Related papers: Order in Partial Markov Categories

We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $\alpha$-admissible unit…

We establish some qualitative properties of minimizers in the fractional Hardy--Sobolev inequalities of arbitrary order.

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 abstraction of the Kleisli category and then defines deterministic…

Chopin (2007) introduced a sequentially ordered hidden Markov model, for which states are ordered according to their order of appearance, and claimed that such a model is a re-parametrisation of a standard Markov model. This note gives a…

We introduce a framework for benchmarking optimizers according to multiple criteria over various test functions. Based on a recently introduced union-free generic depth function for partial orders/rankings, it fully exploits the ordinal…

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of…

In this paper we generalise the notion of linearity (in the sense of Lawvere) to a category C equipped with a compatible sum structure and product structure. In this context, any morphism f from an n-fold sum to an n-fold product has a…

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…

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…

This work introduces a novel cause-effect relation in Markov decision processes using the probability-raising principle. Initially, sets of states as causes and effects are considered, which is subsequently extended to regular path…

The purpose of this paper is to introduce a notion of causality in Markov decision processes based on the probability-raising principle and to analyze its algorithmic properties. The latter includes algorithms for checking cause-effect…

This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every…

In this note we establish some appropriate conditions for stochastic equality of two random variables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result…

In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an…

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…

We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Mikl\'os Schweitzer competition in 2022. We introduce $\ell$-Markov partitions and apply ideas of symbolic dynamics to our systems,…

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion.…

We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent…

The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely…

The categorical distribution is a natural representation of uncertainty in multi-class segmentations. In the two-class case the categorical distribution reduces to the Bernoulli distribution, for which grayscale morphology provides a range…