This paper develops a novel Bayesian approach for nonlinear regression with symmetric matrix predictors, often used to encode connectivity of different nodes. Unlike methods that vectorize matrices as predictors that result in a large number of model parameters and unstable estimation, we propose a Bayesian multi-index regression method, resulting in a projection-pursuit-type estimator that leverages the structure of matrix-valued predictors. We establish the model identifiability conditions and impose a sparsity-inducing prior on the projection directions for sparse sampling to prevent overfitting and enhance interpretability of the parameter estimates. Posterior inference is conducted through Bayesian backfitting. The performance of the proposed method is evaluated through simulation studies and a case study investigating the relationship between brain connectivity features and cognitive scores.

翻译：本文针对对称矩阵预测因子的非线性回归问题提出了一种新颖的贝叶斯方法，此类预测因子常用于编码不同节点间的连接性。与将矩阵向量化作为预测因子的方法不同——这类方法会导致模型参数过多且估计不稳定，我们提出了一种贝叶斯多指标回归方法，从而得到一种投影追踪型估计量，该估计量能够利用矩阵值预测因子的结构。我们建立了模型的可识别性条件，并对投影方向施加了稀疏诱导先验以进行稀疏采样，从而防止过拟合并增强参数估计的可解释性。后验推断通过贝叶斯后向拟合进行。通过模拟研究和一项探究大脑连接特征与认知评分之间关系的案例研究，评估了所提方法的性能。