We consider the problem of symmetrising a neural network along a group
homomorphism: given a homomorphism $\varphi : H \to G$, we would like a
procedure that converts $H$-equivariant neural networks into $G$-equivariant
ones. We formulate this in terms of Markov categories, which allows us to
consider neural networks whose outputs may be stochastic, but with
measure-theoretic details abstracted away. We obtain a flexible, compositional,
and generic framework for symmetrisation that relies on minimal assumptions
about the structure of the group and the underlying neural network
architecture. Our approach recovers existing methods for deterministic
symmetrisation as special cases, and extends directly to provide a novel
methodology for stochastic symmetrisation also. Beyond this, we believe our
findings also demonstrate the utility of Markov categories for addressing
problems in machine learning in a conceptual yet mathematically rigorous way.

将提出的方法应用于神经网络，通过统计学的抽象化，使用 Markov 范畴模型，将 H - 等变神经网络转化为 G - 等变神经网络，可以灵活、组合且适用于确定性和随机对称化方法。

马尔可夫范畴中的随机神经网络对称化

Stochastic Neural Network Symmetrisation in Markov Categories

A growing body of research on probabilistic programs and causal models has
highlighted the need to reason compositionally about model classes that extend
directed graphical models. Both probabilistic programs and causal models define
a joint probability density over a set of random variables, and exhibit sparse
structure that can be used to reason about causation and conditional
independence. This work builds on recent work on Markov categories of
probabilistic mappings to define a category whose morphisms combine a joint
density, factorized over each sample space, with a deterministic mapping from
samples to return values. This is a step towards closing the gap between recent
category-theoretic descriptions of probability measures, and the operational
definitions of factorized densities that are commonly employed in probabilistic
programming and causal inference.

通过 Markov categories 定义无向图模型上的联合概率密度和样本结果的确定性映射的组合的范畴，这是将概率编程和因果推断中的因子化密度的操作定义与概率测度的范畴描述结合的一步。

具分解密度的字符串图

String Diagrams with Factorized Densities

We develop Markov categories as a framework for synthetic probability and
statistics, following work of Golubtsov as well as Cho and Jacobs. This means
that we treat the following concepts in purely abstract categorical terms:
conditioning and disintegration; various versions of conditional independence
and its standard properties; conditional products; almost surely; sufficient
statistics; versions of theorems on sufficient statistics due to
Fisher--Neyman, Basu, and Bahadur.
Besides the conceptual clarity offered by our categorical setup, its main
advantage is that it provides a uniform treatment of various types of
probability theory, including discrete probability theory, measure-theoretic
probability with general measurable spaces, Gaussian probability, stochastic
processes of either of these kinds, and many others.

使用马尔可夫范畴作为框架，以纯抽象范畴术语处理标准概率统计概念，并为各种类型的概率理论提供统一的处理方法，包括离散概率、测度论概率、高斯概率、这些种类的随机过程等。