This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and joint probabilities, in the Kleisli category of the distribution monad. The conditional probabilities are induced by a map together with a predicate (the condition). The latter is a predicate in the logic of effect modules on this Kleisli category.
This same approach can be transferred to the category of C*-algebras (with positive unital maps), whose predicate logic is also expressed in terms of effect modules. Conditional probabilities can again be expressed via a triangle-fill-in property. In the literature, there are several proposals for what quantum conditional probability should be, and also there are extra difficulties not present in the classical case. At this stage, we only describe quantum systems with classical parametrization.
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 trea…
Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, il…
Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy
Quantinuum（昆蒂努姆）；Association for Mathematical Consciousness Science（数学意识科学协会）；Ludwig Maximilian University of Munich（慕尼黑路德维希-马克西米利安大学）；University of Bamberg（班贝格大学）；Topos Institute（托波斯研究所）；VERSES Research（VERSES研究院）
Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of arbitrary von Neumann algebras. We give a 2-{$C^*$}-categorical/planar algebraic description of normal faithful conditional expectations $E:\mathcal{M}\to\mathcal{N}\subset\mathcal{M}$ with finite index and their duals $E':\mathcal{N}'\to\mathcal{M}'\subset\mathcal{N}'$ by means of the solutions of the conjugate equations for the inclusion morphism $ι:\mathcal{N}\to\mathcal{M}$ and its conjugate morphism $\overlineι:\mathcal{M}\to\mathcal{N}$. In particular, the theory of index for conditional expectations admits a 2-{$C^*$}-categori…
The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply both to measure-theoretic probabi…
This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the derived $\infty$-category of a commutative ring, before turning to our main example, the $\infty$-category of spectra. We then go on to consider ring spectra and their $\infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and …
Given two measurable spaces $H$ and $D$ with countably generated $σ$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $μ: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ μ$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measu…
Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in $μ$-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible - and useful - to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a…
University of Oxford（牛津大学）；Queen Mary University of London（伦敦大学玛丽皇后学院）
Originally inspired by categorical quantum mechanics (Abramsky and Coecke, LiCS'04), the categorical compositional distributional model of natural language meaning of Coecke, Sadrzadeh and Clark provides a conceptually motivated procedure to compute the meaning of a sentence, given its grammatical structure within a Lambek pregroup and a vectorial representation of the meaning of its parts. The predictions of this first model have outperformed that of other models in mainstream empirical language processing tasks on large scale data. Moreover, just like CQM allows for varying the model in whic…
In recent work, symmetric dagger-monoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of "quantization" of a classical theory to "classicization" of …
