José R. Berrendero, Antonio Coín and Antonio Cuevas
The compiled PDF version of this preprint is available here. It can also be generated via the command latexmk -pdf paper (requires pdflatex). The template used is a slightly modified version of arxiv-style.
We propose a novel Bayesian methodology for inference in functional linear and logistic regression models based on the theory of reproducing kernel Hilbert spaces (RKHS's). We introduce general models that build upon the RKHS generated by the covariance function of the underlying stochastic process, and whose formulation includes as particular cases all finite-dimensional models based on linear combinations of marginals of the process, which can collectively be seen as a dense subspace made of simple approximations. By imposing a suitable prior distribution on this dense functional space we can perform data-driven inference via standard Bayes methodology, estimating the posterior distribution through reversible jump Markov chain Monte Carlo methods. In this context, our contribution is two-fold. First, we derive a theoretical result that guarantees posterior consistency, based on an application of a classic theorem of Doob to our RKHS setting. Second, we show that several prediction strategies stemming from our Bayesian procedure are competitive against other usual alternatives in both simulations and real data sets, including a Bayesian-motivated variable selection method.
Keywords: functional data analysis, functional regression, reproducing kernel Hilbert space, reversible jump MCMC, Bayesian inference, posterior consistency.
The associated Python code can be consulted in this repository.
Grollemund, P.-M., Abraham, C., Baragatti, M., and Pudlo, P. (2019). "Bayesian Functional Linear Regression with Sparse Step Functions". In: Bayesian Analysis 14(1), pp. 111–135 (preprint | doi).
Berrendero, J. R., Bueno-Larraz, B., and Cuevas, A. (2023). "On functional logistic regression: some conceptual issues". In: TEST 32, pp. 321-349 (preprint | doi).
Miller, J. W. (2023). "Consistency of mixture models with a prior on the number of components". In: Dependence Modeling 11(1), pp. 20220150 (preprint | doi).
Karnesis, N., Katz, M. L., Korsakova, N., Gair, J. R., and Stergioulas, N. (2023). "Eryn: a multipurpose sampler for Bayesian inference." In: Monthly Notices of the Royal Astronomical Society 526(4), pp. 4814–4830 (preprint | doi).
Berrendero, J. R., Cholaquidis, A., and Cuevas, A. (2024). "On the functional regression model and its finite-dimensional approximations.". In: Statistical Papers, pp. 1-35 (preprint | doi).