Related papers: A Third Information-Theoretic Approach to Finite d…
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 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…
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…
We review old and recent finite de Finetti theorems in total variation distance and in relative entropy, and we highlight their connections with bounds on the difference between sampling with and without replacement. We also establish two…
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires…
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…
We develop a framework for the operationalization of models and parameters by combining de Finetti's representation theorem with a conditional form of Sanov's theorem. This synthesis, the tilted de Finetti theorem, shows that conditioning…
Our capacity to process information depends on the computational power at our disposal. Information theory captures our ability to distinguish states or communicate messages when it is unconstrained with unrivaled beauty and elegance. For…
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…
Motivated by recent interests in predictive inference under distribution shift, we study the problem of approximating finite weighted exchangeable sequences by a mixture of finite sequences with independent terms. Various bounds are derived…
Mixtures of product distributions are a powerful device for learning about heterogeneity within data populations. In this class of latent structure models, de Finetti's mixing measure plays the central role for describing the uncertainty…
A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti's Theorem. In its original form it applies to infinite 0-1 valued exchangeable sequences. Later it was extended and…
De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only…
In this note, we show that the relative entropy of an empirical distribution of $n$ samples drawn from a set of size $k$ with respect to the true underlying distribution is exponentially concentrated around its expectation, with central…
Let $K_n$ denote the number of distinct values among the first $n$ terms of an infinite exchangeable sequence of random variables $(X_1,X_2,\ldots)$. We prove for $n=3$ that the extreme points of the convex set of all possible laws of $K_3$…
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 extend de Finetti's (1937) notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We prove…
The de Finetti theorem and its extensions concern the structure of multipartite probability distributions with certain symmetry properties, the paradigmatic original example being permutation symmetry. These theorems assert that such…
We extend de Finetti's [Ann. Inst. H. Poincar\'{e} 7 (1937) 1--68] notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than…
Information theory provides tools to predict the performance of a learning algorithm on a given dataset. For instance, the accuracy of learning an unknown parameter can be upper bounded by reducing the learning task to hypothesis testing…