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Stationary points embedded in the derivatives are often critical for a modelto be interpretable and may be considered as key features of interest in manyapplications. We propose a semiparametric Bayesian model to efficiently inferthe locations of stationary points of a nonparametric function, while treatingthe function itself as a nuisance parameter. We use Gaussian processes as aflexible prior for the underlying function and impose derivative constraints tocontrol the function's shape via conditioning. We develop an inferentialstrategy that intentionally restricts estimation to the case of at least onestationary point, bypassing possible mis-specifications in the number ofstationary points and avoiding the varying dimension problem that often bringsin computational complexity. We illustrate the proposed methods usingsimulations and then apply the method to the estimation of event-relatedpotentials (ERP) derived from electroencephalography (EEG) signals. We show howthe proposed method automatically identifies characteristic components andtheir latencies at the individual level, which avoids the excessive averagingacross subjects which is routinely done in the field to obtain smooth curves.By applying this approach to EEG data collected from younger and older adultsduring a speech perception task, we are able to demonstrate how the time courseof speech perception processes changes with age.

Bayesian Inference for Stationary Points in Gaussian Process Regression Models for Event-Related Potentials Analysis