titleA Synthetic Approach to Markov Kernels, Conditional Independence and Theorems on Sufficient Statistics
18D10,62A01,68Q55,act2022,cs.LO,dphil,math.CT,math.PR,math.ST,Primary: 60A05,Secondary: 62B05,stat.TH

abstractWe generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian inversion.
eprintclasscs, math, stat
shorttitleDependent Bayesian Lenses
titleDependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion
il,Mathematics - Category Theory,Mathematics - Probability,Mathematics - Statistics Theory

abstractEffectus theory is a new branch of categorical logic that aims to capture the essentials of quantum logic, with probabilistic and Boolean logic as special cases. Predicates in effectus theory are not subobjects having a Heyting algebra structure, like in topos theory, but `characteristic' functions, forming effect algebras. Such effect algebras are algebraic models of quantitative logic, in which double negation holds. Effects in quantum theory and fuzzy predicates in probability theory form examples of effect algebras. This text is an account of the basics of effectus theory. It includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. A basic result says that effectuses can be described equivalently in both `total' and `partial' form. So-called `commutative' and `Boolean' effectuses are distinguished, for probabilistic and classical models. It is shown how these Boolean effectuses are essentially extensive categories. A large part of the theory is devoted to the logical notions of comprehension and quotient, which are described abstractly as right adjoint to truth, and as left adjoint to falisity, respectively. It is illustrated how comprehension and quotients are closely related to measurement. The paper closes with a section on `non-commutative' effectus theory, where the appropriate formalisation is not entirely clear yet.
titleAn Introduction to Effectus Theory

Springer Berlin Heidelberg

abstractWe show that a measure-based denotational semantics for probabilistic programming is commutative. The idea underlying probabilistic programming languages (Anglican, Church, Hakaru, etc.) is that programs express statistical models as a combination of prior distributions and likelihood of observations. The product of prior and likelihood is an unnormalized posterior distribution, and the inference problem is to find the normalizing constant. One common semantic perspective is thus that a probabilistic program is understood as an unnormalized posterior measure, in the sense of measure theory, and the normalizing constant is the measure of the entire semantic domain. A programming language is said to be commutative if only data flow is meaningful; control flow is irrelevant, and expressions can be re-ordered. It has been unclear whether probabilistic programs are commutative because it is well-known that Fubini-Tonelli theorems for reordering integration fail in general. We show that probabilistic programs are in fact commutative, by characterizing the measures/kernels that arise from programs as ‘s-finite’, i.e. sums of finite measures/kernels. The result is of theoretical interest, but also of practical interest, because program transformations based on commutativity help with symbolic inference and can improve the efficiency of simulation.
booktitleProgramming Languages and Systems
titleCommutative Semantics for Probabilistic Programming

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleCategorical Aspects of Topology and Analysis
isbn978-3-540-11211-2 978-3-540-39041-1
titleA Categorical Approach to Probability Theory

abstractIn this note, we develop some of the basic theory of s-finite (measures and) kernels, a little-studied class of kernels which Staton has recently convincingly argued is precisely the semantic counterpart of (first-order) probabilistic programs. We discuss their Carathéodory extension and extend Staton's analysis of their product measures. We give various characterisations of such kernels and discuss their relationship to the more commonly studied classes of σ-finite, subprobability and probability kernels. We use these characterisations to establish suitable Radon-Nikodým, Lebesgue decomoposition and disintegration theorems for s-finite kernels. We discuss s-finite analogues of the classical randomisation lemma for probability kernels. Throughout, we give some examples to explain the connection with (first-order) probabilistic programming. Finally, we briefly explore how some of these results extend to quasi-Borel spaces, hence how they apply to higher-order probabilistic programming.
titleOn S-Finite Measures and Kernels

Elsevier BV

abstractEffectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These ‘probabilistic’ models are called commutative effectuses, and are the focus of attention here. The paper describes the main known ‘probability’ monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate- and state-transformers, normalisation and conditioning of states.
journaltitleJournal of Logical and Algebraic Methods in Programming
titleFrom Probability Monads to Commutative Effectuses

abstractThis book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
title2-Dimensional Categories
il,Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - Category Theory,Mathematics - Quantum Algebra,Mathematics - Representation Theory,Primary: 18-01,Secondary: 18A99,18C15,18D05,18D10,18D30,18D50

abstractFibrations over a category B, introduced to category theory by Grothendieck, encode pseudo-functors Bᵒᵖ ⇝Cat, while the special case of discrete fibrations encode presheaves Bᵒᵖ →Set. A two-sided discrete variation encodes functors Bᵒᵖ ×A →Set, which are also known as profunctors from A to B. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to Cat, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model V-profunctors internally to V-Cat.
titleCategorical Notions of Fibration

abstractWe lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
journaltitleTheory and Applications of Categories
titleMonoidal Grothendieck Construction

booktitleProceedings of the Twenty-Fifth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems
titleRelational Lenses: A Language for Updatable Views

Association for Computing Machinery (ACM)

abstractWe propose a novel approach to the view-update problem for tree-structured data: a domain-specific programming language in which all expressions denote bidirectional transformations on trees. In one direction, these transformations—dubbed lenses—map a concrete tree into a simplified abstract view; in the other, they map a modified abstract view, together with the original concrete tree, to a correspondingly modified concrete tree. Our design emphasizes both robustness and ease of use, guaranteeing strong well-behavedness and totality properties for well-typed lenses. We begin by identifying a natural space of well-behaved bidirectional transformations over arbitrary structures, studying definedness and continuity in this setting. We then instantiate this semantic framework in the form of a collection of lens combinators that can be assembled to describe bidirectional transformations on trees. These combinators include familiar constructs from functional programming (composition, mapping, projection, conditionals, recursion) together with some novel primitives for manipulating trees (splitting, pruning, merging, etc.). We illustrate the expressiveness of these combinators by developing a number of bidirectional list-processing transformations as derived forms. An extended example shows how our combinators can be used to define a lens that translates between a native HTML representation of browser bookmarks and a generic abstract bookmark format.
journaltitleACM Transactions on Programming Languages and Systems
titleCombinators for Bidirectional Tree Transformations

abstractLenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category LensF for any category C and functor F: Cop →Cat, using a variant of the Grothendieck construction. All of the mathematics in this note is straightforward; the purpose is simply to see lenses in a broader context where some closely-related examples, such as ringed spaces and open continuous dynamical systems, can be included.
titleGeneralized Lens Categories via Functors C Op → Cat

abstractOptics are a data representation for compositional data access, with lenses as a popular special case. Hedges has presented a diagrammatic calculus for lenses, but in a way that does not generalize to other classes of optic. We present a calculus that works for all optics, not just lenses; this is done by embedding optics into their presheaf category, which naturally features string diagrams. We apply our calculus to the common case of lenses, and explore how the laws of optics manifest in this setting.
titleString Diagrams for Optics

abstractI present a uniform derivation of profunctor optics: isos, lenses, prisms, and grates based on the Yoneda lemma in the (enriched) profunctor category. In particular, lenses and prisms correspond to Tambara modules with the cartesian and cocartesian tensor product.
titleProfunctor Optics: The Categorical View

abstractBidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor optic laws.
titleCategories of Optics

abstractMorphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal category and, more importantly, how should they be composed. Examples of this situation include lenses or learners. We propose a description of these non-square boxes, which we call open diagrams, using the monoidal bicategory of profunctors. A graphical coend calculus can then be used to reason about open diagrams and their compositions. This is work in progress.
titleOpen Diagrams via Coend Calculus
18A30 (Primary),18D10 (Secondary),cs.LO,cs.PL,dphil,math.CT

abstractLenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category. In this paper, we show that the two concepts are tightly linked. We show both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functor embedding the category of learners into a category of symmetric lenses.
journaltitleIn: J. Cheney, H-S. Ko (eds.): Proceedings of the Eighth International Workshop on Bidirectional Transformations (Bx 2019), Philadelphia, PA, USA, June 4, 2019, published at http://ceur-ws.org
titleLenses and Learners

abstractThis paper generalises the treatment of compositional game theory as introduced by the second and third authors with Ghani and Winschel, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the existing notion of an open game is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issue all at once. To achieve this we make significant use of category theory, especially the 'coend optics' of Riley.
titleBayesian Open Games

abstractDynamic programming is a class of algorithms used to compute optimal control policies for Markov decision processes. Dynamic programming is ubiquitous in control theory, and is also the foundation of reinforcement learning. In this paper, we show that value improvement, one of the main steps of dynamic programming, can be naturally seen as composition in a category of optics, and intuitively, the optimal value function is the limit of a chain of optic compositions. We illustrate this with three classic examples: the gridworld, the inverted pendulum and the savings problem. This is a first step towards a complete account of reinforcement learning in terms of parametrised optics.
titleValue Iteration Is Optic Composition

abstractWe implement the semantics of server operations using parameterised lenses. They allow us to define endpoints and extend them using classical lens composition. The parameterised nature of lenses models state updates while the lens laws mimic properties expected from HTTP. This first approach to server development is extended to use dependent parameterised lenses. An upgrade necessary to model not only endpoints, but entire servers, unlocking the ability to compose them together.
titleLenses for Composable Servers

abstractA (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment - a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical systems. In this extended abstract, we put forward a general definition of open dynamical system. We define two general sorts of morphisms between these systems: covariant morphisms which include trajectories, steady states, and periodic orbits; and contravariant morphisms which allow for plugging variables of some systems into parameters of other systems. We define an indexed double category of open dynamical systems indexed by their interface and use a double Grothendieck construction to construct a double category of open dynamical systems. In our main theorem, we construct covariantly representable indexed double functors from the indexed double category of dynamical systems to an indexed double category of spans. This shows that all covariantly representable structures of dynamical systems - including trajectories, steady states, and periodic orbits - compose according to the laws of matrix arithmetic.
journaltitleElectronic Proceedings in Theoretical Computer Science
shortjournalElectron. Proc. Theor. Comput. Sci.
titleDouble Categories of Open Dynamical Systems (Extended Abstract)
il,Mathematics - Category Theory,Mathematics - Dynamical Systems

