Exchangeable random measures

By Tim Austin. 2013

47 citations recorded by the source corpus.

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 permutations of \bbN. The main result is a representation theorem for such `exchangeable' random measures, obtained using the classical representation theorems for exchangeable arrays due to de Finetti, Hoover, Aldous and Kallenberg. After proving this representation, two applications of exchangeable random measures are given. The first is a short new proof of the Dovbysh-Sudakov Representation Theorem for exchangeable PSD matrices. The second is in the formulation of a natural class of limit objects for dilute mean-field spin glass models, retaining more information than just the limiting Gram-de Finetti matrix used in the study of the Sherrington-Kirkpatrick model.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3543d2a026be682e9138f8c573524920194bbbdc

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In Artificial Intelligence

In Discrete Mathematics and Combinatorics

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Citations in the corpus

A hierarchical version of the de Finetti and Aldous-Hoover representations (2013, result): “[2], Chapters 7–9 in [11], or [3], [4] and [5] for recent overviews and results) and the g oal of this paper is to study another notion of exchangeability that is motivated by spin glass mo dels and, in particular, by the work of Mézard and Parisi on diluted models, [12].”
Probabilistic Symmetries and Invariant Neural Networks (2020, background): “The noise variable η acts as a generic source of randomness that is “outsourced”, a term borrowed from Austin (2015).”