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In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {it axiom} for the probability of the disjunction of two incompatible events becomes a {it consequence} of de Finettis logic-operational consistency notion. Working in the context of boolean algebras, we prove de Finettis theorem. The mathematical background required is little more than that which is taught in high school. As a preliminary step we prove what de Finetti called ``the Fundamental Theorem of Probability, his main contribution both to Booles probabilistic inference problem on the object of probability theory, and to its modern reformulation known as the optimization version of the probabilistic satisfiability problem. In a final section, we give a self-contained combinatorial proof of de Finettis exchangeability theorem.
We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative rand
We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchange
The aim of device-independent quantum key distribution (DIQKD) is to study protocols that allow the generation of a secret shared key between two parties under minimal assumptions on the devices that produce the key. These devices are merely modeled
We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symme
An introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS, https://oeis.org) for graduate students in mathematics