Applied Category Theory 2020

abstractDynamical systems—by which we mean machines that take time-varying input, change their state, and produce output—can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to reconfigure their wiring diagram dynamically, based on their collective state. This notion was called "mode dependence", and while the framework was compositional (forming an operad of re-wiring diagrams and algebra of mode-dependent dynamical systems on it), the formulation itself was more "creative" than it was natural. In this paper we show that the theory of mode-dependent dynamical systems can be more naturally recast within the category Poly of polynomial functors. This category is almost superlatively abundant in its structure: for example, it has four interacting monoidal structures (+,×,⊗,∘), two of which (×,⊗) are monoidal closed, and the comonoids for ∘ are precisely categories in the usual sense. We discuss how the various structures in Poly show up in the theory of dynamical systems. We also show that the usual coalgebraic formalism for dynamical systems takes place within Poly. Indeed one can see coalgebras as special dynamical systems—ones that do not record their history—formally analogous to contractible groupoids as special categories.
titlePoly: An Abundant Categorical Setting for Mode-Dependent Dynamics

University of Oxford

titleStructural Foundations for Probabilistic Programming Languages

abstractWe give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.
eprintclasscs, math
journaltitleElectronic Notes in Theoretical Computer Science
shortjournalElectronic Notes in Theoretical Computer Science
titleBimonoidal Structure of Probability Monads
il,Mathematics - Category Theory,Mathematics - Probability,Mathematics - Quantum Algebra

titleFibrations of Bicategories

abstractLenses, optics and dependent lenses (or equivalently morphisms of containers, or equivalently natural transformations of polynomial functors) are all widely used in applied category theory as models of bidirectional processes. From the definition of lenses over a finite product category, optics weaken the required structure to actions of monoidal categories, and dependent lenses make use of the additional property of finite completeness (or, in case of polynomials, even local cartesian closure). This has caused a split in the applied category theory literature between those using optics and those using dependent lenses. The goal of this paper is to unify optics with dependent lenses, by finding a definition of fibre optics admitting both as special cases.
titleFibre Optics

abstractA wide variety of bidirectional data accessors, ranging from mixed optics to functor lenses, can be formalized within a unique framework–dependent optics. Starting from two indexed categories, which encode what maps are allowed in the forward and backward directions, we define a category of dependent optics and establish under what assumptions it has coproducts. Different choices of indexed categories correspond to different families of optics: we discuss dependent lenses and prisms, as well as closed dependent optics. We introduce the notion of Tambara representation and use it to classify contravariant functors from the category of optics, thus generalizing the profunctor encoding of optics to the dependent case.
titleDependent Optics

abstractTambara modules are strong profunctors between monoidal categories. They've been defined by Tambara in the context of representation theory, but quickly found their way in applications when it was understood Tambara modules provide a useful encoding of modular data accessors known as mixed optics. To suit the needs of these applications, Tambara theory has been extended to profunctors between categories receiving an action of a monoidal category. Motivated by the generalization of optics to dependently-typed contexts, we sketch a further extension of the theory of Tambara modules in the setting of actions of double categories (thus doubly indexed categories), by defining them as horizontal natural transformations. The theorems and constructions in Pastro-Street theory relevant to profunctor representation theorem for mixed optics are reobtained in this context. This reproduces the definition of dependent optics recently put forward by Vertechi and Milewski, and hinted at by previous work of the author and his collaborators.
titleSeeing Double through Dependent Optics

Elsevier BV

abstractEffect algebras have been introduced in the 1990s in the study of the foundations of quantum mechanics, as part of a quantum-theoretic version of probability theory. This paper is part of that programme and gives a systematic account of Lebesgue integration for [0,1]-valued functions in terms of effect algebras and effect modules. The starting point is the 'indicator' function for a measurable subset. It gives a homomorphism from the effect algebra of measurable subsets to the effect module of [0,1]-valued measurable functions which preserves countable joins. It is shown that the indicator is free among these maps: any such homomorphism from the effect algebra of measurable subsets can be thought of as a generalised probability measure and can be extended uniquely to a homomorphism from the effect module of [0,1]-valued measurable functions which preserves joins of countable chains. The extension is the Lebesgue integral associated to this probability measure. The preservation of joins by it is the monotone convergence theorem.
journaltitleElectronic Notes in Theoretical Computer Science
titleAn Effect-Theoretic Account of Lebesgue Integration

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleTheoretical Computer Science
isbn978-3-642-15239-9 978-3-642-15240-5
titleConvexity, Duality and Effects
family=nLab authors,

titleEssential Image

abstractThis chapter offers an accessible introduction to the channel-based approach to Bayesian probability theory. This framework rests on algebraic and logical foundations, inspired by the methodologies of programming language semantics. It offers a uniform, structured and expressive language for describing Bayesian phenomena in terms of familiar programming concepts, like channel, predicate transformation and state transformation. The introduction also covers inference in Bayesian networks, which will be modelled by a suitable calculus of string diagrams.
titleThe Logical Essentials of Bayesian Reasoning

issn0001-1541, 1547-5905
journaltitleAIChE Journal
shortjournalAIChE J.
titleNonlinear Principal Component Analysis Using Autoassociative Neural Networks

abstractHow can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.
titleAuto-Encoding Variational Bayes

MIT Press - Journals

abstractDiscovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning. One fruitful approach is to build a parameterized stochastic generative model, independent draws from which are likely to produce the patterns. For all but the simplest generative models, each pattern can be generated in exponentially many ways. It is thus intractable to adjust the parameters to maximize the probability of the observed patterns. We describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations. Our method can be viewed as a form of hierarchical self-supervised learning that may relate to the function of bottom-up and top-down cortical processing pathways.
journaltitleNeural Computation
titleThe Helmholtz Machine

University of Amsterdam

abstractIn this thesis, Variational Inference and Deep Learning: A New Synthesis, we propose novel solutions to the problems of variational (Bayesian) inference, generative modeling, representation learning, semi-supervised learning, and stochastic optimization.
titleVariational Inference & Deep Learning. A New Synthesis

Institute of Mathematical Statistics

journaltitleStatistical Science
titleMemoir on the Probability of the Causes of Events

issn0162-1459, 1537-274X
journaltitleJournal of the American Statistical Association
shortjournalJournal of the American Statistical Association
titleBayes Factors

journaltitleIEEE Transactions on Pattern Analysis and Machine Intelligence
shortjournalIEEE Trans. Pattern Anal. Machine Intell.
titleBayesian Classification with Gaussian Processes

Oxford University Press

eventtitleBayesian Statistics
titleThe Variational Bayesian EM Algorithm for Incomplete Data: With Application to Scoring Graphical Model Structures

abstractThis note derives the variational free energy under the Laplace approximation, with a focus on accounting for additional model complexity induced by increasing the number of model parameters. This is relevant when using the free energy as an approximation to the log-evidence in Bayesian model averaging and selection. By setting restricted maximum likelihood (ReML) in the larger context of variational learning and expectation maximisation (EM), we show how the ReML objective function can be adjusted to provide an approximation to the log-evidence for a particular model. This means ReML can be used for model selection, specifically to select or compare models with different covariance components. This is useful in the context of hierarchical models because it enables a principled selection of priors that, under simple hyperpriors, can be used for automatic model selection and relevance determination (ARD). Deriving the ReML objective function, from basic variational principles, discloses the simple relationships among Variational Bayes, EM and ReML. Furthermore, we show that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. Finally, we also consider, briefly, dynamic models and how these inform the regularisation of free energy ascent schemes, like EM and ReML.
titleVariational Free Energy and the Laplace Approximation

Cambridge, UK ; New York

Cambridge University Press

titleInformation Theory, Inference, and Learning Algorithms
dphil,Information theory

Cambridge Univ. Press

edition1. paperback ed., 8. printing
isbn978-0-521-78450-4 978-0-521-49603-2
seriesCambridge Series in Statistical and Probabilistic Mathematics
titleAsymptotic Statistics

New York

Dover Publications

editionDover ed
titleAsymptotic Methods in Analysis
Approximation theory,dphil,Numerical analysis

Wellesley, Mass

A.K. Peters

seriesAKP Classics
titleAsymptotics and Special Functions
Asymptotic expansions,Differential equations,dphil,Functions,Special,Numerical solutions

issn0036-0279, 1468-4829
journaltitleRussian Mathematical Surveys
shortjournalRuss. Math. Surv.
titleThe Laplace Method for Probability Measures in Banach Spaces

Elsevier Science Publishers B.V. (North-Holland)

booktitleBayesian and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard
titleThe Validity of Posterior Expansions Based on Laplace's Method

issn0162-1459, 1537-274X
journaltitleJournal of the American Statistical Association
shortjournalJournal of the American Statistical Association
titleAccurate Approximations for Posterior Moments and Marginal Densities

issn0273-0979, 1088-9485
journaltitleBulletin of the American Mathematical Society
shortjournalBull. Amer. Math. Soc.
titleThe Theory of Dynamic Programming

titleCategorical Systems Theory (Draft)

San Francisco California USA

booktitlePapers from the 1995 ACM SIGPLAN Workshop on Intermediate Representations
eventtitlePOPL95: 22nd ACM Symposium on Principles of Programming Languages
titleA Correspondence between Continuation Passing Style and Static Single Assignment Form

abstractWe share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.
titleEntropy as a Topological Operad Derivation

abstractThis book brings new mathematical rigour to the ongoing vigorous debate on how to quantify biological diversity. The question "what is diversity?" has surprising mathematical depth, and breadth too: this book involves parts of mathematics ranging from information theory, functional equations and probability theory to category theory, geometric measure theory and number theory. It applies the power of the axiomatic method to a biological problem of pressing concern, but the new concepts and theorems are also motivated from a purely mathematical perspective. The main narrative thread requires no more than an undergraduate course in analysis. No familiarity with entropy or diversity is assumed.
eprintclasscs, math, q-bio
shorttitleEntropy and Diversity
titleEntropy and Diversity: The Axiomatic Approach
eory,dphil,Mathematics - Category Theory,Mathematics - Classical Analysis and ODEs,Quantitative Biology - Populations and Evolution,Quantitative Biology - Quantitative Methods

