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arXiv · 2304.03927

De Finetti's theorem and related results for infinite weighted exchangeable sequences

Rina Foygel Barber·Emmanuel J. Candes·Aaditya Ramdas·Ryan J. Tibshirani
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Abstract

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 of independent and identically distributed (i.i.d.) sequences of random variables. In this paper, we consider a weighted generalization of exchangeability that allows for weight functions to modify the individual distributions of the random variables along the sequence, provided that -- modulo these weight functions -- there is still some common exchangeable base measure. We study conditions under which a de Finetti-type representation exists for weighted exchangeable sequences, as a mixture of distributions which satisfy a weighted form of the i.i.d. property. Our approach establishes a nested family of conditions that lead to weighted extensions of other well-known related results as well, in particular, extensions of the zero-one law and the law of large numbers.

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Rina Foygel Barber, Emmanuel J. Candes, Aaditya Ramdas, Ryan J. Tibshirani. 2023-04-08. De Finetti's theorem and related results for infinite weighted exchangeable sequences. https://arxiv.org/abs/2304.03927

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