@paolop@mathstodon.xyz

Paolo Perrone @paolop@mathstodon.xyz

· 3mo ago

At Warwick they prefer non-standard analysis.

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Paolo Perrone @paolop@mathstodon.xyz

· 5mo ago

Replying to

@joannako@mathstodon.xyz It's the 'mathematics of mathematics' because when you tell other mathematicians you are a category theorist, they say "OMG I hated category theory in grad school! What's the utility of that in real math? I'm so proud to have forgotten everything about it."

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Paolo Perrone @paolop@mathstodon.xyz

· 14mo ago

The document that started categorical probability, part of secret work from 1962, has reappeared, together with new commentaries of its author, Bill Lawvere.
https://lawverearchives.com/wp-content/uploads/2025/07/1962.probmap.pdf
Thanks to Tobias Fritz and to the Lawvere Archives for the work.

lawverearchives.com

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Paolo Perrone @paolop@mathstodon.xyz

· 8mo ago

Metropolis-Hastings using Markov categories!
A work by Rob Cornish and Andi Wang.
https://arxiv.org/abs/2601.22911

arXiv.org

A categorical account of the Metropolis-Hastings algorithm

Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard CD categories. To go further, we then study e

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Paolo Perrone @paolop@mathstodon.xyz

· 5mo ago

Replying to

@skewray@mathstodon.xyz @eigil@mathstodon.xyz The introductory section of this paper may be easier to probabilists and information theorists: https://arxiv.org/abs/2212.11719

arXiv.org

Markov Categories and Entropy

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 quantitative notions of entropy, mutual information, and data processing inequalities. We show that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. As it is customary in in

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Paolo Perrone @paolop@mathstodon.xyz

· 10mo ago

More on the relationship between string diagrams and probabilistic graphical models.
New work by Antonio Lorenzin and Fabio Zanasi.
https://arxiv.org/abs/2512.09908

arXiv.org

Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective

Moralisation and Triangulation are transformations allowing to switch between different ways of factoring a probability distribution into a graphical model. Moralisation allows to view a Bayesian network (a directed model) as a Markov network (an undirected model), whereas triangulation addresses the opposite direction. We present a categorical framework where these transformations are modelled as functors between a category of Bayesian networks and one of Markov networks. The two kinds of netwo

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Paolo Perrone @paolop@mathstodon.xyz

· 11mo ago

Descent in Probability Theory: the first steps downward
https://youtu.be/VG2RTE1R0BY?si=mgRJ2cBgPyrH919n

[Oxford Seminar] Paolo Perrone | Descent in Probability Theory: the first steps downward

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Paolo Perrone @paolop@mathstodon.xyz

· 10mo ago

New work by Bart Jacobs, Márk Széles and Dario Stein.
https://arxiv.org/abs/2512.00209

arXiv.org

Compositional Inference for Bayesian Networks and Causality

Inference is a fundamental reasoning technique in probability theory. When applied to a large joint distribution, it involves updating with evidence (conditioning) in one or more components (variables) and computing the outcome in other components. When the joint distribution is represented by a Bayesian network, the network structure may be exploited to proceed in a compositional manner -- with great benefits. However, the main challenge is that updating involves (re)normalisation, making it an

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Paolo Perrone @paolop@mathstodon.xyz

· 14mo ago

New paper out!
https://arxiv.org/abs/2508.01146

arXiv.org

Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories

Several categories look like categories of relations, but do not fit the established theory of relations in regular categories. They include the category of surjective multivalued functions, the category of injective partial functions, the category of finite probability spaces and stochastic matrices, and the category of Hilbert spaces and linear contractions. To explain these anomalous examples, we develop a parallel theory of relations in epi-regular independence categories. Just as regular ca

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Paolo Perrone @paolop@mathstodon.xyz

· 11mo ago

Anyone in Milan tomorrow?

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Paolo Perrone @paolop@mathstodon.xyz

· 5mo ago

If you are in Oxford and would like to learn about applied category theory, I'm giving a course at the Maths department.
It's on Fridays, at 11 am, in L5.

We start this week with monoidal categories and string diagrams.

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Paolo Perrone @paolop@mathstodon.xyz

· 11mo ago

"Independent States Are Orthogonal", a talk at GSI 2025 bridging probability and geometry.
https://youtu.be/mUPJEt3FeiU

Paolo Perrone - Independent States Are Orthogonal - GSI 2025

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Paolo Perrone @paolop@mathstodon.xyz

· 13mo ago

Great work by Areeb Shah-Mohammed on partial morphisms in Markov categories.
https://arxiv.org/abs/2509.05094

arXiv.org

Partializations of Markov categories

The present work develops a construction of a CD category of partial kernels from a particular type of Markov category called a partializable Markov category. These are a generalization of earlier models of categories of partial morphisms such as p-categories, dominical categories, restriction categories, etc. to a non-deterministic/non-cartesian setting. Here all morphisms are quasi-total, with a natural poset enrichment corresponding to one morphism being a restriction of the other. Furthermor

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Paolo Perrone @paolop@mathstodon.xyz

· 13mo ago

We should call it 'connecting the DOTS'.

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Paolo Perrone @paolop@mathstodon.xyz

· 14mo ago

A categorical definition of independence!
Here is a recording of the talk I gave at CT 2025, for anyone who might have missed it.
https://youtu.be/ls6zOX8L1eI