An Introduction to Inductive Statistical Inference: from Parameter Estimation to Decision-Making

These lecture notes aim at a post-Bachelor audience with a background at an introductory level in Applied Mathematics and Applied Statistics. They discuss the logic and methodology of the Bayes-Laplace approach to inductive statistical inference that places common sense and the guiding lines of the scientific method at the heart of systematic analyses of quantitative-empirical data. Following an exposition of exactly solvable cases of single- and two-parameter estimation problems, the main focus is laid on Markov Chain Monte Carlo (MCMC) simulations on the basis of Hamiltonian Monte Carlo sampling of posterior joint probability distributions for regression parameters occurring in generalised linear models for a univariate outcome variable. The modelling of fixed effects as well as of correlated varying effects via multi-level models in non-centred parametrisation is considered. The simulation of posterior predictive distributions is outlined. The assessment of a model's relative out-of-sample posterior predictive accuracy with information entropy-based criteria WAIC and LOOIC and model comparison with Bayes factors are addressed. Concluding, a conceptual link to the behavioural subjective expected utility representation of a single decision-maker's choice behaviour in static one-shot decision problems is established. Vectorised codes for MCMC simulations of multi-dimensional posterior joint probability distributions with the Stan probabilistic programming language implemented in the statistical software R are provided. The lecture notes are fully hyperlinked. They direct the reader to original scientific research papers, online resources on inductive statistical inference, and to pertinent biographical information.Comment: 161 pages, 22 *.eps figures, LaTeX2e, hyperlinked references. First thorough revision, extended list of reference

Deviation of geodesics in FLRW spacetime geometries

The geodesic deviation equation (`GDE') provides an elegant tool to investigate the timelike, null and spacelike structure of spacetime geometries. Here we employ the GDE to review these structures within the Friedmann--Lema\^{\i}tre--Robertson--Walker (`FLRW') models, where we assume the sources to be given by a non-interacting mixture of incoherent matter and radiation, and we also take a non-zero cosmological constant into account. For each causal case we present examples of solutions to the GDE and we discuss the interpretation of the related first integrals. The de Sitter spacetime geometry is treated separately.Comment: 17 pages, LaTeX 2.09, 3 *.eps figures, Contribution to the forthcoming Engelbert Sch\"{u}cking Festschrift (Springer Verlag

Decision-theoretic approaches to non-knowledge in economics

The aim of this contribution is to provide an overview of conceptual approaches to incorporating a decision maker’s non-knowledge into economic theory. We will focus here on the particular kind of non-knowledge which we consider to be one of the most important for economic discussions: non-knowledge of possible consequence-relevant uncertain events which a decision maker would have to take into account when selecting between different strategies

Partially locally rotationally symmetric perfect fluid cosmologies

We show that there are no new consistent cosmological perfect fluid solutions when in an open neighbourhood ${\cal U}$ of an event the fluid kinematical variables and the electric and magnetic Weyl curvature are all assumed rotationally symmetric about a common spatial axis, specialising the Weyl curvature tensor to algebraic Petrov type D. The consistent solutions of this kind are either locally rotationally symmetric, or are subcases of the Szekeres dust models. Parts of our results require the assumption of a barotropic equation of state. Additionally we demonstrate that local rotational symmetry of perfect fluid cosmologies follows from rotational symmetry of the Riemann curvature tensor and of its covariant derivatives only up to second order, thus strengthening a previous result.Comment: 20 pages, LaTeX2.09 (10pt), no figures; shortened revised version, new references; accepted for publication in Classical and Quantum Gravit

Geometrical order-of-magnitude estimates for spatial curvature in realistic models of the Universe

The thoughts expressed in this article are based on remarks made by J\"urgen Ehlers at the Albert-Einstein-Institut, Golm, Germany in July 2007. The main objective of this article is to demonstrate, in terms of plausible order-of-magnitude estimates for geometrical scalars, the relevance of spatial curvature in realistic models of the Universe that describe the dynamics of structure formation since the epoch of matter-radiation decoupling. We introduce these estimates with a commentary on the use of a quasi-Newtonian metric form in this context.Comment: 11 pages. Fully hyperlinked. Dedicated to the memory of J\"urgen Ehlers. To appear in the upcoming Special Issue of "General Relativity and Gravitation