Related papers: On the Mathematics of the Jeffreys-Lindley Paradox

Simpson's paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson's paradox helps a statistician to draw correct and meaningful conclusions, the…

We propose a convergence mode for positive Radon measures which allows a sequence of probability measures to have an improper limiting measure. We define a sequence of vague priors as a sequence of probability measures that converges to an…

For in vivo research experiments with small sample sizes and available historical data, we propose a sequential Bayesian method for the Behrens-Fisher problem. We consider it as a model choice question with two models in competition: one…

An ultimate universal theory -- a complete theory that accounts, via few and simple first principles, for all the phenomena already observed and that will ever be observed -- has been, and still is, the aspiration of most physicists and…

One of the key elements of probabilistic seismic risk assessment studies is the fragility curve, which represents the conditional probability of failure of a mechanical structure for a given scalar measure derived from seismic ground…

The basic problem of the calculus of variations consists of finding a function that minimizes an energy, like finding the fastest trajectory between two points for a point mass in a gravity field moving without friction under the influence…

Parametric complexity is a central concept in MDL model selection. In practice it often turns out to be infinite, even for quite simple models such as the Poisson and Geometric families. In such cases, MDL model selection as based on NML…

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the…

In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases,…

Published exactly seventy years ago, Jeffreys's Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the…

It has been proposed that the ability to perform joint weak measurements on post-selected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the…

Every philosophy has holes, and it is the responsibility of proponents of a philosophy to point out these problems. Here are a few holes in Bayesian data analysis: (1) the usual rules of conditional probability fail in the quantum realm,…

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy"…

While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the…

The occurrence of Simpson's paradox (SP) in $2\times 2$ contingency tables has been well studied. The present work comprehensively revisits this problem using a combination of philosophical reflections, causal considerations, and…

The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero.…

This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…

How to form priors that do not seem artificial or arbitrary is a central question in Bayesian statistics. The case of forming a prior on the truth of a proposition for which there is no evidence, and the definte evidence that the event can…

We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem dealing with the absence of the Lavrentiev phenomenon, a cornerstone issue in the calculus of variations. We point out some inconsistencies in…

In this study, we provide mathematical and practice-driven justification for using $[0,1]$ normalization of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization,…