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In many applications that involve the inference of an unknown smoothfunction, the inference of its derivatives will often be just as important asthat of the function itself. To make joint inferences of the function and itsderivatives, a class of Gaussian processes called $p^{\text{th}}$ orderIntegrated Wiener's Process (IWP), is considered. Methods for constructing afinite element (FEM) approximation of an IWP exist but have focused only on theorder $p = 2$ case which does not allow appropriate inference for derivatives,and their computational feasibility relies on additional approximation to theFEM itself. In this article, we propose an alternative FEM approximation,called overlapping splines (O-spline), which pursues computational feasibilitydirectly through the choice of test functions, and mirrors the construction ofan IWP as the Ospline results from the multiple integrations of these same testfunctions. The O-spline approximation applies for any order $p \in\mathbb{Z}^+$, is computationally efficient and provides consistent inferencefor all derivatives up to order $p-1$. It is shown both theoretically, andempirically through simulation, that the O-spline approximation converges tothe true IWP as the number of knots increases. We further provide a unified andinterpretable way to define priors for the smoothing parameter based on thenotion of predictive standard deviation (PSD), which is invariant to the order$p$ and the placement of the knot. Finally, we demonstrate the practical use ofthe O-spline approximation through simulation studies and an analysis of COVIDdeath rates where the inference is carried on both the function and itsderivatives where the latter has an important interpretation in terms of thecourse of the pandemic.

Model-based Smoothing with Integrated Wiener Processes and Overlapping Splines