释的字符串图表达方式，其中包括生成模型、贝叶斯更新、感知、规划、主动推理和自由能量的图表表示。我们通过自由能量最小化的图表导出了主动推理的公式，并建立了自由能量的组合性属性，使得自由能量可以应用于代理模型的所有层次。除了为熟悉主动推理的人提供一个有用的图形语言外，我们也希望本文能够提供一个简洁的框架和介绍。

弦图中的主动推理：预测处理和自由能的范畴学解析

Active Inference in String Diagrams: A Categorical Account of Predictive  Processing and Free Energy

Finding parameters that minimise a loss function is at the core of many
machine learning methods. The Stochastic Gradient Descent algorithm is widely
used and delivers state of the art results for many problems. Nonetheless,
Stochastic Gradient Descent typically cannot find the global minimum, thus its
empirical effectiveness is hitherto mysterious. We derive a correspondence
between parameter inference and free energy minimisation in statistical
physics. The degree of undersampling plays the role of temperature. Analogous
to the energy-entropy competition in statistical physics, wide but shallow
minima can be optimal if the system is undersampled, as is typical in many
applications. Moreover, we show that the stochasticity in the algorithm has a
non-trivial correlation structure which systematically biases it towards wide
minima. We illustrate our argument with two prototypical models: image
classification using deep learning, and a linear neural network where we can
analytically reveal the relationship between entropy and out-of-sample error.

研究发现，针对机器学习中许多问题，大多数采用随机梯度下降算法，并能够在实践中实现最佳结果，但通常无法达到全局最小值，其实际效果至今仍是一个谜，本文研究了参数推断和统计物理中的自由能最小化之间的对应关系，证明了宽而浅的极小值在系统欠采样时是最优的，同时还显示该算法的随机性具有非平凡的相关结构，会使其有偏地偏向于宽的极小值。