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Abstract:
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Bayesian testing is studied asymptotically on a high-dimensional normal means model, in which the null hypothesis of zero means in all dimensions is tested against general alternatives. This is known to serve as a canonical model for smooth goodness-of-fit testing. The asymptotic setup is such that prior mass placed on the null hypothesis is allowed to decrease as dimensionality increases, while at the same time the dispersion of the prior placed on the alternative is allowed to increase, thereby tending toward a noninformative specification. The interest is to deduce appropriate rates of change for the prior parameters in order to assure the test's reasonable behavior in large dimensions. When geometric constraints representing smoothness are imposed, it becomes sensible to weight the prior, and an expanded objective is to deduce suitable configurations for those weights.
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