@article{jacobs_causal_2019,
title = {Causal {Inference} by {String} {Diagram} {Surgery}},
url = {http://arxiv.org/abs/1811.08338},
abstract = {Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endofunctor which performs `string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.},
urldate = {2019-11-21},
journal = {arXiv:1811.08338 [cs, math]},
author = {Jacobs, Bart and Kissinger, Aleks and Zanasi, Fabio},
month = jul,
year = {2019},
note = {ZSCC: 0000003
arXiv: 1811.08338},
keywords = {Bayesianism, Categorical probability theory, Implementation}
}
@article{fritz_probability_2019,
title = {A {Probability} {Monad} as the {Colimit} of {Spaces} of {Finite} {Samples}},
url = {http://arxiv.org/abs/1712.05363},
abstract = {We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces. We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension. The colimit characterization allows the development of integration theory and the treatment of measures on spaces of measures, without measure theory. We also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as morphisms.},
urldate = {2019-11-28},
journal = {arXiv:1712.05363 [cs, math]},
author = {Fritz, Tobias and Perrone, Paolo},
month = mar,
year = {2019},
note = {ZSCC: NoCitationData[s1]
arXiv: 1712.05363},
keywords = {Categorical probability theory, Purely theoretical}
}
@article{jacobs_disintegration_2019,
title = {Disintegration and {Bayesian} {Inversion} via {String} {Diagrams}},
volume = {29},
issn = {0960-1295, 1469-8072},
url = {http://arxiv.org/abs/1709.00322},
doi = {10/ggdf9v},
abstract = {The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.},
number = {7},
urldate = {2019-11-21},
journal = {Mathematical Structures in Computer Science},
author = {Jacobs, Bart and Cho, Kenta},
month = aug,
year = {2019},
note = {ZSCC: 0000007
arXiv: 1709.00322},
keywords = {Bayesianism, Categorical probability theory},
pages = {938--971}
}
@article{jacobs_categorical_2018,
title = {Categorical {Aspects} of {Parameter} {Learning}},
url = {http://arxiv.org/abs/1810.05814},
abstract = {Parameter learning is the technique for obtaining the probabilistic parameters in conditional probability tables in Bayesian networks from tables with (observed) data --- where it is assumed that the underlying graphical structure is known. There are basically two ways of doing so, referred to as maximal likelihood estimation (MLE) and as Bayesian learning. This paper provides a categorical analysis of these two techniques and describes them in terms of basic properties of the multiset monad M, the distribution monad D and the Giry monad G. In essence, learning is about the reltionships between multisets (used for counting) on the one hand and probability distributions on the other. These relationsips will be described as suitable natural transformations.},
urldate = {2019-11-21},
journal = {arXiv:1810.05814 [cs]},
author = {Jacobs, Bart},
month = oct,
year = {2018},
note = {ZSCC: 0000001
arXiv: 1810.05814},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Machine learning}
}
@article{jacobs_logical_2018,
title = {The {Logical} {Essentials} of {Bayesian} {Reasoning}},
url = {http://arxiv.org/abs/1804.01193},
abstract = {This 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.},
urldate = {2019-11-21},
journal = {arXiv:1804.01193 [cs]},
author = {Jacobs, Bart and Zanasi, Fabio},
month = apr,
year = {2018},
note = {ZSCC: 0000016
arXiv: 1804.01193},
keywords = {Bayesianism, Categorical probability theory}
}
@article{jacobs_probability_2018,
title = {From probability monads to commutative effectuses},
volume = {94},
issn = {23522208},
url = {https://linkinghub.elsevier.com/retrieve/pii/S2352220816301122},
doi = {10/gct2wr},
abstract = {Eﬀectuses 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.},
language = {en},
urldate = {2019-11-28},
journal = {Journal of Logical and Algebraic Methods in Programming},
author = {Jacobs, Bart},
month = jan,
year = {2018},
note = {ZSCC: 0000028},
keywords = {Categorical probability theory, Effectus theory},
pages = {200--237}
}
@article{scibior_denotational_2017,
title = {Denotational validation of higher-order {Bayesian} inference},
volume = {2},
issn = {24751421},
url = {http://arxiv.org/abs/1711.03219},
doi = {10.1145/3158148},
abstract = {We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.},
number = {POPL},
urldate = {2019-10-10},
journal = {Proceedings of the ACM on Programming Languages},
author = {Ścibior, Adam and Kammar, Ohad and Vákár, Matthijs and Staton, Sam and Yang, Hongseok and Cai, Yufei and Ostermann, Klaus and Moss, Sean K. and Heunen, Chris and Ghahramani, Zoubin},
