634 research outputs found
Sharp Total Variation Bounds for Finitely Exchangeable Arrays
In this article we demonstrate the relationship between finitely exchangeable
arrays and finitely exchangeable sequences. We then derive sharp bounds on the
total variation distance between distributions of finitely and infinitely
exchangeable arrays
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
Bayesian nonparametrics for Sparse Dynamic Networks
We propose a Bayesian nonparametric prior for time-varying networks. To each
node of the network is associated a positive parameter, modeling the
sociability of that node. Sociabilities are assumed to evolve over time, and
are modeled via a dynamic point process model. The model is able to (a) capture
smooth evolution of the interaction between nodes, allowing edges to
appear/disappear over time (b) capture long term evolution of the sociabilities
of the nodes (c) and yield sparse graphs, where the number of edges grows
subquadratically with the number of nodes. The evolution of the sociabilities
is described by a tractable time-varying gamma process. We provide some
theoretical insights into the model and apply it to three real world datasets.Comment: 10 pages, 8 figure
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