Statistical Geometry in Quantum Mechanics

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement theor

Truth Discovery in Crowdsourced Detection of Spatial Events

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Introducing Vibration for use in Interaction Designs to support Human Performance: A Pilot Study

While vibration is a well-used output signal in HCI as part of haptic interaction, vibration outside HCI is used in many other modes to support human performance, from rehabilitation to cognition. In this late breaking work, we present preliminary positive results of a novel protocol that informs how vibration might be used to enrich HCI interventions for aspects of both health and intellectual performance. We also present a novel apparatus specifically designed to help HCI researchers explore different vibration amplitudes and frequencies for such applications.Comment: 10 pages, 5 figures; pilot study repor

On the Equivalence of Three-Particle Scattering Formalisms

In recent years, different on-shell $\mathbf{3}\to\mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in Phys.~Rev.~D92, 114509 (2015) and subsequently Brice\~no, Hansen, and Sharpe in Phys.~Rev.~D95, 074510 (2017) can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai {\em et al.} in Eur.~Phys.~J.~A53, 177 (2017) and Jackura {\em et al.} in Eur.~Phys.~J.~C79, 56 (2019). Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the non-relativistic limit of both representations.Comment: 13 pages, 5 figure

Continuous dependence of an invariant measure on the jump rate of a piecewisedeterministic Markov process

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity . The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say . The aim of this paper is to prove that the map 7! is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression

The Eurace@Unibi Model: An Agent-Based Macroeconomic Model for Economic Policy Analysis

Dawid H, Gemkow S, Harting P, van der Hoog S, Neugart M. The Eurace@Unibi Model: An Agent-Based Macroeconomic Model for Economic Policy Analysis. Working Papers in Economics and Management. Vol 05-2012. Bielefeld: Bielefeld University, Department of Business Administration and Economics; 2012.This document provides a description of the modeling assumptions and economic features of the Eurace@Unibi model. Furthermore, the document shows typical patterns of the output generated by this model and compares it to empirically observable stylized facts. The Eurace@Unibi model provides a representation of a closed macroeconomic model with spatial structure. The main objective is to provide a micro-founded macroeconomic model that can be used as a unified framework for policy analysis in different economic policy areas and for the examination of generic macroeconomic research questions. In spite of this general agenda the model has been constructed with certain specific research questions in mind and therefore certain parts of the model, e.g. the mechanisms driving technological change, have been worked out in more detail than others. The purpose of this document is to give an overview over the model itself and its features rather than discussing how insights into particular economic issues can be obtained using the Eurace@Unibi model. The model has been designed as a framework for economic analysis in various domains of economics. A number of economic issues have been examined using (prior versions of) the model (see Dawid et al. (2008), Dawid et al. (2009), Dawid et al. (2011a), Dawid and Harting (2011), van der Hoog and Deissenberg (2011), Cincotti et al. (2010)) and recent extensions of the model have substantially extended its applicability in various economic policy domains, however results of such policy analyses will be reported elsewhere. Whereas the overall modeling approach, the different modeling choices and the economic rationale behind these choices is discussed in some detail in this document, no detailed description of the implementation is given. Such a detailed documentation is provided in the accompanying document Dawid et al. (2011b)

Criteria of efficiency for conformal prediction

We study optimal conformity measures for various criteria of efficiency of classification in an idealised setting. This leads to an important class of criteria of efficiency that we call probabilistic; it turns out that the most standard criteria of efficiency used in literature on conformal prediction are not probabilistic unless the problem of classification is binary. We consider both unconditional and label-conditional conformal prediction.Comment: 31 page