University of Oxford / University of Oxford

abstractDuality abounds in science, both pure and applied: in mathematics there is duality between space and algebra; in physics there is duality between states and observables; in information science there is duality between systems and properties/behaviours; in logic there is duality between models and theories; and even in philosophy there is duality between realism and antirealism (Dummett), or substance metaphysics and process/function metaphysics (Whitehead/Cassirer). They look akin at a level of abstraction, and yet, taking a closer look at subtleties involved, one may find a parting of the ways. The present thesis builds upon category theory and universal algebra to explicate and articulate the dynamics of duality, the mechanism of how duality emerges, changes, and breaks. A generic duality theory is first pursued in the abstract via categorical topology and algebra, and subtler duality theories are then explored to the end of shedding light on more nuanced facets of concrete dualities such as coalgebraic dualities. Each duality theory thus developed is cashed out to uncover formerly unknown dualities; e.g., the first duality theory tells us a dual equivalence between domains and convex structures, which remained an open problem before. Dualities are often induced by Janusian objects, which sometimes form truth value objects as in topos theory. To elucidate this link between duality and categorical logic, categorical universal logic is developed on the basis of Lawvere's hyperdoctrine and Hyland-Johnstone-Pitts' tripos, thereby expanding the realm of (first-order/higher-order) categorical logic so as to encompass, inter alia, classical, intuitionistic, quantum, fuzzy, relevant, and linear logics. Finally, duality meets symmetry in quantum physics, yielding a categorical understanding of the Wigner theorem, i.e., a purely coalgebraic characterisation of the quantum symmetry groupoid. In passing, operational quantum duality is systematised via Chu space theory. In light of these, duality would arguably serve as a unifying principle to overcome the fragmentation of science, and to soothe the dualistic divide of philosophy between realist and antirealist worldviews.
titleMeaning and Duality: From Categorical Logic to Quantum Physics

New York

editionSecond edition
seriesGraduate Texts in Mathematics
titleSmooth Manifolds and Observables

abstractIn the semantics of programming languages one can view programs as state transformers, or as predicate transformers. Recently the author has introduced state-and-effect triangles which capture this situation categorically, involving an adjunction between state- and predicate-transformers. The current paper exploits a classical result in category theory, part of Jon Beck's monadicity theorem, to systematically construct such a state-and-effect triangle from an adjunction. The power of this construction is illustrated in many examples, covering many monads occurring in program semantics, including (probabilistic) power domains.
journaltitleLogical Methods in Computer Science
titleA Recipe for State-and-Effect Triangles
year2017-05-17, 2017

abstractWe present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new dualities between preordered structures and locales or topological spaces can be generated through our topos-theoretic machinery in a uniform way. We then apply our topos-theoretic interpetation to obtain results connecting properties of preorders and properties of the corresponding locales or topological spaces, and we establish adjunctions between various kinds of categories as natural applications of our general methodology. In the last part of the paper, we exploit the theory developed in the previous parts to obtain a topos-theoretic interpretation of the problem of finding explicit descriptions of models of 'ordered algebraic theories' presented by generators and relations, and give several examples which illustrate the effectiveness of our methodology. In passing, we provide a number of other applications of our theory to Algebra, Topology and Logic.
titleA Topos-Theoretic Approach to Stone-type Dualities
il,Mathematics - Category Theory,Mathematics - General Topology,Mathematics - Logic

Cambridge, UK New York

Cambridge University Press

titleConceptual Mathematics : A First Introduction to Categories

abstractThis article describes recent work on the topic of specifying properties of transition systems. By giving a suitably abstract description of transition systems as coalgebras, it is possible to derive logics for capturing properties of these transition systems in an elegant way.
shorttitleLogic Column 15
titleLogic Column 15: Coalgebras and Their Logics

Springer Berlin Heidelberg

abstractWe propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a connection that models machines on the one hand, and the data they manipulate on the other, testing is used to capture the interactions of each with the objects on the other side: just as the data that are input into a machine can be viewed as tests that the machine can be subjected to, the machine can be viewed as a test that can be used to distinguish data. This approach is based on generalizing from duality theories that now are common in semantics to logical connections, which are simply contravariant adjunctions. In the process, it accomplishes much more than simply moving from one side of a duality to the other; it faithfully represents the interactions that embody what is happening as the computation proceeds. Our basic philosophy is that tests can be used as a basis for modeling interactions, as well as processes and the data on which they operate. In more abstract terms, tests can be viewed as formulas of process logics, and testing semantics connects processes and process logics, and assigns computational meanings to both.
booktitleInternational Conference on Algebraic Methodology and Software Technology
titleTesting Semantics: Connecting Processes and Process Logics

Cambridge University Press

titleIntroduction to Coalgebra
ematics/logic-categories-and-sets/introduction-coalgebra-towards-mathematics-states-and-observation?format=HB   isbn=9781107177895

journaltitleElectronic Notes in Theoretical Computer Science
titleAn Algebra-Coalgebra Framework for System Specification

abstractRecently I’ve been looking into the coalgebra community’s take on modal logic, which in a phrase states simply that Modal Logics are Coalgebraic. The key idea is to start from a dual adjunction between the semantics and syntax of propositional theories, as explained on page 3 of Exemplaric Expressivity of Modal Logics. Typically this is a Stone-style duality between (Lindenbaum algebras of) propositions and spaces of models or valuations for a propositional theory. Modalities are then introduced on the syntactic side in the shape of an endofunctor, where the new modal propositional theories are algebras for this functor. Meanwhile, on the semantic side, we have a endofunctor whose coalgebras provide the semantics for the modal propositional theories. If all goes well, we end up lifting the Stone adjunction to one between coalgebras on one side and algebras on the other. It would be handy at this point if I’d learned how to draw suitable diagrams. As I haven’t you can see what’s going on on page 2 of Coalgebras and Their Logics.
titleCoalgebraic Modal Logic

abstractProfunctor optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. They are modular, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have been studied only using Sets as the enriching category and in the non-mixed case. However, functional programming languages are arguably better described by enriched categories and we have found that some structures in the literature are actually mixed optics. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals that solves an open problem posed by Milewski. Finally, we discuss a Haskell implementation.
titleProfunctor Optics, a Categorical Update

Springer Berlin Heidelberg

abstractThis book is the first systematic presentation of the theory of random dynamical systems, i.e. of dynamical systems under the influence of some kind of randomness. The theory comprises products of random mappings as well as random and stochastic differential equations. The author's approach is based on Oseledets'multiplicative ergodic theorem for linear random systems, for which a detailed proof is presented. This theorem provides us with a random substitute of linear algebra and hence can serve as the basis of a local theory of nonlinear random systems. In particular, global and local random invariant manifolds are constructed and their regularity is proved. Techniques for simplifying a system by random continuous or smooth coordinate tranformations are developed (random Hartman-Grobman theorem, random normal forms). Qualitative changes in families of random systems (random bifurcation theory) are also studied. A dynamical approach is proposed which is based on sign changes of Lyapunov exponents and which extends the traditional phenomenological approach based on the Fokker-Planck equation. Numerous instructive examples are treated analytically or numerically. The main intention is, however, to present a reliable and rather complete source of reference which lays the foundations for future works and applications.
titleRandom Dynamical Systems

abstractIn "Backprop as functor", the authors show that the fundamental elements of deep learning – gradient descent and backpropagation – can be conceptualized as a strong monoidal functor Para(Euc)→Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism Learn≅Para(SLens), where SLens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that SLens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor A↦AyA. Using the fact that (Poly,⊗) is monoidal closed, we show that a map A→B in Para(SLens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [AyA,ByB]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p∈Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.
journaltitleElectronic Proceedings in Theoretical Computer Science
shortjournalElectron. Proc. Theor. Comput. Sci.
titleLearners' Languages

Providence, Rhode Island

American Mathematical Society

booktitleContemporary Mathematics
isbn978-0-8218-0607-4 978-0-8218-7794-4
titleInformation Geometry

New York, NY

Springer New York

abstractIn this chapter, we begin by introducing the simplest type of manifolds, the topological manifolds, which are topological spaces with three special properties that encode what we mean when we say that they “locally look like ℝn.” We then prove some important topological properties of manifolds that we use throughout the book. In the second section we introduce an additional structure, called a smooth structure, that can be added to a topological manifold to enable us to do calculus. Following the basic definitions, we introduce a number of examples of manifolds, so you can have something concrete in mind as you read the general theory. At the end of the chapter we introduce the concept of a smooth manifold with boundary, an important generalization of smooth manifolds that will have numerous applications throughout the book.
booktitleIntroduction to Smooth Manifolds
titleSmooth Manifolds

MIT Press

titleActive Inference. The Free Energy Principle in Mind, Brain, and Behavior

abstractIn this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/CategoricalStochasticProcessesₐndLikelihood
journaltitleCompositionality 3, 1 (2021)
titleCategorical Stochastic Processes and Likelihood

John Wiley & Sons

titleThe Organization of Behavior: A Neuropsychological Approach

Springer Science + Business Media

abstractSeveral formulations of correlation-based Hebbian learning are reviewed. On the presynaptic side, activity is described either by a firing rate or by presynaptic spike arrival. The state of the postsynaptic neuron can be described by its membrane potential, its firing rate, or the timing of backpropagating action potentials (BPAPs). It is shown that all of the above formulations can be derived from the point of view of an expansion. In the absence of BPAPs, it is natural to correlate presynaptic spikes with the postsynaptic membrane potential. Time windows of spike-time-dependent plasticity arise naturally if the timing of postsynaptic spikes is available at the site of the synapse, as is the case in the presence of BPAPs. With an appropriate choice of parameters, Hebbian synaptic plasticity has intrinsic normalization properties that stabilizes postsynaptic firing rates and leads to subtractive weight normalization.
journaltitleBiological Cybernetics
titleMathematical Formulations of Hebbian Learning

abstractSynaptic plasticity was first proposed as a mechanism for learning and memory on the basis of theoretical analysis (Hebb, 1949). The plasticity rule proposed by Hebb postulates that when one neuron drives the activity of another neuron, the connection between these neurons is potentiated. Theoretical analysis indicates that not only Hebbian like synaptic potentiation is necessary but also depression between two neurons that are not sufficiently coactive (Stent, 1973, Sejnowski 1977). Depression is necessary for several reasons, among them to prevent all synapses from saturating to their maximal values and thereby loosing their selectivity, and to prevent a positive feedback loop between network activity and synaptic weights. The experimental correlates of these theoretically proposed forms of synaptic plasticity are called long-term potentiation (LTP) and long-term depression (LTD). Two broad classes of models of synaptic plasticity can be described: 1) Phenomenological models: These are very simple models that are typically based on an input-output relationship between neuronal activity and synaptic plasticity. Phenomenological models are typically used in simulations to account for higher level phenomena such as the formation of memory, or the development of neuronal selectivity. 2) Biophysical models: These more detailed models incorporate more of the cellular and synaptic biophysics of neurons, and are typically used to account for controlled synaptic plasticity experiments.
titleModels of Synaptic Plasticity

Oxford University Press, USA

abstractThis book describes the types of computation that can be performed by biologically plausible neural networks and shows how they may be implemented in different systems of the brain. It is structured in three sections, each of which addresses a different need. The first introduces and analyzes the operation of several fundamental types of neural networks. The second discusses real neural networks in several brain systems, and shows how it is becoming possible to construct theories about the way different parts of the brain work. This section also offers an indication of the various neuroscience and neurocomputation techniques that will need to be combined to ensure further progress in understanding the mechanism of processes in the brain. The third section, a collection of appendices. introduces the formal quantitative approaches to many of the networks described. Neural Networks and Brain Function is an accessible, clear introduction for researchers and students in neuroscience and artificial intelligence to the fascinating problems of how the brain works and how behavior is determined.
titleNeural Networks and Brain Function
action,associative,attention,attractor networks,backpropagation,basal ganglia,binding,capacity,cerebellum,circuit models,coding,competitive,computational,decision,dphil,dynamics,episodic,feed-forward,feedback,human,mechanistic,memory,monkey,motor,neurophysiology,path integration,primate,probabilistic,reinforcement learning,reward,rodent,spatial,spiking,statistical,textbook,uncertainty,vision

abstractCellular sheaves equip graphs with a "geometrical" structure by assigning vector spaces and linear maps to nodes and edges. Graph Neural Networks (GNNs) implicitly assume a graph with a trivial underlying sheaf. This choice is reflected in the structure of the graph Laplacian operator, the properties of the associated diffusion equation, and the characteristics of the convolutional models that discretise this equation. In this paper, we use cellular sheaf theory to show that the underlying geometry of the graph is deeply linked with the performance of GNNs in heterophilic settings and their oversmoothing behaviour. By considering a hierarchy of increasingly general sheaves, we study how the ability of the sheaf diffusion process to achieve linear separation of the classes in the infinite time limit expands. At the same time, we prove that when the sheaf is non-trivial, discretised parametric diffusion processes have greater control than GNNs over their asymptotic behaviour. On the practical side, we study how sheaves can be learned from data. The resulting sheaf diffusion models have many desirable properties that address the limitations of classical graph diffusion equations (and corresponding GNN models) and obtain competitive results in heterophilic settings. Overall, our work provides new connections between GNNs and algebraic topology and would be of interest to both fields.
eprintclasscs, math
shorttitleNeural Sheaf Diffusion
titleNeural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs
ine Learning,dphil,Mathematics - Algebraic Topology

abstractThe automated segmentation of cortical areas has been a long-standing challenge in medical image analysis. The complex geometry of the cortex is commonly represented as a polygon mesh, whose segmentation can be addressed by graph-based learning methods. When cortical meshes are misaligned across subjects, current methods produce significantly worse segmentation results, limiting their ability to handle multi-domain data. In this paper, we investigate the utility of E(n)-equivariant graph neural networks (EGNNs), comparing their performance against plain graph neural networks (GNNs). Our evaluation shows that GNNs outperform EGNNs on aligned meshes, due to their ability to leverage the presence of a global coordinate system. On misaligned meshes, the performance of plain GNNs drop considerably, while E(n)-equivariant message passing maintains the same segmentation results. The best results can also be obtained by using plain GNNs on realigned data (co-registered meshes in a global coordinate system).
titleUtility of Equivariant Message Passing in Cortical Mesh Segmentation
ine Learning,dphil

University of Pennsylvania

abstractCellular sheaves are a discrete model for the theory of sheaves on cell complexes. They carry a canonical cochain complex computing their cohomology. This thesis develops the theory of the Hodge Laplacians of this complex, as well as avenues for their application to concrete engineering and data analysis problems. The sheaf Laplacians so developed are a vast generalization of the graph Laplacians studied in spectral graph theory. As such, they admit generalizations of many results from spectral graph theory and the spectral theory of discrete Hodge Laplacians. A theory of approximation of cellular sheaves is developed, and algorithms for producing spectrally good approximations are given, as well as a generalization of the notion of expander graphs. Sheaf Laplacians allow development of various dynamical systems associated with sheaves, and their behavior is studied. Finally, applications to opinion dynamics, extracting network structure from data, linear control systems, and distributed optimization are outlined.
titleLaplacians of Cellular Sheaves: Theory and Applications

journaltitleElectronic Proceedings in Theoretical Computer Science
shortjournalElectron. Proc. Theor. Comput. Sci.
titleInternal Lenses as Functors and Cofunctors

abstractCofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched categories, enriched functors, and enriched cofunctors, whose horizontal and vertical 2-categories have 2-cells given by enriched natural transformations between functors and cofunctors, respectively. Enriched lenses are defined as a compatible enriched functor and enriched cofunctor pair; weighted lenses, which were introduced by Perrone, are precisely lenses enriched in weighted sets. Several other examples are also studied in detail.
titleAn Introduction to Enriched Cofunctors
il,Mathematics - Category Theory

The Royal Society

abstractWe establish a strong link between two apparently unrelated topics: the study of conflicting information in the formal framework of valuation algebras, and the phenomena of non-locality and contextuality. In particular, we show that these peculiar features of quantum theory are mathematically equivalent to a general notion of disagreement between information sources. This result vastly generalizes previously observed connections between contextuality, relational databases, constraint satisfaction problems and logical paradoxes, and gives further proof that contextual behaviour is not a phenomenon limited to quantum physics, but pervades various domains of mathematics and computer science. The connection allows to translate theorems, methods and algorithms from one field to the other, and paves the way for the application of generic inference algorithms to study contextuality. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
journaltitlePhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
titleNon-Locality, Contextuality and Valuation Algebras: A General Theory of Disagreement

University of Oxford / University of Oxford / University of Oxford

abstractThe main subjects of this thesis are non-locality and contextuality, two fundamental features of quantum mechanics that constitute valuable resources for quantum computation. Our analysis is based on Abramsky & Brandenburger's sheaf theoretic framework, which captures both these phenomena in a unified treatment and in a very general setting. This high-level description transcends quantum physics and allows to precisely characterise the notion of contextuality as the apparent paradox realised by data being locally consistent but globally inconsistent. More specifically, we aim to develop a deeper understanding of so-called logical forms of contextuality, i.e. situations where the phenomenon can be witnessed using purely logical arguments, disregarding probabilities. The sheaf theoretic description of logical contextuality has recently inspired the development of a topological treatment of the phenomenon based on sheaf cohomology. In this thesis, we embark on a detailed analysis of the cohomology of contextuality, exposing key shortcomings in the current methods, and introducing an (almost) complete cohomological characterisation of logical forms of contextuality. More specifically, we show that, in its current formulation, sheaf cohomology does not constitute a complete invariant for contextuality, not even in its strongest forms, and that higher cohomology groups cannot be used to study the phenomenon. Then, we solve these issues by introducing a novel construction, which derives refined versions of the presheaves describing empirical models to expose their deeper extendability properties, resulting in a sheaf cohomological invariant which is applicable to the vast majority of empirical models, and conjectured to work universally. We propose a general theory of contextual semantics using the language of valuation algebras. In particular, we give a general definition of contextual behaviour as a fundamental gap between local agreement and global disagreement of information sources. Not only does this formalism aptly capture and generalise the known instances of contextuality beyond quantum theory, but it also provides inspiration for further applications of the phenomenon, and paves the way for the transfer of results and techniques between different fields. We give a prime example of this potential by developing faster algorithms to detect contextuality based on mainstream methods of generic inference. Finally, we turn our attention back to instances of contextuality in quantum physics, and study strong contextuality in multi-qubit states. We give a complete combinatorial characterisation of All-vs-Nothing proofs of strong contextuality in stabiliser quantum mechanics. This allows to produce the complete list of all stabiliser states exhibiting this kind of contextuality, which consitututes an important resource in certain models of quantum computation. Then, we extend our search for strongly contextual behaviour beyond stabiliser states, and identify the minimum quantum resources needed to realise strong non-locality. Additional results include a partial classification of strongly non-local models comprised of three-qubit states and local projective measurements, and the introduction of a new infinite family of strongly non-local three-qubit states.
titleLogical and Topological Contextuality in Quantum Mechanics and Beyond

Cambridge University Press

abstractMuch of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, ćontextuality,́ is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The second, q́uantum entanglement,́ allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision.
titleQuantum Models of Cognition and Decision

Public Library of Science (PLoS)

abstractThere has been a strong recent interest in applying quantum mechanics (QM) outside physics, including in cognitive science. We analyze the applicability of QM to two basic properties in opinion polling. The first property (response replicability) is that, for a large class of questions, a response to a given question is expected to be repeated if the question is posed again, irrespective of whether another question is asked and answered in between. The second property (question order effect) is that the response probabilities frequently depend on the order in which the questions are asked. Whenever these two properties occur together, it poses a problem for QM. The conventional QM with Hermitian operators can handle response replicability, but only in the way incompatible with the question order effect. In the generalization of QM known as theory of positive-operator-valued measures (POVMs), in order to account for response replicability, the POVMs involved must be conventional operators. Although these problems are not unique to QM and also challenge conventional cognitive theories, they stand out as important unresolved problems for the application of QM to cognition . Either some new principles are needed to determine the bounds of applicability of QM to cognition, or quantum formalisms more general than POVMs are needed.
journaltitlePLoS ONE
titleQuantum Models for Psychological Measurements: An Unsolved Problem

Springer International Publishing

abstractThe ‘quantum cognition’ paradigm was recently challenged by its proven impossibility to simultaneously model ‘question order effects’ and ‘response replicability’. In the present article we describe sequential dichotomic measurements within an operational and realistic framework for human cognition, and represent them in a quantum-like ‘extended Bloch representation’, where the Born rule of quantum probability does not necessarily hold. We then apply this mathematical framework to successfully model question order effects, response replicability and unpacking effects, thus opening the way toward ‘quantum cognition beyond Hilbert space’.
journaltitleLecture Notes in Computer Science
titleQuantum Cognition Beyond Hilbert Space: Fundamentals and Applications

abstractRecent advances in theoretical neuroscience have provided new insights into information processing within large brain-like networks operating in an uncertain world. The computational framework can overcome some of the complexity within the object of study by predicting how basic changes in neural architecture may lead to systems-level changes that translate into changes in behavior. Computational models offer ways to unify basic neurochemical findings with data from more macroscopic levels and to start to apply these findings to cognitive sciences and psychiatry. Some of these approaches have been used to investigate the underlying mechanisms of subjective experiences, such as hallucinations, which can spontaneously emerge into consciousness in the absence of any corresponding external stimuli. This chapter describes some recent theoretical studies on four categories of positive symptoms of schizophrenia: neurodynamics, noise, disconnectivity, and Bayesian models of hallucinations. Results from simulations of these neural networks as well as the potential alterations leading to aberrant experiences are presented and discussed.
booktitleThe Neuroscience of Hallucinations
titleComputational Models of Hallucinations

journaltitleJournal of Vision
shortjournalJournal of Vision
titleCircular Inference in Bistable Perception

abstractContextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.
journaltitle24th EACSL Annual Conference on Computer Science Logic (CSL 2015), Leibniz International Proceedings in Informatics (LIPIcs), 41: 211-228, 2015
titleContextuality, Cohomology and Paradox

abstractThis thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.
titleSheaves, Cosheaves and Applications

American Mathematical Society (AMS)

abstractCohomology groups Hq(X, E) are defined, where X is a topological space and E is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the "homotopical algebra" of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the "local to global" spectral sequence (in sheaf cohomology theory).
journaltitleTransactions of the American Mathematical Society
titleAbstract Homotopy Theory and Generalized Sheaf Cohomology

abstractCellular sheaves are a mathematical structure specifying consistency relations for data associated to vertices and edges of a graph, generalizing connection graphs and matrix weighted graphs. We consider the problem of learning such a sheaf from a collection of highly consistent or smooth signals associated to the vertices of the underlying graph.
booktitleICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
titleLearning Sheaf Laplacians from Smooth Signals

abstractWe introduce a novel class of Laplacians and diffusion dynamics on discourse sheaves as a model for network dynamics, with application to opinion dynamics on social networks. These sheaves are algebraic data structures tethered to a network (or more general space) that can represent various modes of communication, including selective opinion modulation and lying. After introducing the sheaf model, we develop a sheaf Laplacian in this context and show how to evolve both opinions and communications with diffusion dynamics over the network. Issues of controllability, reachability, bounded confidence, and harmonic extension are addressed using this framework.
titleOpinion Dynamics on Discourse Sheaves
55N30 (Secondary),91D30 (Primary),dphil,math.AT,math.DS

Elsevier BV

journaltitleProgress in Biophysics and Molecular Biology
titleA Free Energy Principle for Generic Quantum Systems

abstractWe show in this pedagogical review that far from being "an apparent law of physics that stands by itself" (R. Bousso, Rev. Mod. Phys. 74 (2002), 825-874), the holographic principle (HP) is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Principles equivalent to the HP appear in both computer science and the life sciences, suggesting that the HP is not just a fundamental principle of physics, but of all of science.
eprintclassgr-qc, physics:hep-th, physics:quant-ph
titleThe Physical Meaning of the Holographic Principle
dphil,General Relativity and Quantum Cosmology,High Energy Physics - Theory,Quantum Physics

abstractWe introduce the variational graph auto-encoder (VGAE), a framework for unsupervised learning on graph-structured data based on the variational auto-encoder (VAE). This model makes use of latent variables and is capable of learning interpretable latent representations for undirected graphs. We demonstrate this model using a graph convolutional network (GCN) encoder and a simple inner product decoder. Our model achieves competitive results on a link prediction task in citation networks. In contrast to most existing models for unsupervised learning on graph-structured data and link prediction, our model can naturally incorporate node features, which significantly improves predictive performance on a number of benchmark datasets.
eprintclasscs, stat
titleVariational Graph Auto-Encoders
ine Learning,dphil,Statistics - Machine Learning

abstractWe present a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs. We motivate the choice of our convolutional architecture via a localized first-order approximation of spectral graph convolutions. Our model scales linearly in the number of graph edges and learns hidden layer representations that encode both local graph structure and features of nodes. In a number of experiments on citation networks and on a knowledge graph dataset we demonstrate that our approach outperforms related methods by a significant margin.
eprintclasscs, stat
titleSemi-Supervised Classification with Graph Convolutional Networks
ine Learning,dphil,Statistics - Machine Learning

abstractLots of learning tasks require dealing with graph data which contains rich relation information among elements. Modeling physics system, learning molecular fingerprints, predicting protein interface, and classifying diseases require a model to learn from graph inputs. In other domains such as learning from non-structural data like texts and images, reasoning on extracted structures, like the dependency tree of sentences and the scene graph of images, is an important research topic which also needs graph reasoning models. Graph neural networks (GNNs) are connectionist models that capture the dependence of graphs via message passing between the nodes of graphs. Unlike standard neural networks, graph neural networks retain a state that can represent information from its neighborhood with arbitrary depth. Although the primitive GNNs have been found difficult to train for a fixed point, recent advances in network architectures, optimization techniques, and parallel computation have enabled successful learning with them. In recent years, systems based on variants of graph neural networks such as graph convolutional network (GCN), graph attention network (GAT), gated graph neural network (GGNN) have demonstrated ground-breaking performance on many tasks mentioned above. In this survey, we provide a detailed review over existing graph neural network models, systematically categorize the applications, and propose four open problems for future research.
titleGraph Neural Networks: A Review of Methods and Applications

abstractThe last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach – such as computer vision, playing Go, or protein folding – are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.
eprintclasscs, stat
shorttitleGeometric Deep Learning
titleGeometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
ine Learning,dphil,Statistics - Machine Learning

New York, NY, USA

booktitleProbabilistic and Causal Inference
shorttitleReverend Bayes on Inference Engines
titleReverend Bayes on Inference Engines: A Distributed Hierarchical Approach

journaltitleIEEE Transactions on Information Theory
shortjournalIEEE Trans. Inform. Theory
titleConstructing Free-Energy Approximations and Generalized Belief Propagation Algorithms

abstractWe discuss a categorical version of the celebrated belief propagation algorithm. This provides a way to prove that some algorithms which are known or suspected to be analogous, are actually identical when formulated generically. It also highlights the computational point of view in monoidal categories.
booktitleProceedings of the 11th Workshop on Quantum Physics and Logic
titleBelief Propagation in Monoidal Categories
dphil,Mathematics - Category Theory

Frontiers Media SA

abstractActive inference is a corollary of the Free Energy Principle that prescribes how self-organizing biological agents interact with their environment. The study of active inference processes relies on the definition of a generative probabilistic model and a description of how a free energy functional is minimized by neuronal message passing under that model. This paper presents a tutorial introduction to specifying active inference processes by Forney-style factor graphs (FFG). The FFG framework provides both an insightful representation of the probabilistic model and a biologically plausible inference scheme that, in principle, can be automatically executed in a computer simulation. As an illustrative example, we present an FFG for a deep temporal active inference process. The graph clearly shows how policy selection by expected free energy minimization results from free energy minimization per se, in an appropriate generative policy model.
journaltitleFrontiers in Computational Neuroscience
titleA Factor Graph Description of Deep Temporal Active Inference

abstractWe propose a theoretical framework for non redundant reconstruction of a global loss from a collection of local ones under constraints given by a functor; we call this loss the regionalized loss in honor to Yedidia, Freeman, Weiss' celebrated article `Constructing free-energy approximations and generalized belief propagation algorithms' where a first example of regionalized loss, for entropy and the marginal functor, is built. We show how one can associate to these regionalized losses message passing algorithms for finding their critical points. It is a natural mathematical framework for optimization problems where there are multiple points of views on a dataset and replaces message passing algorithms as canonical ways of finding the optima of these problems. We explain how Generalized Belief propagation algorithms fall into the framework we propose and propose novel message passing algorithms for noisy channel networks.
eprintclasscs, math
titleRegionalized Optimization
il,Mathematics - Optimization and Control

Curran Associates, Inc.

booktitleAdvances in Neural Information Processing Systems
titleDynamic Routing Between Capsules

titleSheaves in Geometry and Logic: A First Introduction to Topos Theory

Elsevier BV

abstractDespite significant advances in neuroscience, the neural bases of intelligence remain poorly understood. Arguably the most elusive aspect of intelligence is the ability to make robust inferences that go far beyond oneś experience. Animals categorize objects, learn to vocalize and may even estimate causal relationships — all in the face of data that is often ambiguous and sparse. Such inductive leaps are thought to result from the brainś ability to infer latent structure that governs the environment. However, we know little about the neural computations that underlie this ability. Recent advances in developing computational frameworks that can support efficient structure learning and inductive inference may provide insight into the underlying component processes and help pave the path for uncovering their neural implementation.
journaltitleCurrent Opinion in Neurobiology
titleToward the Neural Implementation of Structure Learning

Elsevier BV

abstractWe propose a theory of structure learning in the primate brain. We argue that the parietal cortex is critical for learning about relations among the objects and categories that populate a visual scene. We suggest that current deep learning models exhibit poor global scene understanding because they fail to perform the relational inferences that occur in the primate dorsal stream. We review studies of neural coding in primate posterior parietal cortex (PPC), drawing the conclusion that neurons in this brain area represent potentially high-dimensional inputs on a low-dimensional manifold that encodes the relative position of objects or features in physical space, and relations among entities in abstract conceptual space. We argue that this low-dimensional code supports generalisation of relational information, even in nonspatial domains. Finally, we propose that structure learning is grounded in the actions that primates take when they reach for objects or fixate them with their eyes. We sketch a model of how this might occur in neural circuits.
journaltitleProgress in Neurobiology
titleStructure Learning and the Posterior Parietal Cortex

