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Register a new userDe Finetti for mathematics undergraduates
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Authors
Daniele Mundici
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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.
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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 random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of de Finetti theorem on exchangeable sequences. In the end of the paper, we will discuss the other two invariance conditions which lead to some trivial results.
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 exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $mu_k$ of the first $k$ components has a representation of the form $mu_k=int_{{mathcal P}_{frac{1}{N}}(X)} F_{N,k}(lambda) , mbox{d} alpha(lambda)$ for some probability measure $alpha$ on the set of $1/N$-quantized probability measures on $X$ and certain universal polynomials $F_{N,k}$. The latter consist of a leading term $N^{k-1}! /{small prod_{j=1}^{k-1}(N! -! j), lambda^{otimes k}}$ and a finite, exponentially decaying series of correlated corrections of order $N^{-j}$ ($j=1,...,k$). The $F_{N,k}(lambda)$ are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals $lambda$. Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis-Freedman between finite and infinite exchangeable laws.
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 as black boxes and mathematically described as conditional probability distributions. A major obstacle in the analysis of DIQKD protocols is the huge space of possible black box behaviors. De Finetti theorems can help to overcome this problem by reducing the analysis to black boxes that have an iid structure. Here we show two new de Finetti theorems that relate conditional probability distributions in the quantum set to de Finetti distributions (convex combinations of iid distributions), that are themselves in the quantum set. We also show how one of these de Finetti theorems can be used to enforce some restrictions onto the attacker of a DIQKD protocol. Finally we observe that some desirable strengthenings of this restriction, for instance to collective attacks only, are not straightforwardly possible.
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 symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.
An introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS, https://oeis.org) for graduate students in mathematics
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