Mathematical Foundations for a Compositional Account of the Bayesian Brain

This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment of statistical games. A number of errors also fixed, and some presentation improved. Comments most welcom

Coalgebraic Behavior Analysis — From Qualitative To Quantitative Analyses

In order to specify and analyze the behavior of systems (computer programs, circuits etc.) it is important to have a suitable specification language. Although it is possible to define such a language separately for each type of system, it is desirable to have a standard toolbox that allows to do this in a generic way for various – possibly quite different – systems. Coalgebra, a concept of category theory, has proven to be a suitable framework to model transition systems. This class of systems includes many well-known examples like deterministic automata, nondeterministic automata or probabilistic systems. All these systems are coalgebras and their behavior can be analyzed via the notion of final coalgebra or other category theoretic constructions. This thesis investigates how to improve and build upon existing results to explore the expressive power of category theory and in particular coalgebra in behavioral analysis. The three main parts of the thesis all have a different focus but are strongly connected by the coalgebraic concepts used. Part one discusses adjunctions in the context of coalgebras. Here, well-known automata constructions such as the powerset-construction are (re)discovered as liftings of simple and well-known basic adjunctions. The second part deals with continuous generative probabilistic systems. It is shown that their trace semantics can be captured by a final coalgebra in a category of stochastic relations. The final contribution is a shift from qualitative to quantitative reasoning. Via the development of methods to lift functors on the category of sets and functions to functors on pseudometric spaces and nonexpansive functions it is possible to define a canonical, coalgebraic framework for behavioral pseudometrics.Um das Verhalten von Systemen (Computerprogrammen, Schaltkreisen etc.) zu spezifizieren und zu analysieren, ist es wichtig eine geeignete Spezifizierungssprache zu finden. Obwohl es möglich ist, eine solche Sprache separat für jeden Typ von System zu definieren, ist es erstrebenswert einen standardisierten Ansatz zu haben, welcher es ermöglicht, dies in einer generischen Weise für diverse – möglicherweise stark unterschiedliche – Systeme zu tun. Koalgebra, ein Konzept der Kategorientheorie, hat sich als geeignetes Modell zur Modellierung von Transitionssystemen herausgestellt. Diese Klasse von Systemen umfasst viele bekannte Beispiele wie deterministische, nichtdeterministische oder probabilistische Automaten. Alle diese Systeme sind Koalgebren und ihr Verhalten kann mittels finaler Koalgebren oder anderer kategorientheoretischer Konstruktionen analysiert werden. Diese Dissertation beschäftigt sich mit der Frage, wie existierende Resultate verbessert und erweitert werden können, um die Ausdrucksmächtigkeit von Kategorientheorie und besonders Koalgebra in der Verhaltensanalyse zu untersuchen. Die drei Hauptteile dieser Arbeit haben alle eine unterschiedliche Ausrichtung, sind aber durch die verwendeten koalgebraischen Methoden stark miteinander verbunden. Der erste Teil behandelt Adjunktionen im Kontext von Koalgebren. Hierbei werden übliche Automatenkonstruktionen wie die Potenzmengenkonstruktion als Lifting einfacher und wohlbekannter Basisadjunktionen (wieder-)entdeckt. Im zweiten Teil werden kontinuierliche, generative probabilististische Systeme betrachtet. Es wird gezeigt, dass deren lineares Verhalten von einer finalen Koalgebra in einer Kategorie stochastischer Relationen erfasst werden kann. Der letzte Beitrag ist ein Wechsel von qualitativer zu quantitativer Argumentation. Durch die Entwicklung von Methoden, die dazu dienen Funktoren von der Kategorie der Mengen und Funktionen zu Funktoren auf pseudometrischen Räumen und nicht-expansiven Funktionen zu erweitern, ist es möglich eine kanonische, koalgebraische Herangehensweise für Verhaltensmetriken zu entwerfen

Measures and all that --- A Tutorial

This tutorial gives an overview of some of the basic techniques of measure theory. It includes a study of Borel sets and their generators for Polish and for analytic spaces, the weak topology on the space of all finite positive measures including its metrics, as well as measurable selections. Integration is covered, and product measures are introduced, both for finite and for arbitrary factors, with an application to projective systems. Finally, the duals of the Lp-spaces are discussed, together with the Radon-Nikodym Theorem and the Riesz Representation Theorem. Case studies include applications to stochastic Kripke models, to bisimulations, and to quotients for transition kernels