month = dec,
year = {2017},
note = {arXiv: 1711.03219},
pages = {1--29}
}
@article{jacobs_quantum_2017,
title = {Quantum effect logic in cognition},
volume = {81},
issn = {0022-2496},
url = {http://www.sciencedirect.com/science/article/pii/S0022249617300378},
doi = {10/gcnkcj},
abstract = {This paper illustrates applications of a new, modern version of quantum logic in quantum cognition. The new logic uses ‘effects’ as predicates, instead of the more restricted interpretation of predicates as projections — which is used so far in this area. Effect logic involves states and predicates, validity and conditioning, and also state and predicate transformation via channels. The main aim of this paper is to demonstrate the usefulness of this effect logic in quantum cognition, via many high-level reformulations of standard examples. The usefulness of the logic is greatly increased by its implementation in the programming language Python.},
language = {en},
urldate = {2019-11-22},
journal = {Journal of Mathematical Psychology},
author = {Jacobs, Bart},
month = dec,
year = {2017},
note = {ZSCC: 0000002},
keywords = {Categorical probability theory, Effectus theory, Psychology},
pages = {1--10}
}
@article{heunen_convenient_2017,
title = {A {Convenient} {Category} for {Higher}-{Order} {Probability} {Theory}},
url = {http://arxiv.org/abs/1701.02547},
abstract = {Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.},
urldate = {2019-10-10},
journal = {arXiv:1701.02547 [cs, math]},
author = {Heunen, Chris and Kammar, Ohad and Staton, Sam and Yang, Hongseok},
month = jan,
year = {2017},
note = {arXiv: 1701.02547}
}
@unpublished{clerc_pointless_2017,
title = {Pointless learning (long version)},
url = {https://hal.archives-ouvertes.fr/hal-01429663},
abstract = {Bayesian inversion is at the heart of probabilistic programming and more generally machine learning. Understanding inversion is made difficult by the pointful (kernel-centric) point of view usually taken in the literature. We develop a pointless (kernel-free) approach to inversion. While doing so, we revisit some foundational objects of probability theory, unravel their category-theoretical underpinnings and show how pointless Bayesian inversion sits naturally at the centre of this construction .},
urldate = {2019-11-24},
author = {Clerc, Florence and Danos, Vincent and Dahlqvist, Fredrik and Garnier, Ilias},
month = jan,
year = {2017},
note = {ZSCC: 0000000},
keywords = {Bayesianism, Categorical probability theory, Purely theoretical}
}
@article{jacobs_formal_2017,
title = {A {Formal} {Semantics} of {Influence} in {Bayesian} {Reasoning}},
url = {http://drops.dagstuhl.de/opus/volltexte/2017/8089/},
doi = {10/ggdgbc},
abstract = {This paper proposes a formal deﬁnition of inﬂuence in Bayesian reasoning, based on the notions of state (as probability distribution), predicate, validity and conditioning. Our approach highlights how conditioning a joint entwined/entangled state with a predicate on one of its components has ‘crossover’ inﬂuence on the other components. We use the total variation metric on probability distributions to quantitatively measure such inﬂuence. These insights are applied to give a rigorous explanation of the fundamental concept of d-separation in Bayesian networks.},
language = {en},
urldate = {2019-11-24},
journal = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany},
author = {Jacobs, Bart and Zanasi, Fabio},
year = {2017},
note = {ZSCC: 0000012},
keywords = {Bayesianism, Categorical probability theory, Programming language theory, Semantics}
}
@article{jacobs_predicate/state_2016,
series = {The {Thirty}-second {Conference} on the {Mathematical} {Foundations} of {Programming} {Semantics} ({MFPS} {XXXII})},
title = {A {Predicate}/{State} {Transformer} {Semantics} for {Bayesian} {Learning}},
volume = {325},
issn = {1571-0661},
url = {http://www.sciencedirect.com/science/article/pii/S1571066116300883},
doi = {10/ggdgbb},
abstract = {This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These definitions are illustrated in many examples in discrete and continuous probability theory and also in quantum theory.},
language = {en},
urldate = {2019-11-24},
journal = {Electronic Notes in Theoretical Computer Science},
author = {Jacobs, Bart and Zanasi, Fabio},
month = oct,
year = {2016},
note = {ZSCC: 0000030},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Effectus theory, Programming language theory, Semantics},
pages = {185--200}
}
@article{cho_introduction_2015,
title = {An {Introduction} to {Effectus} {Theory}},
url = {http://arxiv.org/abs/1512.05813},
abstract = {Effectus 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.},
urldate = {2019-11-23},
journal = {arXiv:1512.05813 [quant-ph]},
author = {Cho, Kenta and Jacobs, Bart and Westerbaan, Bas and Westerbaan, Abraham},
month = dec,
year = {2015},
note = {ZSCC: 0000039
arXiv: 1512.05813},
keywords = {Effectus theory}
}
@article{jacobs_towards_2015,
title = {Towards a {Categorical} {Account} of {Conditional} {Probability}},
volume = {195},
issn = {2075-2180},
url = {http://arxiv.org/abs/1306.0831},
doi = {10/ggdf9w},
abstract = {This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and joint probabilities, in the Kleisli category of the distribution monad. The conditional probabilities are induced by a map together with a predicate (the condition). The latter is a predicate in the logic of effect modules on this Kleisli category. This same approach can be transferred to the category of C*-algebras (with positive unital maps), whose predicate logic is also expressed in terms of effect modules. Conditional probabilities can again be expressed via a triangle-fill-in property. In the literature, there are several proposals for what quantum conditional probability should be, and also there are extra difficulties not present in the classical case. At this stage, we only describe quantum systems with classical parametrization.},
urldate = {2019-11-21},
journal = {Electronic Proceedings in Theoretical Computer Science},
author = {Jacobs, Bart and Furber, Robert},
month = nov,
year = {2015},
note = {ZSCC: NoCitationData[s0]
arXiv: 1306.0831},
keywords = {Categorical probability theory, Effectus theory},
pages = {179--195}
}
@article{culbertson_bayesian_2013,
title = {Bayesian machine learning via category theory},
url = {http://arxiv.org/abs/1312.1445},
abstract = {From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.},
urldate = {2019-11-22},
journal = {arXiv:1312.1445 [math]},
author = {Culbertson, Jared and Sturtz, Kirk},
month = dec,
year = {2013},
note = {ZSCC: 0000006
arXiv: 1312.1445},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Purely theoretical}
}
@article{varacca_distributing_2006,
title = {Distributing probability over non-determinism},
volume = {16},
issn = {0960-1295, 1469-8072},
url = {http://www.journals.cambridge.org/abstract_S0960129505005074},
doi = {10/czs9sx},
language = {en},
number = {01},
urldate = {2019-11-26},
journal = {Mathematical Structures in Computer Science},
author = {Varacca, Daniele and Winskel, Glynn},
month = feb,
year = {2006},
note = {ZSCC: 0000108},
keywords = {Categorical probability theory, Denotational semantics, Programming language theory},
pages = {87}
}
@article{mccullagh_what_2002,
title = {What is a statistical model?},
volume = {30},
url = {http://projecteuclid.org/euclid.aos/1035844977},
doi = {10/bkts3m},
language = {en},
number = {5},
urldate = {2019-11-22},
journal = {The Annals of Statistics},
author = {McCullagh, Peter},
month = oct,
year = {2002},
note = {ZSCC: 0000230},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Compendium, Purely theoretical, Statistical learning theory},
pages = {1225--1310}
}
@inproceedings{de_vink_bisimulation_1997,
address = {Berlin, Heidelberg},
series = {Lecture {Notes} in {Computer} {Science}},
title = {Bisimulation for probabilistic transition systems: {A} coalgebraic approach},
isbn = {978-3-540-69194-5},
shorttitle = {Bisimulation for probabilistic transition systems},
doi = {10/fcqzmk},
abstract = {The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendier in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation.},
language = {en},
booktitle = {Automata, {Languages} and {Programming}},
publisher = {Springer},
author = {de Vink, E. P. and Rutten, J. J. M. M.},
editor = {Degano, Pierpaolo and Gorrieri, Roberto and Marchetti-Spaccamela, Alberto},
year = {1997},
note = {ZSCC: NoCitationData[s1]},
keywords = {Categorical probability theory, Coalgebras, Denotational semantics, Probabilistic transition systems, Transition systems},
pages = {460--470}
}
@inproceedings{giry_categorical_1982,
address = {Berlin, Heidelberg},
series = {Lecture {Notes} in {Mathematics}},
title = {A categorical approach to probability theory},
isbn = {978-3-540-39041-1},
doi = {10/dtx5t5},
language = {en},
booktitle = {Categorical {Aspects} of {Topology} and {Analysis}},
publisher = {Springer},
author = {Giry, Michèle},
editor = {Banaschewski, B.},
year = {1982},
note = {ZSCC: NoCitationData[s1]},
keywords = {Categorical probability theory},
pages = {68--85}
}