  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 address these
challenges by suggesting an encompassing strategy that eliminates the need to
specify the number of local extrema, which leads to a remarkably simple, fast
semiparametric Bayesian approach for inference on local extrema. We provide
closed-form characterization of the posterior distribution and study its large
sample behaviors under this encompassing regime. We show a multi-modal
Bernstein-von Mises phenomenon in which 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. We
illustrate the method through simulations and a real data application to
event-related potential analysis.
