Rates of contraction of posterior distributions based on p-exponential priors
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We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call p-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with p-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in VZ08, namely we use prior concentration to verify prior mass and entropy conditions sufficient for posterior contraction. However, the situation is more convoluted compared to Gaussian priors leading to a more complex entropy bound which can influence negatively the obtained rate of contraction, depending on the topology of the parameter space. Subject to the more complex entropy bound, we show that the rate of contraction depends on the position of the true parameter relative to a certain Banach space associated to p-exponential measures and on the small ball probabilities of these measures. For example, we compute these quantities for α-regular p-exponential priors in separable Hilbert spaces under Besov-type regularity of the truth, in which case the entropy bound is verified to be benign.
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