arXiv subjects

Márk Széles

Publications and source records attributed to Márk Széles.

4 recordsLinked to original sources

arXiv2026

Normalized Probabilistic Semantics is Not Associative

Normalization, $D(X + 1) \to D(X) + 1$, fails to form a distributive law, forcing the composition of normalized stochastic kernels to be non-associative. We introduce a first normalized-by-construction probabilistic semantics: the non-associative monad - the magmad - of normalized distributions. Non-associativity allows this semantics to express both observations and interventions, both Pearl's and Jeffrey's updates, and both evidential and causal interpretations. Within this framework, we derive causality results including Pearl's front-door and back-door criteria.

math.CT↗

arXiv2026

Order in Partial Markov Categories

Partial Markov categories are a recent framework for categorical probability theory that provide an abstract account of partial probabilistic computation with updating semantics. In this article, we discuss two order relations on the morphisms of a partial Markov category. In particular, we prove that every partial Markov category is canonically preorder-enriched, recovering several well-known order enrichments. We also demonstrate that the existence of codiagonal maps (comparators) is closely related to order properties of partial Markov categories. Finally, we introduce a synthetic version of the Cauchy--Schwarz inequality and, from it, we prove that updating increases validity.

cs.LO↗

arXiv2026

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 operation that interacts badly with other operations. String diagrams are becoming popular as a graphical technique for probabilistic (and quantum) reasoning. Conditioning has appeared in string diagrams, in terms of a disintegration, using bent wires and shaded (or dashed) normalisation boxes. It has become clear that such normalisation boxes do satisfy certain compositional rules. This paper takes a decisive step in this development by adding a removal rule to the formalism, for the deletion of shaded boxes. Via this removal rule one can get rid of shaded boxes and terminate an inference argument. This paper illustrates via many (graphical) examples how the resulting compositional inference technique can be used for Bayesian networks, causal reasoning and counterfactuals.

cs.LO↗

arXiv2025

Combs, Causality and Contractions in Atomic Markov Categories

We present a counterexample showing that Markov categories with conditionals (such as BorelStoch) need not validate a natural scheme of axioms which we call contraction identities. These identities hold in every traced monoidal category, so in particular this shows that BorelStoch cannot be embedded in any traced monoidal category. We remedy this under the additional assumption of atomicity: Atomic Markov categories validate all contraction identities, and furthermore admit a notion of trace defined for non-signalling morphisms. We conclude that atomic Markov categories admit an intrinsic calculus of combs without having to assume an embedding into a compact-closed category.

math.CT↗