The Royal Society

journaltitlePhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
titleStructure Learning in Coupled Dynamical Systems and Dynamic Causal Modelling

journaltitleInformation and Computation
titleFunctorial Data Migration

abstractAggregating data in a database could also be called "integrating along fibers": given functions $\pi\colon E\to D$ and $s\colon E\to R$, where $(R,\circledast)$ is a commutative monoid, we want a new function $(\circledast s)_\pi$ that sends each $d\in D$ to the "sum" of all $s(e)$ for which $\pi(e)=d$. The operation lives alongside querying – or more generally data migration – in typical database usage: one wants to know how much Canadians spent on cell phones in 2021, for example, and such requests typically require both aggregation and querying. But whereas querying has an elegant category-theoretic treatment in terms of parametric right adjoints between copresheaf categories, a categorical formulation of aggregation – especially one that lives alongside that for querying – appears to be completely absent from the literature. In this paper we show how both querying and aggregation fit into the "polynomial ecosystem". Starting with the category $\mathbf{Poly}$ of polynomial functors in one variable, we review the relatively recent results of Ahman-Uustalu and Garner, which showed that the framed bicategory $\mathbb{C}\mathbf{at}\̂sharp$ of comonads in $\mathbf{Poly}$ is precisely the right setting for data migration: its objects are categories and its bicomodules are parametric right adjoints between their copresheaf categories. We then develop a great deal of theory, compressed for space reasons, including local monoidal closed structures, a coclosure to bicomodule composition, and an understanding of adjoints in $\mathbb{C}\mathbf{at}\̂sharp$. Doing so allows us to derive interesting mathematical results, e.g.\ that the ordinary operation of transposing a span can be decomposed into the composite of two more primitive operations, and then finally to explain how aggregation arises, alongside querying, in $\mathbb{C}\mathbf{at}\̂sharp$.
eprintclasscs, math
titleFunctorial Aggregation
il,H.0,Mathematics - Category Theory

annotationOCLC: ocn930997330
numbervolume 194
seriesApplied Mathematical Sciences
titleInformation Geometry and Its Applications
dphil,Géométrie différentielle,Geometry,Differential,Information theory,Information theory in mathematics,Information,Théorie de l',Mathematical statistics,Mathematics,Statistique mathématique

abstractWe describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some uses of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential geometry with proofs omitted for brevity.
titleAn Elementary Introduction to Information Geometry

abstractThe extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, the extended Kalman filter relies on linearizations which have been debated. We recover the exact extended Kalman filter equations from first principles in statistical learning: the extended Kalman filter is equal to Amariś online natural gradient, applied in the space of trajectories of the system. Namely, each possible trajectory of the dynamical system defines a probability law over possible observations. In principle this makes it possible to treat the underlying trajectory as the parameter of a statistical model of the observations. Then the parameter can be learned by gradient ascent on the log-likelihood of observations, as they become available. Using Amariś natural gradient from information geometry (a gradient descent preconditioned with the Fisher matrix, which provides parameterization-invariance) exactly recovers the extended Kalman filter. This applies only to a particular choice of process noise in the Kalman filter, namely, taking noise proportional to the posterior covariance - a canonical choice in the absence of specific model information.
titleThe Extended Kalman Filter Is a Natural Gradient Descent in Trajectory Space

abstractWe use the geometric framework describing gauge theories to enrich our understanding of the principle of maximum entropy, a variational method appearing in statistical inference and the analysis of stochastic dynamical systems. Using the connection on a principal $G$-bundle, the gradient flows found in the calculus of functional optimisation are grounded in a geometric picture of constraint functions interacting with the dynamics of the probabilistic degrees of freedom of the process. From this, we can describe the point of maximum entropy as parallel transport over the state space. A reinterpretation of splitting results in stochastic dynamical systems is also suggested. Beyond stochastic analysis, we indicate a collection of geometric structures surrounding energy-based inference.
eprintclasscond-mat, physics:math-ph
titleA Constraint Geometry for Inference and Integration
anics,dphil,Mathematical Physics,Mathematics - Dynamical Systems,Mathematics - Probability,Primary 37A50,70S15,Secondary 37K58

eLife Sciences Organisation, Ltd.

abstractThe hippocampal–entorhinal system encodes a map of space that guides spatial navigation. Goal-directed behaviour outside of spatial navigation similarly requires a representation of abstract forms of relational knowledge. This information relies on the same neural system, but it is not known whether the organisational principles governing continuous maps may extend to the implicit encoding of discrete, non-spatial graphs. Here, we show that the human hippocampal–entorhinal system can represent relationships between objects using a metric that depends on associative strength. We reconstruct a map-like knowledge structure directly from a hippocampal–entorhinal functional magnetic resonance imaging adaptation signal in a situation where relationships are non-spatial rather than spatial, discrete rather than continuous, and unavailable to conscious awareness. Notably, the measure that best predicted a behavioural signature of implicit knowledge and blood oxygen level-dependent adaptation was a weighted sum of future states, akin to the successor representation that has been proposed to account for place and grid-cell firing patterns.
titleA Map of Abstract Relational Knowledge in the Human Hippocampal–Entorhinal Cortex

titleWhat Is a Cognitive Map? Organizing Knowledge for Flexible Behavior

Springer Science and Business Media LLC

abstractRelations between task elements often follow hidden underlying structural forms such as periodicities or hierarchies, whose inferences fosters performance. However, transferring structural knowledge to novel environments requires flexible representations that are generalizable over particularities of the current environment, such as its stimuli and size. We suggest that humans represent structural forms as abstract basis sets and that in novel tasks, the structural form is inferred and the relevant basis set is transferred. Using a computational model, we show that such representation allows inference of the underlying structural form, important task states, effective behavioural policies and the existence of unobserved state-trajectories. In two experiments, participants learned three abstract graphs during two successive days. We tested how structural knowledge acquired on Day-1 affected Day-2 performance. In line with our model, participants who had a correct structural prior were able to infer the existence of unobserved state-trajectories and appropriate behavioural policies.
journaltitleNature Communications
titleTransferring Structural Knowledge across Cognitive Maps in Humans and Models

Cold Spring Harbor Laboratory

abstractAbstraction can be defined as a cognitive process that identifies common features - abstract variables, or concepts - shared by many examples. Such conceptual knowledge enables subjects to generalize upon encountering new examples, an ability that supports inferential reasoning and cognitive flexibility. To confer the ability to generalize, the brain must represent variables in a particular ‘abstract’ format. Here we show how to construct neural representations that encode multiple variables in an abstract format simultaneously, and we characterize their geometry. Neural representations conforming to this geometry were observed in dorsolateral pre-frontal cortex, anterior cingulate cortex and the hippocampus in monkeys performing a serial reversal-learning task. Similar representations are observed in a simulated multi-layer neural network trained with back-propagation. These findings provide a novel framework for characterizing how different brain areas represent abstract variables that are critical for flexible conceptual generalization.
titleThe Geometry of Abstraction in Hippocampus and Pre-Frontal Cortex

American Association for the Advancement of Science

abstractThe hippocampal formation has long been suggested to underlie both memory formation and spatial navigation. We discuss how neural mechanisms identified in spatial navigation research operate across information domains to support a wide spectrum of cognitive functions. In our framework, place and grid cell population codes provide a representational format to map variable dimensions of cognitive spaces. This highly dynamic mapping system enables rapid reorganization of codes through remapping between orthogonal representations across behavioral contexts, yielding a multitude of stable cognitive spaces at different resolutions and hierarchical levels. Action sequences result in trajectories through cognitive space, which can be simulated via sequential coding in the hippocampus. In this way, the spatial representational format of the hippocampal formation has the capacity to support flexible cognition and behavior.
journaltitleScience (New York, N.Y.)
titleNavigating Cognition: Spatial Codes for Human Thinking
year2018, 2018-11

abstractThis is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and Mathieu Anel).
booktitleNew Spaces for Mathematics and Physics
titleHomotopy Type Theory: The Logic of Space
year2017-03-08, 2017

MIT Press - Journals

abstractEstimation of returns over time, the focus of temporal difference (TD) algorithms, imposes particular constraints on good function approximators or representations. Appropriate generalization between states is determined by how similar their successors are, and representations should follow suit. This paper shows how TD machinery can be used to learn such representations, and illustrates, using a navigation task, the appropriately distributed nature of the result.
journaltitleNeural Computation
titleImproving Generalization for Temporal Difference Learning: The Successor Representation

Curran Associates, Inc.

abstractHippocampal place fields have been shown to reflect behaviorally relevant aspects of space. For instance, place fields tend to be skewed along commonly traveled directions, they cluster around rewarded locations, and they are constrained by the geometric structure of the environment. We hypothesize a set of design principles for the hippocampal cognitive map that explain how place fields represent space in a way that facilitates navigation and reinforcement learning. In particular, we suggest that place fields encode not just information about the current location, but also predictions about future locations under the current transition distribution. Under this model, a variety of place field phenomena arise naturally from the structure of rewards, barriers, and directional biases as reflected in the transition policy. Furthermore, we demonstrate that this representation of space can support efficient reinforcement learning. We also propose that grid cells compute the eigendecomposition of place fields in part because is useful for segmenting an enclosure along natural boundaries. When applied recursively, this segmentation can be used to discover a hierarchical decomposition of space. Thus, grid cells might be involved in computing subgoals for hierarchical reinforcement learning.
booktitleAdvances in Neural Information Processing Systems 27
titleDesign Principles of the Hippocampal Cognitive Map

Cold Spring Harbor Laboratory

abstractAs we navigate the world we learn about associations among events, extract relational structures, and store them in memory. This relational knowledge, in turn, enables generalization, inference, and hierarchical planning. Here we investigated relational knowledge during spatial navigation as multiscale predictive representations in the brain. We hypothesized that these representations are organized at multiple scales along posterior to anterior hierarchies in prefrontal and hippocampal regions. To test this, we conducted model based representational similarity analyses of neuroimaging data measured during virtual reality navigation of familiar and unfamiliar paths with realistically long distances. We tested the pattern similarity of each point, along each navigational path, to a weighted sum of its successor points within different temporal horizons. Predictive similarity was significantly higher for familiar paths. Overall, anterior PFC showed predictive horizons at the largest scales ( 875m) and posterior hippocampus at the lowest ( 25m), while the anterior hippocampus ( 175m), prepolar PFC, and orbitofrontal regions ( 350m) were in between. These findings support the idea that predictive representations are maintained at higher scales of abstraction in the anterior PFC, and unfolded at lower scales by prepolar PFC and hippocampal regions. This representational hierarchy can support generalization, hierarchical planning, and subgoals at multiple scales.
journaltitlebioRxiv : the preprint server for biology
titlePredictive Representations in Hippocampal and Prefrontal Hierarchies

Hindawi Limited

journaltitleMathematical Problems in Engineering
titleGeneralised Filtering

ASME International

abstractThe classical filtering and prediction problem is re-examined using the Bode-Shannon representation of random processes and the ÃÂ¢ÃÂÃÂstate-transitionÃÂ¢ÃÂÃÂ method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modification to stationary and nonstationary statistics and to growing-memory and infinite-memory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the co-efficients of the difference (or differential) equation of the optimal linear filter are obtained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.
journaltitleJournal of Basic Engineering
titleA New Approach to Linear Filtering and Prediction Problems

abstractWe incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.
titleAutomatic Backward Filtering Forward Guiding for Markov Processes and Graphical Models
il,Primary 62H22,62M20,secondary 60J05,60J25,Statistics - Computation,Statistics - Methodology

abstractIn this document I aim to give an informal treatment of automatic Backward Filtering Forward Guiding, a general algorithm for conditional sampling from a Markov process on a directed acyclic graph. I'll show that the underlying ideas can be understood with a basic background in probability and statistics. The more technical treatment is the paper Automatic backward filtering forward guiding for Markov processes and graphical models (Van der Meulen and Schauer, 2021). I specifically assume some background knowledge on likelihood based inference and Bayesian statistics. The final sections are more demanding and assume familiarity with continuous-time stochastic processes constructed from their infinitesimal generator. Clearly, all work discussed here is the result of research carried out over the past decade together with various collaborators, most importantly Moritz Schauer (Chalmers University of Technology and University of Gothenburg, Sweden). Section 8 is based on joint work with Marcin Mider (Trium Analysis Online GmbH, Germany) and Frank Sch\"afer (University of Basel, Switzerland) as well.
eprintclassmath, stat
titleIntroduction to Automatic Backward Filtering Forward Guiding
il,Mathematics - Probability,Primary 60J60,65C05,secondary 62F15,Statistics - Computation

abstractThe Expected Free Energy (EFE) is a central quantity in the theory of active inference. It is the quantity that all active inference agents are mandated to minimize through action, and its decomposition into extrinsic and intrinsic value terms is key to the balance of exploration and exploitation that active inference agents evince. Despite its importance, the mathematical origins of this quantity and its relation to the Variational Free Energy (VFE) remain unclear. In this paper, we investigate the origins of the EFE in detail and show that it is not simply "the free energy in the future". We present a functional that we argue is the natural extension of the VFE, but which actively discourages exploratory behaviour, thus demonstrating that exploration does not directly follow from free energy minimization into the future. We then develop a novel objective, the Free-Energy of the Expected Future (FEEF), which possesses both the epistemic component of the EFE as well as an intuitive mathematical grounding as the divergence between predicted and desired futures.
titleWhence the Expected Free Energy?

Springer Nature

abstractThe ćognitive map ́hypothesis proposes that brain builds a unified representation of the spatial environment to support memory and guide future action. Forty years of electrophysiological research in rodents suggest that cognitive maps are neurally instantiated by place, grid, border and head direction cells in the hippocampal formation and related structures. Here we review recent work that suggests a similar functional organization in the human brain and yields insights into how cognitive maps are used during spatial navigation. Specifically, these studies indicate that (i) the human hippocampus and entorhinal cortex support map-like spatial codes, (ii) posterior brain regions such as parahippocampal and retrosplenial cortices provide critical inputs that allow cognitive maps to be anchored to fixed environmental landmarks, and (iii) hippocampal and entorhinal spatial codes are used in conjunction with frontal lobe mechanisms to plan routes during navigation. We also discuss how these three basic elements of cognitive map based navigation—spatial coding, landmark anchoring and route planning—might be applied to nonspatial domains to provide the building blocks for many core elements of human thought.
journaltitleNature Neuroscience
titleThe Cognitive Map in Humans: Spatial Navigation and Beyond

abstractSingle units were recorded from the CA1 field of the hippocampus in the freely-moving rat. They were classified as place units, displace units or others. Place units were defined as those for which the ratś position on the maze was a necessary condition for maximal unit firing. Some of these place units (misplace units) fired maximally when the animal sniffed in a place, either because it found something new there or failed to find something which was usually there. Displace units increased their rates during behaviors associated with theta activity in the hippocampal slow waves. In general these were behaviors which changed the ratś position relative to the environment. The influence of various environmental manipulations (e.g., turning off the room lights) on the firing pattern of the place units was tested and the results suggest that they were not responding to a simple sensory stimulus nor to a specific motor behavior. Nor could the unit firing be due purely to motivational or incentive factors. The results are interpreted as strong support for the cognitive map theory of hippocampal function.
journaltitleExperimental neurology
titlePlace Units in the Hippocampus of the Freely Moving Rat
coding,computational,dphil,hippocampus,mechanistic,neurophysiology,place cells,primary,rodent,spatial location

abstractThe cognitive map theory of hippocampal function (OḰeefe & Nadel, 1978, The Hippocampus as a Cognitive Map, Oxford: Oxford Univ. Press) has recently been challenged by the assertion that this neural structure is involved in “working memory” rather than mapping (Olton, Becker, & Handelmann, 1979, The Behavioral and Brain Sciences, 2, 313–365). In a within-subjects design where each lesioned animal served as its own control, rats were trained on two versions (PLACE and CUE) of a maze task designed to simultaneously assess long-term and short-term (working) memory function. After lesions in the hippocampus, rats rapidly regained criterion in the CUE task but not the PLACE task. Their performance was uninfluenced by the putative working memory aspects of the tasks. These data support cognitive map theory and its assertion that the defect following damage to the hippocampus is, at least in rats, selectively spatial.
journaltitleBehavioral and Neural Biology
titleHippocampus: Cognitive Map or Working Memory?

abstractBrain connectivity refers to a pattern of anatomical links ("anatomical connectivity"), of statistical dependencies ("functional connectivity") or of causal interactions ("effective connectivity") between distinct units within a nervous system. The units correspond to individual neurons, neuronal populations, or anatomically segregated brain regions. The connectivity pattern is formed by structural links such as synapses or fiber pathways, or it represents statistical or causal relationships measured as cross-correlations, coherence, or information flow. Neural activity, and by extension neural codes, are constrained by connectivity. Brain connectivity is thus crucial to elucidating how neurons and neural networks process information.
titleBrain Connectivity

Cold Spring Harbor Laboratory

abstractBiological self-organisation is a process of spontaneous pattern formation; namely the emergence of coherent and stable systemic configurations that distinguish themselves from their environment. This process can occur at various spatial scales: from the microscopic (giving rise to cells) to the macroscopic (the emergence of organisms). Self-organisation at each level is essential to account for the hierarchical organisation of living organisms (organelles within cells, within tissues, within organs, etc.). In this paper, we pursue the idea that Markov blankets – statistical boundaries separating states that are external to a system from its internal states – emerge at every possible level of the description of the (living) system. Through simulations, we show that the concept of a Markov blanket is fundamental in defining biological systems and underwrites the nature and form of interactions between successive levels of hierarchical structure. We demonstrate the validity of our argument using simulations, based on the normative principle of variational free energy minimisation. Specifically, we adopt a top-down approach to provide a proof of concept for the claim that the self-organisation of Markov blankets (and blankets of blankets) underwrites the self-evidencing, autopoietic behaviour of living systems.
titleBiological Self-organisation and Markov Blankets

The Royal Society

abstractThis work addresses the autonomous organization of biological systems. It does so by considering the boundaries of biological systems, from individual cells to Home sapiens, in terms of the presence of Markov blankets under the active inference scheme—a corollary of the free energy principle. A Markov blanket defines the boundaries of a system in a statistical sense. Here we consider how a collective of Markov blankets can self-assemble into a global system that itself has a Markov blanket; thereby providing an illustration of how autonomous systems can be understood as having layers of nested and self-sustaining boundaries. This allows us to show that: (i) any living system is a Markov blanketed system and (ii) the boundaries of such systems need not be co-extensive with the biophysical boundaries of a living organism. In other words, autonomous systems are hierarchically composed of Markov blankets of Markov blankets—all the way down to individual cells, all the way up to you and me, and all the way out to include elements of the local environment.
journaltitleJournal of The Royal Society Interface
titleThe Markov Blankets of Life: Autonomy, Active Inference and the Free Energy Principle

abstractActive inference is a probabilistic framework for modelling the behaviour of biological and artificial agents, which derives from the principle of minimising free energy. In recent years, this framework has successfully been applied to a variety of situations where the goal was to maximise reward, offering comparable and sometimes superior performance to alternative approaches. In this paper, we clarify the connection between reward maximisation and active inference by demonstrating how and when active inference agents perform actions that are optimal for maximising reward. Precisely, we show the conditions under which active inference produces the optimal solution to the Bellman equation–a formulation that underlies several approaches to model-based reinforcement learning and control. On partially observed Markov decision processes, the standard active inference scheme can produce Bellman optimal actions for planning horizons of 1, but not beyond. In contrast, a recently developed recursive active inference scheme (sophisticated inference) can produce Bellman optimal actions on any finite temporal horizon. We append the analysis with a discussion of the broader relationship between active inference and reinforcement learning.
titleReward Maximisation through Discrete Active Inference

isbn978-0-471-61977-2 978-0-470-31688-7
seriesWiley Series in Probability and Statistics
shorttitleMarkov Decision Processes
titleMarkov Decision Processes: Discrete Stochastic Dynamic Programming

Cambridge University Press

booktitleProceedings of the Fifth International Congress of Mathematicians
titleÜber Eine Anwendung Der Mengenlehre Auf Die Theorie Des Schachspiels

