# Integrated Model Selection and Scalability in Functional Data Analysis Through Bayesian Learning
## 1 Introduction  
The emergence of big data across diverse fields, such as biomedicine, finance, and physical modeling, has catalyzed the need for advanced analytical methodologies capable of handling complex, high-dimensional datasets that conventional discrete data analysis approaches cannot always process effectively. Such datasets often require analysis that captures and interprets their continuous and potentially high-dimensional complexities--a central promise of _functional data analysis_ (FDA) [1, 2]. Foundational work established FDA's capacity to treat each observation as an entire function [3], be it a curve, surface, or higher-dimensional structure, thereby extracting richer insights than conventional discrete point analyses. Over the past decade, FDA's scope has widened significantly to accommodatehigh-dimensional applications, with theoretical and computational advances emerging across various contexts [4; 5]. A pivotal technique within FDA is _functional principal component analysis_ (fPCA), which serves as a dimension reduction tool similar to classical PCA and factor analysis. Unlike classical PCA, however, fPCA operates, in principle, in an infinite-dimensional function space to capture dominant modes of variation and reduce complexity [6]. Despite its conceptual elegance, existing fPCA and similar FDA models often assume that data are observed on a shared, finite grid, often relying on heuristic imputation or posterior estimation to handle any missing entries [7; 8; 9]. This assumption conveniently facilitates the adoption of established linear algebraic methods but compromises the integrity of FDA by introducing significant information loss and high-computational demands in high-dimensional applications. An ideal approach would represent each function at its naturally sampled measurement points rather than forcing all observations onto a shared grid, thus preserving crucial information and avoiding the need for heuristic resampling. This point is critical when considering that, given only a finite number of data points, infinitely many functions can interpolate these points, each reflecting different inductive biases about smoothness or shape [10]. Conventional smoothing or regridding methods (e.g., polynomial interpolation) introduce biases that may distort the underlying function's actual behavior. In contrast, we will achieve more accurate and unbiased predictions by concurrently updating the function estimation and the population-level statistics governing the estimation, such as those encoded in the covariance operator. Such an approach requires the direct modeling of the function from its original measurement points, rather than imposing artificial grids. To mitigate these limitations, several studies have proposed alternative strategies. For instance, [PERSON]. [3] developed a nonparametric technique for the estimation of the mean and covariance for functional data under smoothness assumptions while also discussing a continuous formulation and the necessary discretization in practical applications. In [11; 12], the authors extended fPCA to sparse and irregular longitudinal designs by smoothing the covariance estimate and then discretizing. Nonetheless, classical discretization steps often result in significant information loss and computational burdens. As functional data's size and complexity grew, researchers turned to flexible basis expansions, including sinusoids (Fourier), wavelets, polynomials, and B-splines, for a finite-dimensional representation of functional data that is convenient and accurate in computation, avoiding the drawbacks of explicit approximation and resampling [2; 13; 14]. However, a core challenge remains in selecting a suitable model. For instance, researchers must choose the number and form (e.g., smoothness), along with the dimensionality of the representational subspace. In approximation, the placement of basis functions is also essential. Evenly spaced nodes remain popular for their simplicity but may be suboptimal.