794 research outputs found

### Stochastic order on metric spaces and the ordered Kantorovich monad

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 further to a certain class of ordered metric
spaces, by endowing the spaces of probability measures with the usual
stochastic order. It can be considered a metric analogue of the probabilistic
powerdomain.
The spaces we consider, which we call L-ordered, are spaces where the order
satisfies a mild compatibility condition with the metric itself, rather than
merely with the underlying topology. As we show, this is related to the theory
of Lawvere metric spaces, in which the partial order structure is induced by
the zero distances.
We show that the algebras of the ordered Kantorovich monad are the closed
convex subsets of Banach spaces equipped with a closed positive cone, with
algebra morphisms given by the short and monotone affine maps. Considering the
category of L-ordered metric spaces as a locally posetal 2-category, the lax
and oplax algebra morphisms are exactly the concave and convex short maps,
respectively.
In the unordered case, we had identified the Wasserstein space as the colimit
of the spaces of empirical distributions of finite sequences. We prove that
this extends to the ordered setting as well by showing that the stochastic
order arises by completing the order between the finite sequences, generalizing
a recent result of Lawson. The proof holds on any metric space equipped with a
closed partial order.Comment: 49 pages. Removed incorrect statement (Theorem 6.1.10 of previous
version

### Monads, partial evaluations, and rewriting

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 expression partially": for example, "2+3"
can be obtained as a partial evaluation of "2+2+1". This construction can be
given for any monad, and it is linked to the famous bar construction, of which
it gives an operational interpretation: the bar construction induces a
simplicial set, and its 1-cells are partial evaluations.
We study the properties of partial evaluations for general monads. We prove
that whenever the monad is weakly cartesian, partial evaluations can be
composed via the usual Kan filler property of simplicial sets, of which we give
an interpretation in terms of substitution of terms.
In terms of rewritings, partial evaluations give an abstract reduction system
which is reflexive, confluent, and transitive whenever the monad is weakly
cartesian.
For the case of probability monads, partial evaluations correspond to what
probabilists call conditional expectation of random variables.
This manuscript is part of a work in progress on a general rewriting
interpretation of the bar construction.Comment: Originally written for the ACT Adjoint School 2019. To appear in
Proceedings of MFPS 202

### Lifting couplings in Wasserstein spaces

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-theoretic concept of
a lens, which can be interpreted as a consistent choice of arrow liftings. The
category we study is the one of probability measures over a given standard
Borel space, with morphisms given by the couplings, or transport plans.
The geometrical picture is even more apparent once we equip the arrows of the
category with weights, which one can interpret as "lengths" or "costs", forming
a so-called weighted category, which unifies several concepts of category
theory and metric geometry.
Indeed, we show that the weighted version of a lens is tightly connected to
the notion of submetry in geometry.
Every weighted category gives rise to a pseudo-quasimetric space via
optimization over the arrows. In particular, Wasserstein spaces can be obtained
from the weighted categories of probability measures and their couplings, with
the weight of a coupling given by its cost.
In this case, conditionals allow one to form weighted lenses, which one can
interpret as "lifting transport plans, while preserving their cost".Comment: 27 page

### 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 information theory, mutual information can be defined
as a measure of how far a joint source is from displaying independence of its
components.
More strikingly, Markov categories give a notion of determinism for sources
and channels, and we can define entropy exactly by measuring how far a source
or channel is from being deterministic. This recovers Shannon and R\'enyi
entropies, as well as the Gini-Simpson index used in ecology to quantify
diversity, and it can be used to give a conceptual definition of generalized
entropy.Comment: 54 page

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