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Abstract:
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We provide a unified theoretical framework for Bayesian sparse regression when a possibly infinite-dimensional nuisance parameter is involved in a model. A mixture of a point mass at zero and a Laplace distribution is used for the prior distribution on sparse regression coefficients and an appropriate prior distribution is assigned to a nuisance parameter to yield the optimal posterior contraction rates. A shape approximation to the posterior distribution using a Bernstein-von Mises-type theorem is studied to show the model selection consistency. Numerous examples are discussed including regression with partially sparse coefficients, multiple response models with missing components, multivariate measurement error models, mixed effects models, partial linear models, and nonparametric heteroscedastic regression models.
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