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.

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