Jeffreys's law for general games of prediction: in search of a theory

We are interested in the following version of Jeffreys's law: if two predictors are predicting the same sequence of events and either is doing a satisfactory job, they will make similar predictions in the long run. We give a classification of instances of Jeffreys's law, illustrated with examples.Comment: 12 page

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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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

Spatial interactions in agent-based modeling

Agent Based Modeling (ABM) has become a widespread approach to model complex interactions. In this chapter after briefly summarizing some features of ABM the different approaches in modeling spatial interactions are discussed. It is stressed that agents can interact either indirectly through a shared environment and/or directly with each other. In such an approach, higher-order variables such as commodity prices, population dynamics or even institutions, are not exogenously specified but instead are seen as the results of interactions. It is highlighted in the chapter that the understanding of patterns emerging from such spatial interaction between agents is a key problem as much as their description through analytical or simulation means. The chapter reviews different approaches for modeling agents' behavior, taking into account either explicit spatial (lattice based) structures or networks. Some emphasis is placed on recent ABM as applied to the description of the dynamics of the geographical distribution of economic activities, - out of equilibrium. The Eurace@Unibi Model, an agent-based macroeconomic model with spatial structure, is used to illustrate the potential of such an approach for spatial policy analysis.Comment: 26 pages, 5 figures, 105 references; a chapter prepared for the book "Complexity and Geographical Economics - Topics and Tools", P. Commendatore, S.S. Kayam and I. Kubin, Eds. (Springer, in press, 2014

Inferential models: A framework for prior-free posterior probabilistic inference

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This paper presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but does so with an automatic long-run frequency calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the non-uniqueness issue. Several examples are presented to illustrate this new approach.Comment: 29 pages with 3 figures. Main text is the same as the published version. Appendix B is an addition, not in the published version, that contains some corrections and extensions of two of the main theorem