Related papers: De Finetti's Theorem in Categorical Probability
The classical de Finetti Theorem classifies the $\mathrm{Sym}(\mathbb N)$-invariant probability measures on $[0,1]^{\mathbb N}$. More precisely it states that those invariant measures are combinations of measures of the form…
The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is…
A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finetti's theorem characterizes all $\{0,1\}$-valued exchangeable sequences as a "mixture" of…
The Glivenko--Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform…
We give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random…
Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely…
Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to…
De Finetti's theorem, also called the de Finetti-Hewitt-Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture…
Let $\Om$ be a Borel subset of $S^\Bbb N$ where $S$ is countable. A measure is called exchangeable on $\Om$, if it is supported on $\Om$ and is invariant under every Borel automorphism of $\Om$ which permutes at most finitely many…
We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically…
Let A be a standard Borel space, and consider the space A^{\bbN^{(k)}} of A-valued arrays indexed by all size-k subsets of \bbN. This paper concerns random measures on such a space whose laws are invariant under the natural action of…
We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to…
We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Koml\'{o}s-Berkes theorem, the usual strong law of large numbers for…
Markov categories have recently turned out to be a powerful high-level framework for probability and statistics. They accommodate purely categorical definitions of notions like conditional probability and almost sure equality, as well as…
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical…
A finite form of de Finetti's representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $n\geq k$ is close to a mixture…
We study a category of probability spaces and measure-preserving Markov kernels up to almost sure equality. This category contains, among its isomorphisms, mod-zero isomorphisms of probability spaces. It also gives an isomorphism between…
This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finetti's representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the…
A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n =…
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting…