Adaptive shrinkage of smooth functional effects towards a predefined functional subspace
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In this paper, we propose a new horseshoe-type prior hierarchy for adaptively shrinking spline-based functional effects towards a predefined vector space of parametric functions. Instead of shrinking each spline coefficient towards zero, we use an adapted horseshoe prior to control the deviation from the predefined vector space. For this purpose, the modified horseshoe prior is set up with one scale parameter per spline and not one per coefficient. The presented prior allows for a large number of basis functions to capture all kinds of functional effects while the estimated functional effect is prevented from a highly oscillating overfit. We achieve this by integrating a smoothing penalty similar to the random walk prior commonly applied in Bayesian P-spline priors. In a simulation study, we demonstrate the properties of the new prior specification and compare it to other approaches from the literature. Furthermore, we showcase the applicability of the proposed method by estimating the energy consumption in Germany over the course of a day. For inference, we rely on Markov chain Monte Carlo simulations combining Gibbs sampling for the spline coefficients with slice sampling for all scale parameters in the model.
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