Categorical Properties Of Lattice-valued Convergence Spaces

This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jager [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L = f0; 1g:Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jager [2001]. Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson [1996] when L = [0; 1]: Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gahler [1995]. Secondly, diagonal axioms are defned in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jager convergence and proves that Jager\u27s construct is isomorphic to a bireáective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gahler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided

Convergence and quantale-enriched categories

Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category

Belief propagation in monoidal categories

We discuss a categorical version of the celebrated belief propagation algorithm. This provides a way to prove that some algorithms which are known or suspected to be analogous, are actually identical when formulated generically. It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810

A categorical characterization of relative entropy on standard Borel spaces

We give a categorical treatment, in the spirit of Baez and Fritz, of relative entropy for probability distributions defined on standard Borel spaces. We define a category suitable for reasoning about statistical inference on standard Borel spaces. We define relative entropy as a functor into Lawvere's category and we show convexity, lower semicontinuity and uniqueness.Comment: 16 page

Machine Learning Theory and Practice as a Source of Insight into Universal Grammar

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A categorical foundation for Bayesian probability

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measurement. The main result uses an existence theorem for regular conditional probabilities by Faden, which holds in more generality than the setting of Polish spaces. This less stringent setting then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability.Comment: 15 pages; revised setting to more clearly explain how to incorporate perfect measures and the Giry monad; to appear in Applied Categorical Structure

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs