In a fully-Bayesian Functional Principal Components Analysis (FPCA) the principal components are treated as unknown infinite-dimensional parameters. By projecting the functional principal components on a rich orthonormal spline basis, we show that orthonormality of the principal components is equivalent with the orthonormality of the spline coefficients. A penalty on the integral of the second derivative of the functional principal components can be induced on the spline coefficients, where each function has its own smoothing parameter. Finally, each smoothing parameter is treated as an inverse variance component in the associated mixed effects model. In this paper we provide sufficient conditions to ensure that the posterior distribution is proper. This condition is expressed in terms of the eigenvalues of the smoothing penalty design matrix, which provides a practical and simple choice for the prior on the smoothing parameters.

翻译：在完全贝叶斯函数主成分分析中，主成分被视为未知的无穷维参数。通过将函数主成分投影到丰富的正交样条基上，我们证明主成分的正交性等价于样条系数的正交性。可对样条系数施加基于函数主成分二阶导数积分的惩罚项，其中每个函数拥有独立的平滑参数。进一步地，每个平滑参数被处理为关联混合效应模型中的逆方差分量。本文提供了确保后验分布适定性的充分条件。该条件以平滑惩罚设计矩阵的特征值形式表达，为平滑参数的先验选择提供了实用且简便的方案。