B1669
Title: Optimal plug-in Gaussian processes for inferring functional derivatives and equivalence with kernel methods
Authors: Meng Li - Rice University (United States) [presenting]
Abstract: Functional derivatives are key nonparametric functionals in wide-ranging applications that require the analysis of the rate of change in unknown functions. In the Bayesian paradigm, Gaussian processes (GPs) are routinely used as flexible priors for unknown functions but lack a comprehensive theoretical and methodological basis for derivative estimation. A plug-in strategy is presented by differentiating the posterior distribution with GP priors for derivatives of any order. Contrary to existing perceptions of sub-optimality, it is demonstrated that plug-in GPs offer adaptive and optimal posterior contraction rates. An empirical Bayes approach for data-driven hyperparameter tuning is also introduced. The approach satisfies optimal rate conditions while maintaining computational efficiency. To the knowledge, this constitutes the first positive result for plug-in GPs in the context of inferring derivative functionals and leads to a practically simple nonparametric Bayesian method with optimal and adaptive hyperparameter tuning for simultaneously estimating the regression function and its derivatives. Time permitting, an equivalence connection between GPs and kernel ridge regression will be introduced for function derivatives, which serve as a mathematical foundation.
