Related papers: Quantifiers as Adjoint in Probability
The Giry monad on the category of measurable spaces sends a space to a space of all probability measures on it. There is also a finitely additive Giry monad in which probability measures are replaced by finitely additive probability…
We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…
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…
Giry algebras are barycenters maps, which are coequalizers of contractible coequalizer pairs (like any algebras), and their existence, in general, requires the measurable space be coseparated by the discrete two point space, and the…
Several monads of probability measures have been shown to have presentations as codensity monads over small categories of stochastic maps. This paper studies how three key properties of these probability monads, relevant to categorical…
We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
The notion of effectus from categorical logic is relevant in the emerging field of categorical probability theory. In some cases, stochastic maps are represented by maps in the Kleisli category of some probability monad. Quantum…
We construct a factorization of the Giry monad through the category of convex spaces, and show that, provided that no measurable cardinals exist, probability measures can be viewed as natural transformations. Using the adjunction of this…
A rigorous general definition of quantum probability is given, which is valid for elementary events and for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting…
We introduce a version of probabilistic Kleene algebra with angelic nondeterminism and a corresponding class of automata. Our approach implements semantics via distributions over multisets in order to overcome theoretical barriers arising…
The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category…
For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also…
Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with…
This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and…
We propose a system for the interpretation of anaphoric relationships between unbound pronouns and quantifiers. The main technical contribution of our proposal consists in combining generalized quantifiers with dependent types. Empirically,…
The Principle of Complementarity of Probabilities based on of noncommutative probability is introduced.
Automata with monitor counters, where the transitions do not depend on counter values, and nested weighted automata are two expressive automata-theoretic frameworks for quantitative properties. For a well-studied and wide class of…
The stochastic interpretation of Parikh's game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general approach to probabilistic…
In this paper, a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, is presented and the guidelines to how to extend…
Various categories have been proposed as targets for the denotational semantics of higher-order probabilistic programming languages. One such proposal involves joint probability distributions (couplings) used in Bayesian statistical models…