This note defines cones in homotopy probability theory and demonstrates that a cone over a space is a reasonable replacement for the space. The homotopy Gaussian distribution in one variable is revisited as a cone on the ordinary Gaussian.

The following manuscript was written for my lectures at the special semester ’Free probability theory and operator spaces’, IHP, Paris, 1999. A complementary series of lectures was given by A. Nica. It is planned to combine both manuscripts into a book A. Nica and R. Speicher: Lectures on the Combinatorics of Free Probability, which will be published by Cambridge University Press

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.

These lecture notes were written for MATH 6710 at Cornell University in the Fall semester of 2013. They were revised in the Fall of 2015 and the schedule on the following page re ects that semester. These notes are for personal educational use only and are not to be published or redistributed. Almost all of the material, much of the structure, and some of the language comes directly from the co...

New foundations for probability theory are suggested and investigated, derived from the formalized systems theory and using also some other branches of mathematical logic. The probability measures are supposed to take their values not in the unit interval of reals, but in a special kind of Boolean algebras. Some analogies as well as differences of this approach with respect to the classical Kol...

Coupling is a powerful method in probability theory through which random variables can be compared with each other. Coupling has been applied in a broad variety of contexts, e.g. to prove limit theorems, to derive inequalities, or to obtain approximations. The present course is intended for master students and PhD students. A basic knowledge of probability theory is required, as well as some fa...

In a probability-based reasoning system, Bayes’ theorem and its variations are often used to revise the system’s beliefs. However, if the explicit conditions and the implicit conditions of probability assignments are properly distinguished, it follows that Bayes’ theorem is not a generally applicable revision rule. Upon properly distinguishing belief revision from belief updating, we see that J...

Within the Kolmogorov theory of probability, Bayes’ rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for ...