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Bayesian statistics is based on the subjective definition of probability as {\it ``degree of belief''} and on Bayes' theorem, the basic tool for assigning probabilities to hypotheses combining {\it a priori} judgements and experimental…
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…
Prediction of future observations is an important and challenging problem. The two mainstream approaches for quantifying prediction uncertainty use prediction regions and predictive distributions, respectively, with the latter believed to…
The Principle of Complementarity of Probabilities based on of noncommutative probability is introduced.
Evidence for fine-tuning of physical parameters suitable for life can perhaps be explained by almost any combination of providence, coincidence or multiverse. A multiverse usually includes parts unobservable to us, but if the theory for it…
To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making…
The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $\mu$. After observing each outcome, it is required…
The goal of this note is to provide a geometric setting in which generalized arithmetic means are best predictors in an appropriate metric. This characterization provides a geometric interpretation to the concept of certainty equivalent.…
We define notions of cautiousness and cautious belief to provide epistemic conditions for iterated admissibility in finite games. We show that iterated admissibility characterizes the behavioral implications of "cautious rationality and…
Comprehensible explanations of probabilistic reasoning are a prerequisite for wider acceptance of Bayesian methods in expert systems and decision support systems. A study of human reasoning under uncertainty suggests two different…
The causal (belief) network is a well-known graphical structure for representing independencies in a joint probability distribution. The exact methods and the approximation methods, which perform probabilistic inference in causal networks,…
Performance prediction or forecasting sporting outcomes involves a great deal of insight into the particular area one is dealing with, and a considerable amount of intuition about the factors that bear on such outcomes and performances. The…
In statistical practice, whether a Bayesian or frequentist approach is used in inference depends not only on the availability of prior information but also on the attitude taken toward partial prior information, with frequentists tending to…
An approach is presented treating decision theory as a probabilistic theory based on quantum techniques. Accurate definitions are given and thorough analysis is accomplished for the quantum probabilities describing the choice between…
Propensity scores are often used for stratification of treatment and control groups of subjects in observational data to remove confounding bias when estimating of causal effect of the treatment on an outcome in so-called potential outcome…
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…
Football forecasting models traditionally rate teams on past match results, that is based on the number of goals scored. Goals, however, involve a high element of chance and thus past results often do not reflect the performances of the…
A rigorous general definition of quantum probability is given, which is valid for elementary events and for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
This thesis presents some geometric insights into three different types of two player prediction games -- namely general learning task, prediction with expert advice, and online convex optimization. These games differ in the nature of the…