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We discuss a connection between two areas of mathematics which until recently seemed to be rather distant from each other: (1) noncommutative harmonic analysis on groups and (2) some topics in probability theory related to random point…
Although there are many ideas for the formulations of statistical hypothesis testing, we consider that the likelihood ratio test is the most reasonable and orthodox. However, it is not handy, and thus, it is not usual in elementary books.…
This paper considers the notion of possible events which are insignificant in probabilistic analysis (i.e. events that have zero probability). The paper discusses the method of modal logic based on "possible worlds" and discusses a…
We propose a formalization of the three-tier causal hierarchy of association, intervention, and counterfactuals as a series of probabilistic logical languages. Our languages are of strictly increasing expressivity, the first capable of…
The present communication addresses a set of observations, obeying both deterministic as well as statistical formal requirements, and serving to operate within the framework of the dynamical systems theory, with a certain emphasis placed on…
Quantum theory does not provide a unique definition for the joint probability of two non-commuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were…
There has long been an impression that reliabilism implies externalism and that frequentist statistics, due to its reliabilist nature, is inherently externalist. I argue, however, that frequentist statistics can plausibly be understood as a…
The concept of probability was prominent in the original foundations of quantum mechanics, and continues to be so today. Indeed, the controversies regarding objective and subjective interpretations of probability have again become active. I…
Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the…
In this paper two hypotheses are developed. The first hypothesis is the existence of random phenomena/experiments in which the events cannot generally be assigned a definite probability but that nevertheless admit a class of nearly certain…
In Probabilistic Logic Nilsson uses the device of a probability distribution over a set of possible worlds to assign probabilities to the sentences of a logical language. In his paper Nilsson concentrated on inference and associated…
Robust classification algorithms have been developed in recent years with great success. We take advantage of this development and recast the classical two-sample test problem in the framework of classification. Based on the estimates of…
Statistics has made tremendous advances since the times of Fisher, Neyman, Jeffreys, and others, but the fundamental and practically relevant questions about probability and inference that puzzled our founding fathers remain unanswered. To…
The classical propositional assumption-based model is extended to incorporate probabilities for the assumptions. Then it is placed into the framework of evidence theory. Several authors like Laskey, Lehner (1989) and Provan (1990) already…
Recent decades have seen an interest in prediction problems for which Bayesian methodology has been used ubiquitously. Sampling from or approximating the posterior predictive distribution in a Bayesian model allows one to make inferential…
The reconstruction of the parameter of the model by the measurement of the random variable depending on this parameter is one of the main tasks of statistics. In the paper the notion of the statistically dual distributions is introduced.…
In this paper, the authors first provide an overview of two major developments on complex survey data analysis: the empirical likelihood methods and statistical inference with non-probability survey samples, and highlight the important…
How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences.…
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…
When testing a statistical hypothesis, is it legitimate to deliberate on the basis of initial data about whether and how to collect further data? Game-theoretic probability's fundamental principle for testing by betting says yes, provided…