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

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. (2022) 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.

[1]  Meng Li,et al.  Optimal plug-in Gaussian processes for modelling derivatives , 2022, 2210.11626.

[2]  Debdeep Pati,et al.  Modality-Constrained Density Estimation via Deformable Templates , 2021, Technometrics.

[3]  Zejian Liu,et al.  Equivalence of Convergence Rates of Posterior Distributions and Bayes Estimators for Functions and Nonparametric Functionals , 2020, 2011.13967.

[4]  M. Vannucci,et al.  Bayesian inference for stationary points in Gaussian process regression models for event‐related potentials analysis , 2020, Biometrics.

[5]  John F. Magnotti,et al.  Functional group bridge for simultaneous regression and support estimation , 2020, Biometrics.

[6]  Meng Li,et al.  On the Estimation of Derivatives Using Plug-in KRR Estimators , 2020, 2006.01350.

[7]  Meng Li,et al.  Non-asymptotic Analysis in Kernel Ridge Regression , 2020, ArXiv.

[8]  L. Devroye,et al.  The total variation distance between high-dimensional Gaussians , 2018, 1810.08693.

[9]  Yusha Liu,et al.  Function-on-scalar quantile regression with application to mass spectrometry proteomics data , 2018, The Annals of Applied Statistics.

[10]  Debdeep Pati,et al.  Frequentist coverage and sup-norm convergence rate in Gaussian process regression , 2017, 1708.04753.

[11]  Aad van der Vaart,et al.  Fundamentals of Nonparametric Bayesian Inference , 2017 .

[12]  A. Bhattacharya,et al.  Adaptive Bayesian inference in the Gaussian sequence model using exponential-variance priors , 2015 .

[13]  C. Abraham,et al.  Bayesian regression with B‐splines under combinations of shape constraints and smoothness properties , 2015 .

[14]  Subhashis Ghosal,et al.  Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression , 2014, 1411.6716.

[15]  Dan Cheng,et al.  MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN RANDOM FIELDS. , 2014, Annals of statistics.

[16]  A. V. D. Vaart,et al.  BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS , 2014, 1403.0735.

[17]  R. Nickl,et al.  On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures , 2013, 1310.2484.

[18]  Martin J. Wainwright,et al.  Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates , 2013, J. Mach. Learn. Res..

[19]  J. Rousseau,et al.  A Bernstein–von Mises theorem for smooth functionals in semiparametric models , 2013, 1305.4482.

[20]  V. A. Menegatto,et al.  Reproducing properties of differentiable Mercer‐like kernels , 2012 .

[21]  I. Castillo A semiparametric Bernstein–von Mises theorem for Gaussian process priors , 2012 .

[22]  Armin Schwartzman,et al.  MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN 1D. , 2011, Annals of statistics.

[23]  M. Wainwright,et al.  Sampled forms of functional PCA in reproducing kernel Hilbert spaces , 2011, 1109.3336.

[24]  Thomas S. Shively,et al.  Nonparametric function estimation subject to monotonicity, convexity and other shape constraints , 2011 .

[25]  S. Walker,et al.  A Bayesian approach to non‐parametric monotone function estimation , 2009 .

[26]  Mary C. Meyer INFERENCE USING SHAPE-RESTRICTED REGRESSION SPLINES , 2008, 0811.1705.

[27]  Christian Eggeling,et al.  Fluorescence Nanoscopy in Whole Cells by Asynchronous Localization of Photoswitching Emitters , 2007, Biophysical journal.

[28]  A. Kovac,et al.  Smooth functions and local extreme values , 2007, Comput. Stat. Data Anal..

[29]  A. Egner,et al.  Resolution of λ /10 in fluorescence microscopy using fast single molecule photo-switching , 2007 .

[30]  Ding-Xuan Zhou,et al.  Learning Theory: An Approximation Theory Viewpoint , 2007 .

[31]  Yongdai Kim The Bernstein–von Mises theorem for the proportional hazard model , 2006, math/0611230.

[32]  Rui Liu,et al.  Nonparametric Inference for Local Extrema with Application to Oligonucleotide Microarray Data in Yeast Genome , 2006, Biometrics.

[33]  David B Dunson,et al.  A transformation approach for incorporating monotone or unimodal constraints. , 2005, Biostatistics.

[34]  Yongdai Kim,et al.  A Bernstein–von Mises theorem in the nonparametric right-censoring model , 2004, math/0410083.

[35]  Brian Neelon,et al.  Bayesian Isotonic Regression and Trend Analysis , 2004, Biometrics.

[36]  Miguel Á. Carreira-Perpiñán,et al.  On the Number of Modes of a Gaussian Mixture , 2003, Scale-Space.

[37]  C. Holmes,et al.  Generalized monotonic regression using random change points , 2003, Statistics in medicine.

[38]  Ronald W. Davis,et al.  Replication dynamics of the yeast genome. , 2001, Science.

[39]  P. Davies,et al.  Local Extremes, Runs, Strings and Multiresolution , 2001 .

[40]  T Gasser,et al.  The analysis of the EEG , 1996, Statistical methods in medical research.

[41]  Audris Mockus,et al.  A nonparametric Bayes method for isotonic regression , 1995 .

[42]  G. Wahba Spline Models for Observational Data , 1990 .

[43]  Marina Schmid,et al.  An Introduction To The Event Related Potential Technique , 2016 .

[44]  Karin Rothschild,et al.  A Course In Functional Analysis , 2016 .

[45]  de R René Jonge,et al.  Semiparametric Bernstein-von Mises for the error standard deviation , 2013 .

[46]  Van Der Vaart,et al.  The Bernstein-Von-Mises theorem under misspecification , 2012 .

[47]  J. Ramsay Estimating smooth monotone functions , 1998 .