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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big SPECIAL SEMINAR}
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\centerline{4:15 p.m., Wednesday, December 8, 1999}
\centerline{Sequoia Hall Rm. 200}

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\centerline{\sl Sonia Petrone}
\centerline{\sl Universita' dell'Insubria - Varese. Italy}
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\centerline{\bf An exponential family random scheme for non-parametric priors}
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For Bayesian semi- and non-parametric inference, we need a prior on a
``large'' class of distribution functions on the sample space;  and in
applications where we think of the data as continuous, we want 
a nonparametric prior which selects an absolutely continuous distribution
function.

In this work we describe one way of constructiong a nonparametric prior
with the required  properties, based on the notion of ``Feller-type
approximation.'' Roughly speaking, we model the density of the data as
a mixture of given densities. The components of the mixture have no unknown
parameters and are related to the natural exponential family. The mixing
weights and the number of components of the mixture
(or the order of approximation) are unknown and have a prior distribution.

Applications include density estimation, estimating a mixing distribution
and non parametric regression. A nice property of the proposed estimators
is that the choice of the smoothing parameter is driven by
the data through its posterior distributions.  This is still work in
progress and examples and results about
consistency of the posterior will be restricted to the case of data in [0,1].

Joint work  with Piero Veronese and Larry Wasserman.


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