Novel approaches to uncertainty quantification in nonparametric settings

This thesis contains explorations of uncertainty quantification in a variety of nonparametric statistical settings, focusing on novel uses of uncertainty to enhance inference in ways which may otherwise be overlooked. The first two chapters concern the Laplace approximation for high-dimensional integrals. This approximation is commonly used in complex models — for instance, to obtain marginal likelihoods in hierarchical models for use in optimization. The quality of the approximation may depend intimately on the true shape of the integrand. To assess this, we use probabilistic numerics, recasting the approximation problem and its inherent uncertainty in the framework of probability theory. We develop a diagnostic tool for the Laplace approximation and its underlying shape assumptions with this framework. The tool is decidedly non-asymptotic and is not intended as a full substitute for other quadrature methods. Rather, it is simply meant to test the feasibility of the assumptions underpinning the Laplace approximation with as little computational burden as possible. Next, we provide a comprehensive overview of uncertainty quantification methods for density estimation. There are many methods of estimating an unknown density and constructing "plausible" sets in which it may lie. Examples of the latter include pointwise intervals, simultaneous bands, or balls in a function space; and they may be frequentist or Bayesian in interpretation. Here, we thoroughly review literature on density inference, covering a broad spectrum of ideas ranging from theoretical to practical. Finally, we propose a novel approach to modelling in a micro-macro situation, in which group-level outcomes are dependent on covariates measured at the level of individuals within groups. Although such models are perhaps underrepresented in the literature, they have applications in economics, epidemiology, and the social sciences. Our approach is an empirical Bayesian method which jointly infers group-specific covariate densities and uses them as predictors in a functional linear model. Unlike many similar methods, the assumptions made on the structure of the data are minimal, allowing for better inference and a fuller quantification of uncertainty in a wide variety of situations.