Objective Bayesian trend filtering via adaptive piecewise polynomial regression

Several methods have been developed for nonparametric regression problems, including classical approaches such as kernels, local polynomials, smoothing splines, sieves, and wavelets, as well as relatively new methods such as lasso, generalized lasso, and trend filtering. This study proposes an objective Bayesian trend filtering method based on model selection. The procedure followed in this study estimates the functions based on adaptive piecewise polynomial regression models with two components. First, we determine the intervals with varying trends using Bayesian binary segmentation and then evaluate the most reasonable trend via Bayesian model selection at these intervals. This trend filtering procedure follows Bayesian model selection that uses intrinsic priors, which eliminated any subjective input. Additionally, we prove that the proposed method using these intrinsic priors was consistent when applied to large sample sizes. The behavior of the proposed Bayesian trend filtering procedure is compared with the trend filtering using a simulation study and real examples. Finally, we apply the proposed method to detect the variance change points under mean changes, whereas the existing methods yielded inaccurate estimates of the variance change points when the mean varied smoothly, as the sudden-change assumption was violated in such cases.