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Register a new userDe Finettis construction as a categorical limit
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This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finettis representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the Giry monad. This new limit is used to identify among exchangeable coalgebras the final one.
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We present a novel proof of de Finettis Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finettis Theorem for standard Borel spaces is an instance of this result.
Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partitions; this is an elementary and intuitive presentation of a consequence of the profiniteness of Stone spaces. We then apply this construction to answer a question of Rosinger posed in arXiv:0709.0084v2 in the negative.
Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $bigotimes_{i in J} X_i$ come in t
Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence ${bf LKHaus}simeq{bf CLCA}^{rm op}$ that was obtained by the first author more than a decade ago. Unlike the morphisms of $bf CLCA$, the morphisms of the new category and their composition law are very natural and easy to handle.
A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finettis theorem characterizes all ${0,1}$-valued exchangeable sequences as a mixture of sequences of independent random variables. We present an new, elementary proof of de Finettis Theorem. The purpose of this paper is to make this theorem accessible to a broader community through an essentially self-contained proof.
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