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). These models build upon the RKHS associated with the covariance function of the underlying stochastic process, and can be viewed as a finite-dimensional approximation to the classical functional regression paradigm. The corresponding functional model is determined by a function living on a dense subspace of the RKHS of interest, which has a tractable parametric form based on linear combinations of the kernel. By imposing a suitable prior distribution on this functional space, we can naturally perform data-driven inference via standard Bayes methodology, estimating the posterior distribution through Markov chain Monte Carlo (MCMC) methods. In this context, our contribution is two-fold. First, we derive a theoretical result that guarantees posterior consistency in these models, 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 formulation are competitive against other usual alternatives in both simulations and real data sets, including a Bayesian-motivated variable selection procedure.

翻译：我们提出了一种新的贝叶斯方法，用于函数线性与逻辑回归模型的推断，该方法基于再生核希尔伯特空间（RKHS）理论。这些模型依赖于与底层随机过程协方差函数相关联的RKHS，并可作为经典函数回归范式的一种有限维近似。相应的函数模型由定义在目标RKHS稠密子空间上的函数确定，该函数具有基于核线性组合的可处理参数形式。通过对该函数空间施加适当的先验分布，我们可以通过标准贝叶斯方法自然地进行数据驱动推断，并借助马尔可夫链蒙特卡洛（MCMC）方法估计后验分布。在此背景下，我们的贡献体现在两个方面：首先，我们推导了一个理论结果，保证了这些模型的后验一致性，该结果基于Doob经典定理在RKHS框架下的应用；其次，我们证明，在模拟与真实数据集上，基于我们贝叶斯公式的多种预测策略（包括一种贝叶斯驱动的变量选择程序）在性能上优于其他常用替代方法。