Related papers: Statistics with improper posteriors
It is shown that the standard Kolmogorov model for probability spaces cannot in general allow the elimination but of only a small amount of probabilistic redundancy. This issue, a purely theoretical weakness, not necessarily related to…
Bayesian inference --- although becoming popular in physics and chemistry --- is hampered up to now by the vagueness of its notion of prior probability. Some of its supporters argue that this vagueness is the unavoidable consequence of the…
Probabilities of causation play a crucial role in modern decision-making. Pearl defined three binary probabilities of causation, the probability of necessity and sufficiency (PNS), the probability of sufficiency (PS), and the probability of…
Ray Solomonoff invented the notion of universal induction featuring an aptly termed "universal" prior probability function over all possible computable environments. The essential property of this prior was its ability to dominate all other…
The famous G\"odel incompleteness theorem states that for every consistent sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but…
Education in statistics, the application of statistics in scientific research, and statistics itself as a scientific discipline are in crisis. Within science, the main cause of the crisis is the insufficiently clarified concept of…
The fiducial coincides with the posterior in a group model equipped with the right Haar prior. This result is here generalized. For this the underlying probability space of Kolmogorov is replaced by a $\sigma$-finite measure space and…
Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the…
When performing Bayesian inference, we frequently need to work with conditional probability densities. For example, the posterior function is the conditional density of the parameters given the data. Some might worry that conditional…
Several approaches to causal inference from observational studies have been proposed. Since the proposal of Rubin (1974) many works have developed a counterfactual approach to causality, statistically formalized by potential outcomes. Pearl…
There has been a growing interest, both in physics and psychology, in understanding contextuality in experimentally observed quantities. Different approaches have been proposed to deal with contextual systems, and a promising one is…
Historically, probability theory has been studied for a long time, and Kolmogorov, Levy Ito Kiyoshi, and others have mathematically developed modern probability in conjunction with measurement theory. On the other hand, commutative algebra…
We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized…
The standard conditional probability definition formula is derived as a consequence of the Insufficient Reason Principle expressed as the Maximum Relative Divergence Principle for grading (order-comonotonic) functions on a totally ordered…
Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question…
In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal…
This contribution derives from a rather extensive study on the foundations of probability. We start by discussing critically the two main models of the random event in Probability Theroy and cast light over a number of incongruities. We…
We present a basis for studying questions of cause and effect in statistics which subsumes and reconciles the models proposed by Pearl, Robins, Rubin and others, and which, as far as mathematical notions and notation are concerned, is…
Unexpectedness is a central concept in Simplicity Theory, a theory of cognition relating various inferential processes to the computation of Kolmogorov complexities, rather than probabilities. Its predictive power has been confirmed by…
R. A. Fisher founded modern statistical inference in 1922 and identified its fundamental problems to be: specification, estimation and distribution. Since then the problem of statistical model specification has received scant attention in…