There are a wide range of applications where the local extrema of a function are important quantities. However, there is surprisingly little research on how to use uncertainty quantification to infer local extrema in the presence of noise. By viewing the function as an infinite-dimensional nuisance parameter, the semiparametric formulation of this problem poses a difficult challenge both methodologically and theoretically. Local extrema are very irregular. In this article, we build on the differentially constrained Gaussian process recently proposed by Yu et al. (2022) to derive what is called a comprehensive approach that may index multiple extrema by a single parameter. We provide a closed-form characterization of the posterior distribution and study the behavior of large samples under this unconventional and comprehensive regime. We show that posterior measurements converge on Gaussian mixtures with a number of components consistent with the underlying truth, leading to posterior investigations that account for multimodality. Point and interval estimates of local extrema with frequentist properties are also provided. A comprehensive approach leads to a very simple and fast semiparametric approach for local extrema inference. We illustrate this method by applying simulation and real-world data to potential event-related analyses.
