Abstract:
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We consider the sparse high dimensional linear regression model Y=Xb+Z where b is a sparse vector. For this problem, Bayesian posterior contraction results have been shown, according to which if the prior is suitably chosen, the posterior distribution of b puts most of its mass in a small neighborhood near the truth. However there have been few results about the shape of that posterior distribution. We were able to give a prior distribution for b such that under the truth, the posterior marginal distribution of a single entry b_i is asymptotically normal with its mean equal to an efficient estimator and its variance close to the Fisher information at the truth. Based on this Bernstein-von Mises phenomenon, one could deduce that Bayesian credible interval matches with the frequentist confidence interval constructed from an efficient estimator for b_i.
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