Semiparametric Bayesian Inference for Local Extrema of Functions in the Presence of Noise

TL;DR

This paper introduces a fast, semiparametric Bayesian method for inferring multiple local extrema of noisy functions, providing uncertainty quantification and theoretical guarantees.

Contribution

It develops an encompassing Bayesian approach using derivative-constrained Gaussian processes to infer multiple local extrema with uncertainty quantification in a semiparametric setting.

Findings

Posterior distribution converges to a mixture of Gaussians matching the true number of extrema.

Method provides accurate point and interval estimates with good frequentist properties.

Approach is computationally simple and effective, demonstrated through simulations and real data.

Abstract

There is a wide range of applications where the local extrema of a function are the key quantity of interest. However, there is surprisingly little work on methods to infer local extrema with uncertainty quantification in the presence of noise. By viewing the function as an infinite-dimensional nuisance parameter, a semiparametric formulation of this problem poses daunting challenges, both methodologically and theoretically, as (i) the number of local extrema may be unknown, and (ii) the induced shape constraints associated with local extrema are highly irregular. In this article, we build upon a derivative-constrained Gaussian process prior recently proposed by Yu et al. (2023) to derive what we call an encompassing approach that indexes possibly multiple local extrema by a single parameter. We provide closed-form characterization of the posterior distribution and study its large sample behavior under this unconventional encompassing regime. We show that the posterior measure converges to a mixture of Gaussians with the number of components matching the underlying truth, leading to posterior exploration that accounts for multi-modality. Point and interval estimates of local extrema with frequentist properties are also provided. The encompassing approach leads to a remarkably simple, fast semiparametric approach for inference on local extrema. We illustrate the method through simulations and a real data application to event-related potential analysis.