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Bayesian Multiscale Smoothing for Making Inferences about Features in Scatter Plots

A rather common problem of data analysis is to nd interesting features, such as local minima, maxima and trends in a scatter plot. Variance in the data can then be a problem and inferences about features must be made at some selected level of signi cance. The recently introduced SiZer technique uses a family of nonparametric smooths of the data to uncover features in a whole range of scales. To aid the analysis, a color map is generated that visualizes the inferences made about the signi cance of the features. The purpose of this article is to present Bayesian versions of the SiZer methodology. Both an analytically solvable regression model and a fully Bayesian approach that uses Gibbs sampling are presented. The prior distributions of the smooths are based on a roughness penalty. Both discrete and continuous models are discussed. Simulation based algorithms are proposed for making simultaneous inferences about the features in the data