On Prior Distributions for Orthogonal Function Sequences

We propose a novel class of prior distributions for sequences of orthogonal functions, which are frequently required in various statistical models such as functional principal component analysis (FPCA). Our approach constructs priors sequentially by imposing adaptive orthogonality constraints through a hierarchical formulation of conditionally normal distributions. The orthogonality is controlled via hyperparameters, allowing for flexible trade-offs between exactness and smoothness, which can be learned from the observed data. We illustrate the properties of the proposed prior and show that it leads to nearly orthogonal posterior estimates. The proposed prior is employed in Bayesian FPCA, providing more interpretable principal functions and efficient low-rank representations. Through simulation studies and analysis of human mobility data in Tokyo, we demonstrate the superior performance of our approach in inducing orthogonality and improving functional component estimation.