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.

翻译：我们提出了一种基于再生核希尔伯特空间（RKHS）理论的功能性线性与逻辑回归模型推断新方法。该方法构建于底层随机过程协方差函数生成的RKHS之上，其模型框架将基于过程边际线性组合的所有有限维模型作为特例纳入其中，这些有限维模型可整体视为由简单近似构成的稠密子空间。通过对该稠密函数空间施加合适的先验分布，我们能够通过标准贝叶斯方法进行数据驱动的推断，并利用可逆跳转马尔可夫链蒙特卡洛方法估计后验分布。本研究的贡献主要体现在两个方面：首先，通过将Doob经典定理应用于RKHS框架，我们推导出保证后验一致性的理论结果；其次，在仿真和实际数据集实验中，我们证明基于该贝叶斯框架的多种预测策略（包括一种贝叶斯启发的变量选择方法）相较于其他常规替代方法具有竞争优势。