Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

The topology and geometry of causality

Quantum theory is manifestly in tension with the classical notion of causality. How do we recover causal reasoning in the quantum regime? In this dissertation, we propose a framework where such causal idiosyncrasies are identified as obstructions to the existence of global sections for presheaves of causal data. We do so by extending the Abramsky- Brandenburger framework for non-locality and contextuality [6] to situations where measurement contexts are allowed to be signalling. This results in a theory-independent phenomenology of causality, which can be used to reason about causal structure in any theory exhibiting contextuality. In the first part of this dissertation, we study the specific phenomenology of coherent control of quantum channels, giving rigorous operational meaning to the superposition of causal order. We pursue a bottom-up approach—alternative to the process matrix formalism—by investigating how indefiniteness of causality emerges from specific characteristics of operational theories. This provides the recipe for building processes with indefinite causality, which are then causally analysed using tools described in the second part of the thesis. The second, more substantial part of this dissertation is devoted to building the sheaf- theoretic framework unifying non-locality, contextuality and indefinite causality. We provide a combinatorial description of the operational assumptions underlying definite and indefinite causal order, and characterise the emergent topologies of classical contexts. We explain how to associate causal data to such topologies and detail the relationship between the covers for a topological space and varying degrees of classicality. We develop a complementary geometric understanding of the space of empirical models for this presheaf, and show how it can be used to perform theory-independent causal analysis of empirical data. We conclude by providing novel examples of such causal analysis, showcasing the existence of the phenomenon of contextual causality. Importantly, our examples demonstrate that such phenomenon can be witnessed in quantum theory, as long as coherently control of causal order is allowed for quantum processes

Probability, valuations, hyperspace: Three monads on Top and the support as a morphism

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.Comment: 65 page

Correlated uncertainty arithmetic with application to fusion neutronics

his thesis advances the idea of automatic and rigorous uncertainty propagation for computational science. The aim is to replace the deterministic arithmetic and logical operations composing a function or a computer program with their uncertain equivalents. In this thesis, uncertain computer variables are labelled uncertain numbers, which may be probability distributions, intervals, probability boxes, and possibility distributions. The individual models of uncertainty are surveyed in the context of imprecise probability theory, and their individual arithmetic described and developed, with new results presented in each arithmetic. The presented arithmetic framework allows random variables to be imprecisely characterised or partially defined. It is a common situation that input random variables are unknown or that only certain characteristics of the inputs are known. How uncertain numbers can be rigorously represented by a finite numerical discretisation is described. Further, it is shown how arithmetic operations are computed by numerical convolution, accounting for both the error from the input's discretisation and from the numerical integration, yielding guaranteed bounds on computed uncertain numbers. One of the central topics of this thesis is stochastic dependency. Considering complex dependencies amongst uncertain numbers is necessary, as it plays a key role in operations. An arithmetic operation between two uncertain numbers is a function not only of the input numbers, but also how they are correlated. This is often more important than the marginal information. In the presented arithmetic, dependencies between uncertain numbers may also be partially defined or missing entirely. A major proposition of this thesis are methods to propagate dependence information through functions alongside marginal information. The long-term goal is to solve probabilistic problems with partial knowledge about marginal distributions and dependencies using algorithms which were written deterministically. The developed arithmetic frameworks can be used individually, or may be combined into a larger uncertainty computing framework. We present an application of the developed method to a radiation transport algorithm for nuclear fusion neutronics problems

The Topology of Causality

We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. Our work has its roots in the sheaf-theoretic framework for contextuality by Abramsky and Brandenburger, which it extends to include arbitrary causal orders (be they definite, dynamical or indefinite). We define a notion of causal function for arbitrary spaces of input histories, and we show that the explicit imposition of causal constraints on joint outputs is equivalent to the free assignment of local outputs to the tip events of input histories. We prove factorisation results for causal functions over parallel, sequential, and conditional sequential compositions of the underlying spaces. We prove that causality is equivalent to continuity with respect to the lowerset topology on the underlying spaces, and we show that partial causal functions defined on open sub-spaces can be bundled into a presheaf. In a striking departure from the Abramsky-Brandenburger setting, however, we show that causal functions fail, under certain circumstances, to form a sheaf. We define empirical models as compatible families in the presheaf of probability distributions on causal functions, for arbitrary open covers of the underlying space of input histories. We show the existence of causally-induced contextuality, a phenomenon arising when the causal constraints themselves become context-dependent, and we prove a no-go result for non-locality on total orders, both static and dynamical.Comment: Originally Part 2 of arXiv:2206.08911v2, now extended and published as a stand-alone paper. Introduction shares some material with Part 1 of the trilogy, "The Combinatorics of Causality

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

Pattern Recognition

A wealth of advanced pattern recognition algorithms are emerging from the interdiscipline between technologies of effective visual features and the human-brain cognition process. Effective visual features are made possible through the rapid developments in appropriate sensor equipments, novel filter designs, and viable information processing architectures. While the understanding of human-brain cognition process broadens the way in which the computer can perform pattern recognition tasks. The present book is intended to collect representative researches around the globe focusing on low-level vision, filter design, features and image descriptors, data mining and analysis, and biologically inspired algorithms. The 27 chapters coved in this book disclose recent advances and new ideas in promoting the techniques, technology and applications of pattern recognition