Posterior contraction and credible regions for level sets

Abstract

For a given function f on a multivariate domain, the level sets, given by $\{x:f(x)=c\}$ for different values of c, provide important geometrical insights about the underlying function of interest. The distance on level sets of two functions may be measured by the Hausdorff metric or a metric based on the Lebesgue measure of a discrepancy, both of which can be linked with the $L^{\mathrm{\infty }}$-distance on the underlying functions. In a Bayesian framework, we derive posterior contraction rates and optimal sized credible sets with assured frequentist coverage for level sets in some nonparametric settings by extending some univariate $L^{\mathrm{\infty }}$-posterior contraction rates to the corresponding multivariate settings. For the multivariate Gaussian white noise model, adaptive Hausdorff and Lebesgue contraction rates for levels sets of the signal function and its mixed order partial derivatives are derived using a wavelet series prior on the function. Assuming a known smoothness level of the signal function, an optimal sized credible region for a level set with assured frequentist coverage is derived based on a multidimensional trigonometric series prior. For the nonparametric regression problem, adaptive rates for level sets of the function and its mixed partial derivatives are obtained using a multidimensional wavelet series prior. When the smoothness level is given, optimal sized credible regions with assured frequentist coverage are obtained using a finite random series prior based on tensor products of B-splines. We also derive Hausdorff and Lebesgue contraction rates of a multivariate density function under a known smoothness setting.