28 citations · 159 across the 26 of their papers we have counts for
27 papers
Empirical Measures and Strong Laws of Large Numbers in Categorical Probability
Tobias Fritz, Tomáš Gonda, Antonio Lorenzin +2
The Glivenko--Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges t…
Categorical algebra of conditional probability
Mika Bohinen, Paolo Perrone
In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and s…
How to Represent Non-Representable Functors
Paolo Perrone
The arrows of a category are elements of particular sets, the hom-sets. These sets are functorial, and their functoriality specifies how to compose the arrows with other arrows of…
Convergence of martingales via enriched dagger categories
Paolo Perrone, Ruben Van Belle
We provide a categorical proof of convergence for martingales and backward martingales in mean, using enriched category theory. The enrichment we use is in topological spaces, with…
Categorical probability spaces, ergodic decompositions, and transitions to equilibrium
Noé Ensarguet, Paolo Perrone
We study a category of probability spaces and measure-preserving Markov kernels up to almost sure equality. This category contains, among its isomorphisms, mod-zero isomorphisms of…
Absolute continuity, supports and idempotent splitting in categorical probability
Tobias Fritz, Tomáš Gonda, Antonio Lorenzin +2
Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like con…
Weakly Markov categories and weakly affine monads
Tobias Fritz, Fabio Gadducci, Paolo Perrone +1
Introduced in the 1990s in the context of the algebraic approach to graph rewriting, gs-monoidal categories are symmetric monoidal categories where each object is equipped with the…
Markov Categories and Entropy
Paolo Perrone
Markov categories are a novel framework to describe and treat problems in probability and information theory. In this work we combine the categorical formalism with the traditional…
Dilations and information flow axioms in categorical probability
Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen +3
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in variations thereof for arbitrary sem…
A category-theoretic proof of the ergodic decomposition theorem
Sean Moss, Paolo Perrone
The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergo…
Probability monads with submonads of deterministic states - Extended version
Sean Moss, Paolo Perrone
Probability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstracti…
Lifting couplings in Wasserstein spaces
Paolo Perrone
This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category…
Weak cartesian properties of simplicial sets
Carmen Constantin, Tobias Fritz, Paolo Perrone +1
Many special classes of simplicial sets, such as the nerves of categories or groupoids, the 2-Segal sets of Dyckerhoff and Kapranov, and the (discrete) decomposition spaces of Gálv…
De Finetti's Theorem in Categorical Probability
Tobias Fritz, Tomáš Gonda, Paolo Perrone
We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The…
Kan extensions are partial colimits
Paolo Perrone, Walter Tholen
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by…
Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
Tobias Fritz, Tomáš Gonda, Paolo Perrone +1
Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent…
Partial Evaluations and the Compositional Structure of the Bar Construction
Carmen Constantin, Paolo Perrone, Tobias Fritz +1
The algebraic expression can be evaluated to , but it can also be partially evaluated to . In categorical algebra, such partial evaluations can be defined in…
Notes on Category Theory with examples from basic mathematics
Paolo Perrone
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a…
Probability, valuations, hyperspace: Three monads on Top and the support as a morphism
Tobias Fritz, Paolo Perrone, Sharwin Rezagholi
We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the H…
Monads, partial evaluations, and rewriting
Tobias Fritz, Paolo Perrone
Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of "evaluating an…
A Criterion for Kan Extensions of Lax Monoidal Functors
Tobias Fritz, Paolo Perrone
In this mainly expository note, we state a criterion for when a left Kan extension of a lax monoidal functor along a strong monoidal functor can itself be equipped with a lax monoi…
Stochastic order on metric spaces and the ordered Kantorovich monad
Tobias Fritz, Paolo Perrone
In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad fu…
Bimonoidal Structure of Probability Monads
Tobias Fritz, Paolo Perrone
We 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 mo…
A Probability Monad as the Colimit of Spaces of Finite Samples
Tobias Fritz, Paolo Perrone
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…