Review: Alfred Renyi, Foundations of Probability

Foundations of Probability

Probability theory is that part of mathematics that is concerned with the description and modeling of random phenomena, or in a more general — but not unanimously accepted — sense, of any kind of uncertainty. Probability is assigned to random events, expressing their tendency to occur in a random experiment, or more generally to propositions, characterizing the degree of belief in their truth. ...

Cognitive Foundations of Probability

Prediction is based on past cases. We assume that a predictor can rank eventualities according to their plausibility given any memory that consists of repetitions of past cases. In a companion paper, we show that under mild consistency requirements, these rankings can be represented by numerical functions, such that the function corresponding to each eventuality is linear in the number of case ...

PROBABILITY IS SYMMETRY On Foundations of the Science of Probability

The Foundations of Probability with Black Swans

We extend the foundation of probability in samples with rare events that are potentially catastrophic, called black swans, such as natural hazards, market crashes, catastrophic climate change, and species extinction. Such events are generally treated as “outliers” and disregarded. We propose a new axiomatization of probability requiring equal treatment in the measurement of rare and frequent ev...

Algorithmic Information Theory and Foundations of Probability

The question how and why mathematical probability theory can be applied to the “real world” has been debated for centuries. We try to survey the role of algorithmic information theory (Kolmogorov complexity) in this debate. 1 Probability theory paradox One often describes the natural sciences framework as follows: a hypothesis is used to predict something, and the prediction is then checked aga...