MIT press Cambridge

abstractIn which we try to give a basic intuitive sense of what reinforcement learning is and how it differs and relates to other fields, e.g., supervised learning and neural networks, genetic algorithms and artificial life, control theory. Intuitively, RL is trial and error (variation and selection, search) plus learning (association, memory). We argue that RL is the only field that seriously addresses the special features of the problem of learning from interaction to achieve long-term goals.
titleReinforcement Learning: An Introduction
action selection,credit assignment,decision,dphil,exploitation/exploration,machine learning,reinforcement learning,reward,statistics,textbook,uncertainty,utility

abstractWe provide a compositional coalgebraic semantics for strategic games. In our framework, like in the semantics of functional programming languages, coalgebras represent the observable behaviour of systems derived from the behaviour of the parts over an unobservable state space. We use coalgebras to describe and program stage games, finitely and potentially infinitely repeated hierarchical or parallel games with imperfect and incomplete information based on deterministic, non-deterministic or probabilistic decisions of learning agents in possibly endogenous networks. Our framework is compositional in that arbitrarily complex network of games can be composed. The coalgebraic approach allows to represent self-referential or reflexive structures like institutional dynamics, strategic network formation from within the network, belief formation, learning agents or other self-referential phenomena that characterise complex social systems of cognitive agents. And finally our games represent directly runnable code in functional programming languages that can also be analysed by sophisticated verification and logical tools of software engineering.
titleA Compositional Coalgebraic Semantics of Strategic Games

Public Library of Science (PLoS)

abstractThis paper questions the need for reinforcement learning or control theory when optimising behaviour. We show that it is fairly simple to teach an agent complicated and adaptive behaviours using a free-energy formulation of perception. In this formulation, agents adjust their internal states and sampling of the environment to minimize their free-energy. Such agents learn causal structure in the environment and sample it in an adaptive and self-supervised fashion. This results in behavioural policies that reproduce those optimised by reinforcement learning and dynamic programming. Critically, we do not need to invoke the notion of reward, value or utility. We illustrate these points by solving a benchmark problem in dynamic programming; namely the mountain-car problem, using active perception or inference under the free-energy principle. The ensuing proof-of-concept may be important because the free-energy formulation furnishes a unified account of both action and perception and may speak to a reappraisal of the role of dopamine in the brain.
journaltitlePLoS ONE
titleReinforcement Learning or Active Inference?

abstractAn open question in the study of emergent behaviour in multi-agent Bayesian systems is the relationship, if any, between individual and collective inference. In this paper we explore the correspondence between generative models that exist at two distinct scales, using spin glass models as a sandbox system to investigate this question. We show that the collective dynamics of a specific type of active inference agent is equivalent to sampling from the stationary distribution of a spin glass system. A collective of specifically-designed active inference agents can thus be described as implementing a form of sampling-based inference (namely, from a Boltzmann machine) at the higher level. However, this equivalence is very fragile, breaking upon simple modifications to the generative models of the individual agents or the nature of their interactions. We discuss the implications of this correspondence and its fragility for the study of multiscale systems composed of Bayesian agents.
titleSpin Glass Systems as Collective Active Inference

Boston, MA

Springer US

booktitleFacets of Systems Science
isbn978-1-4899-0720-2 978-1-4899-0718-9
titleAutopoiesis: The Organization of Living Systems, Its Characterization and a Model

abstractWe formulate differential cohomology and Chern-Weil theory – the theory of connections on fiber bundles and of gauge fields – abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections, hence higher gauge fields. We discuss various models of the axioms and applications to fundamental notions and constructions in quantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multi-tiered quantum field theory – of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories. We close with an outlook on the cohomological quantization of such higher boundary prequantum field theories by a kind of cohesive motives.
titleDifferential Cohomology in a Cohesive Infinity-Topos

abstractIn this paper, a simple case of Bayesian mechanics under the free energy principle is formulated in axiomatic terms. We argue that any dynamical system with constraints on its dynamics necessarily looks as though it is performing inference against these constraints, and that in a non-isolated system, such constraints imply external environmental variables embedding the system. Using aspects of classical dynamical systems theory in statistical mechanics, we show that this inference is equivalent to a gradient ascent on the Shannon entropy functional, recovering an approximate Bayesian inference under a locally ergodic probability measure on the state space. We also use some geometric notions from dynamical systems theoryx2014namely, that the constraints constitute a gauge degree of freedomx2014to elaborate on how the desire to stay self-organised can be read as a gauge force acting on the system. In doing so, a number of results of independent interest are given. Overall, we provide a related, but alternative, formalism to those driven purely by descriptions of random dynamical systems, and take a further step towards a comprehensive statement of the physics of self-organisation in formal mathematical language.
titleTowards a Geometry and Analysis for Bayesian Mechanics
37L40,51P05,60K35,70S15,cond-mat.stat-mech,dphil,math-ph,math.DS,math.MP,nlin.AO,physics.bio-ph,Primary 37K58,Secondary 46N55

abstractThe aim of this paper is to introduce a field of study that has emerged over the last decade, called Bayesian mechanics. Bayesian mechanics is a probabilistic mechanics, comprising tools that enable us to model systems endowed with a particular partition (i.e., into particles), where the internal states (or the trajectories of internal states) of a particular system encode the parameters of beliefs about quantities that characterise the system. These tools allow us to write down mechanical theories for systems that look as if they are estimating posterior probability distributions over the causes of their sensory states, providing a formal language to model the constraints, forces, fields, and potentials that determine how the internal states of such systems move in a space of beliefs (i.e., on a statistical manifold). Here we will review the state of the art in the literature on the free energy principle, distinguishing between three ways in which Bayesian mechanics has been applied to particular systems (i.e., path-tracking, mode-tracking, and mode-matching). We will go on to examine the duality of the free energy principle and the constrained maximum entropy principle, both of which lie at the heart of Bayesian mechanics. We also discuss the implications of this duality for Bayesian mechanics and limitations of current treatments.
titleOn Bayesian Mechanics: A Physics of and by Beliefs

journaltitleTransactions of the American Institute of Electrical Engineers
shortjournalTrans. Am. Inst. Electr. Eng.
titleCertain Topics in Telegraph Transmission Theory

journaltitleBell System Technical Journal
titleTransmission of Information 1

journaltitleBell System Technical Journal
titleA Mathematical Theory of Communication

abstractWe give an introduction to the QBist interpretation of quantum mechanics. We note that it removes the paradoxes, conundra, and pseudo-problems that have plagued quantum foundations for the past nine decades. As an example, we show in detail how it eliminates quantum "nonlocality".
journaltitleAm. J. Phys., Vol.
titleAn Introduction to QBism with an Application to the Locality of Quantum Mechanics

abstractWithout Niels Bohr, QBism would be nothing. But QBism is not Bohr. This paper attempts to show that, despite a popular misconception, QBism is no minor tweak to Bohr's interpretation of quantum mechanics. It is something quite distinct. Along the way, we lay out three tenets of QBism in some detail: 1) The Born Rule—the foundation of what quantum theory means for QBism—is a normative statement. It is about the decision-making behavior any individual agent should strive for; it is not a descriptive "law of nature" in the usual sense. 2) All probabilities, including all quantum probabilities, are so subjective they never tell nature what to do. This includes probability-1 assignments. Quantum states thus have no "ontic hold" on the world. 3) Quantum measurement outcomes just are personal experiences for the agent gambling upon them. Particularly, quantum measurement outcomes are not, to paraphrase Bohr, instances of "irreversible amplification in devices whose design is communicable in common language suitably refined by the terminology of classical physics." Finally, an explicit comparison is given between QBism and Bohr with regard to three subjects: a) The issue of the "detached observer" as it arose in a debate between Pauli and Bohr, b) Bohr's reply to Einstein, Podolsky, and Rosen, and c) Bohr's mature notion of "quantum phenomena." At the end, we discuss how Bohr's notion of phenomena may have something to offer the philosophy of William James: A physics from which to further develop his vision of the world—call it an ontology if you will—in which "new being comes in local spots and patches."
journaltitleMind and Matter 15(2), 245-300 (2017)
titleNotwithstanding Bohr, the Reasons for QBism

Cambridge University Press (CUP)

abstractFollowing work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a ‘coalgebra modality’) and a differential combinator satisfying a number of coherence conditions. In such a category one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces. After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad S∞ on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps. Finally, we derive additional properties of differential categories in certain special cases, especially when the comonad is a storage modality, as in linear logic. In particular, we introduce the notion of a categorical model of the differential calculus, and show that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential λ-calculus.
journaltitleMathematical Structures in Computer Science
titleDifferential Categories

abstractThe reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.
booktitleCSL 2020
eventtitle28th International Conference on Computer Science Logic
titleReverse Derivative Categories
year2019-10-15, 2020

Elsevier BV

abstractThis paper generalizes existing connections between automata and logic to a coalgebraic abstraction level. Let F: Set to Set be a standard functor that preserves weak pullbacks. We introduce various notions of F-automata, devices that operate on pointed F-coalgebras. The criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two-player graph game. We also introduce a language of coalgebraic fixed point logic for F-coalgebras, and we provide a game semantics for this language. Finally, we show that the two approaches are equivalent in expressive power. We prove that any coalgebraic fixed point formula can be transformed into an F-automaton that accepts precisely those pointed F-coalgebras in which the formula holds. And conversely, we prove that any F-automaton can be converted into an equivalent fixed point formula that characterizes the pointed F-coalgebras accepted by the automaton.
journaltitleInformation and Computation
titleAutomata and Fixed Point Logic: A Coalgebraic Perspective

Cambridge University Press

abstractOne of the most exciting and potentially rewarding areas of scientific research is the study of the principles and mechanisms underlying brain function. It is also of great promise to future generations of computers. A growing group of researchers, adapting knowledge and techniques from a wide range of scientific disciplines, have made substantial progress understanding memory, the learning process, and self organization by studying the properties of models of neural networks - idealized systems containing very large numbers of connected neurons, whose interactions give rise to the special qualities of the brain. This book introduces and explains the techniques brought from physics to the study of neural networks and the insights they have stimulated. It is written at a level accessible to the wide range of researchers working on these problems - statistical physicists, biologists, computer scientists, computer technologists and cognitive psychologists. The author presents a coherent and clear nonmechanical presentation of all the basic ideas and results. More technical aspects are restricted, wherever possible, to special sections and appendices in each chapter. The book is suitable as a text for graduate courses in physics, electrical engineering, computer science and biology.
titleModeling Brain Function: The World of Attractor Neural Networks
associative,attractor networks,circuit models,coding,computational,dphil,dynamics,mechanistic,physics,textbook