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Abstract:
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Assumed structure in a high-dimensional parameter can be treated as prior information so, naturally, the Bayesian's prior distribution should at least encourage this structure in the posterior distribution. But while information is available about the structure, there are also structure-specific parameters about which there generally is very little information, and, unfortunately, the high-dimensionality implies that the prior tails actually matter in both the asymptotics and finite-sample computations. One recent idea to reduce the impact of prior tails is to use an informative, data-driven prior centering. With such an approach, computationally simple, thin-tailed conjugate priors can be used, without the usual sacrifices in the asymptotic properties associated with thin tails. In this talk, I will describe this empirical prior approach in the context of several applications, including estimation sparse high-dimensional precision matrices in Gaussian graphical models and estimation of a high-dimensional mean vector with a piecewise polynomial structure.
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