arXiv subjects

Toby St Clere Smithe

Publications and source records attributed to Toby St Clere Smithe.

11 recordsLinked to original sources

arXiv2025

Copy-composition for Probabilistic Graphical Models

In probabilistic modelling, joint distributions are often of more interest than their marginals, but the standard composition of stochastic channels is defined by marginalization. Last year at ACT, the notion of 'copy-composition' was introduced in order to circumvent this problem and express the chain rule of the relative entropy fibrationally, but while that goal was achieved, copy-composition lacked a satisfactory origin story. Here, we supply such a story for two standard probabilistic tools: directed and undirected graphical models. We explain that (directed) Bayesian networks may be understood as ''stochastic terms'' of product type, in which context copy-composition amounts to a pull-push operation. Likewise, we show that (undirected) factor graphs compose by copy-composition. In each case, our construction yields a double fibration of decorated (co)spans. Along the way, we introduce a useful bifibration of measure kernels, to provide semantics for the notion of stochastic term, which allows us to generalize probabilistic modelling from product to dependent types.

math.CT↗

arXiv2025

AutoBayes: A Compositional Framework for Generalized Variational Inference

We introduce a new compositional framework for generalized variational inference, clarifying the different parts of a model, how they interact, and how they compose. We explain that both exact Bayesian inference and the loss functions typical of variational inference (such as variational free energy and its generalizations) satisfy chain rules akin to that of reverse-mode automatic differentiation, and we advocate for exploiting this to build and optimize models accordingly. To this end, we construct a series of compositional tools: for building models; for constructing their inversions; for attaching local loss functions; and for exposing parameters. Finally, we explain how the resulting parameterized statistical games may be optimized locally, too. We illustrate our framework with a number of classic examples, pointing to new areas of extensibility that are revealed.

stat.ML↗

arXiv2025

On the Functoriality of Belief Propagation Algorithms on finite Partially Ordered Sets

Undirected graphical models are a widely used class of probabilistic models in machine learning that capture prior knowledge or putative pairwise interactions between variables. Those interactions are encoded in a graph for pairwise interactions; however, generalizations such as factor graphs account for higher-degree interactions using hypergraphs. Inference on such models, which is performed by conditioning on some observed variables, is typically done approximately by optimizing a free energy, which is an instance of variational inference. The Belief Propagation algorithm is a dynamic programming algorithm that finds critical points of that free energy. Recent efforts have been made to unify and extend inference on graphical models and factor graphs to more expressive probabilistic models. A synthesis of these works shows that inference on graphical models, factor graphs, and their generalizations relies on the introduction of presheaves and associated invariants (homology and cohomology groups).We propose to study the impact of the transformation of the presheaves onto the associated message passing algorithms. We show that natural transformations between presheaves associated with graphical models and their generalizations, which can be understood as coherent binning of the set of values of the variables, induce morphisms between associated message-passing algorithms. It is, to our knowledge, the first result on functoriality of the Loopy Belief Propagation.

math.ST↗

arXiv2024

Structured Active Inference (Extended Abstract)

We introduce structured active inference, a large generalization and formalization of active inference using the tools of categorical systems theory. We cast generative models formally as systems "on an interface", with the latter being a compositional abstraction of the usual notion of Markov blanket; agents are then 'controllers' for their generative models, formally dual to them. This opens the active inference landscape to new horizons, such as: agents with structured interfaces (e.g. with 'mode-dependence', or that interact with computer APIs); agents that can manage other agents; and 'meta-agents', that use active inference to change their (internal or external) structure. With structured interfaces, we also gain structured ('typed') policies, which are amenable to formal verification, an important step towards safe artificial agents. Moreover, we can make use of categorical logic to describe express agents' goals as formal predicates, whose satisfaction may be dependent on the interaction context. This points towards powerful compositional tools to constrain and control self-organizing ensembles of agents.

cs.AI↗

arXiv2024

Designing Ecosystems of Intelligence from First Principles

This white paper lays out a vision of research and development in the field of artificial intelligence for the next decade (and beyond). Its denouement is a cyber-physical ecosystem of natural and synthetic sense-making, in which humans are integral participants -- what we call ''shared intelligence''. This vision is premised on active inference, a formulation of adaptive behavior that can be read as a physics of intelligence, and which inherits from the physics of self-organization. In this context, we understand intelligence as the capacity to accumulate evidence for a generative model of one's sensed world -- also known as self-evidencing. Formally, this corresponds to maximizing (Bayesian) model evidence, via belief updating over several scales: i.e., inference, learning, and model selection. Operationally, this self-evidencing can be realized via (variational) message passing or belief propagation on a factor graph. Crucially, active inference foregrounds an existential imperative of intelligent systems; namely, curiosity or the resolution of uncertainty. This same imperative underwrites belief sharing in ensembles of agents, in which certain aspects (i.e., factors) of each agent's generative world model provide a common ground or frame of reference. Active inference plays a foundational role in this ecology of belief sharing -- leading to a formal account of collective intelligence that rests on shared narratives and goals. We also consider the kinds of communication protocols that must be developed to enable such an ecosystem of intelligences and motivate the development of a shared hyper-spatial modeling language and transaction protocol, as a first -- and key -- step towards such an ecology.

cs.AI↗

arXiv2023

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.

q-bio.NC↗

arXiv2023

Approximate Inference via Fibrations of Statistical Games

We characterize a number of well known systems of approximate inference as loss models: lax sections of 2-fibrations of statistical games, constructed by attaching internally-defined loss functions to Bayesian lenses. Our examples include the relative entropy, which constitutes a strict section, and whose chain rule is formalized by the horizontal composition of the 2-fibration. In order to capture this compositional structure, we first introduce the notion of 'copy-composition', alongside corresponding bicategories through which the composition of copy-discard categories factorizes. These bicategories are a variant of the Copara construction, and so we additionally introduce coparameterized Bayesian lenses, proving that coparameterized Bayesian updates compose optically, as in the non-coparameterized case.

math.CT↗

arXiv2023

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

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.

math.CT↗

arXiv2023

The Compositional Structure of Bayesian Inference

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.

math.CT↗

arXiv2022

Polynomial Life: the Structure of Adaptive Systems

We extend our earlier work on the compositional structure of cybernetic systems in order to account for the embodiment of such systems. All their interactions proceed through their bodies' boundaries: sensations impinge on their surfaces, and actions correspond to changes in their configurations. We formalize this morphological perspective using polynomial functors. The 'internal universes' of systems are shown to constitute an indexed category of statistical games over polynomials; their dynamics form an indexed category of behaviours. We characterize 'active inference doctrines' as indexed functors between such categories, resolving a number of open problems in our earlier work, and pointing to a formalization of the 'free energy principle' as adjoint to such doctrines. We illustrate our framework through fundamental examples from biology, including homeostasis, morphogenesis, and autopoiesis, and suggest a formal connection between spatial navigation and the process of proof.

nlin.AO↗

arXiv2021

Cyber Kittens, or Some First Steps Towards Categorical Cybernetics

We define a categorical notion of cybernetic system as a dynamical realisation of a generalized open game, along with a coherence condition. We show that this notion captures a wide class of cybernetic systems in computational neuroscience and statistical machine learning, exposes their compositional structure, and gives an abstract justification for the bidirectional structure empirically observed in cortical circuits. Our construction is built on the observation that Bayesian updates compose optically, a fact which we prove along the way, via a fibred category of state-dependent stochastic channels.

cs.NE↗