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A filament consists of local maximizers of a smooth function $f$ when movingin a certain direction. A filamentary structure is an important feature of theshape of an object and is also considered as an important lower dimensionalcharacterization of multivariate data. There have been some recent theoreticalstudies of filaments in the nonparametric kernel density estimation context.This paper supplements the current literature in two ways. First, we provide aBayesian approach to the filament estimation in regression context and studythe posterior contraction rates using a finite random series of B-splinesbasis. Compared with the kernel-estimation method, this has a theoreticaladvantage as the bias can be better controlled when the function is smoother,which allows obtaining better rates. Assuming that $f: \mathbb{R}^2 \mapsto\mathbb{R}$ belongs to an isotropic H\"{o}lder class of order $\alpha \geq 4$,with the optimal choice of smoothing parameters, the posterior contractionrates for the filament points on some appropriately defined integral curves andfor the Hausdorff distance of the filament are both $(n/\logn)^{(2-\alpha)/(2(1+\alpha))}$. Secondly, we provide a way to construct acredible set with sufficient frequentist coverage for the filaments. Wedemonstrate the success of our proposed method in simulations and oneapplication to earthquake data.

Posterior Contraction and Credible Sets for Filaments of Regression Functions