3,514 research outputs found
Local Exchangeability
Exchangeability---in which the distribution of an infinite sequence is
invariant to reorderings of its elements---implies the existence of a simple
conditional independence structure that may be leveraged in the design of
probabilistic models, efficient inference algorithms, and randomization-based
testing procedures. In practice, however, this assumption is too strong an
idealization; the distribution typically fails to be exactly invariant to
permutations and de Finetti's representation theory does not apply. Thus there
is the need for a distributional assumption that is both weak enough to hold in
practice, and strong enough to guarantee a useful underlying representation. We
introduce a relaxed notion of local exchangeability---where swapping data
associated with nearby covariates causes a bounded change in the distribution.
We prove that locally exchangeable processes correspond to independent
observations from an underlying measure-valued stochastic process. We thereby
show that de Finetti's theorem is robust to perturbation and provide further
justification for the Bayesian modelling approach. Using this probabilistic
result, we develop three novel statistical procedures for (1) estimating the
underlying process via local empirical measures, (2) testing via local
randomization, and (3) estimating the canonical premetric of local
exchangeability. These three procedures extend the applicability of previous
exchangeability-based methods without sacrificing rigorous statistical
guarantees. The paper concludes with examples of popular statistical models
that exhibit local exchangeability
Tractability through Exchangeability: A New Perspective on Efficient Probabilistic Inference
Exchangeability is a central notion in statistics and probability theory. The
assumption that an infinite sequence of data points is exchangeable is at the
core of Bayesian statistics. However, finite exchangeability as a statistical
property that renders probabilistic inference tractable is less
well-understood. We develop a theory of finite exchangeability and its relation
to tractable probabilistic inference. The theory is complementary to that of
independence and conditional independence. We show that tractable inference in
probabilistic models with high treewidth and millions of variables can be
understood using the notion of finite (partial) exchangeability. We also show
that existing lifted inference algorithms implicitly utilize a combination of
conditional independence and partial exchangeability.Comment: In Proceedings of the 28th AAAI Conference on Artificial Intelligenc
On a representation theorem for finitely exchangeable random vectors
A random vector with the taking values in an
arbitrary measurable space is exchangeable if its law is the
same as that of for any permutation
. We give an alternative and shorter proof of the representation result
(Jaynes \cite{Jay86} and Kerns and Sz\'ekely \cite{KS06}) stating that the law
of is a mixture of product probability measures with respect to a signed
mixing measure. The result is "finitistic" in nature meaning that it is a
matter of linear algebra for finite . The passing from finite to an
arbitrary one may pose some measure-theoretic difficulties which are avoided by
our proof. The mixing signed measure is not unique (examples are given), but we
pay more attention to the one constructed in the proof ("canonical mixing
measure") by pointing out some of its characteristics. The mixing measure is,
in general, defined on the space of probability measures on , but for
, one can choose a mixing measure on .Comment: We here give an alternative proof of the measurability of the random
signed-measure underlying the construction. We also add an independent proof
of the main algebraic fact used in the paper. Title update
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