Recent works have shown an interest in investigating the frequentist asymptotic properties of Bayesian procedures for high-dimensional linear models under sparsity constraints. However, there exists a gap in the literature regarding analogous theoretical findings for non-linear models within the high-dimensional setting. The current study provides a novel contribution, focusing specifically on a non-linear mixed-effects model. In this model, the residual variance is assumed to be known, while the covariance matrix of the random effects and the regression vector are unknown and must be estimated. The prior distribution for the sparse regression coefficients consists of a mixture of a point mass at zero and a Laplace distribution, while an Inverse-Wishart prior is employed for the covariance parameter of the random effects. First, the effective dimension of this model is bounded with high posterior probabilities. Subsequently, we derive posterior contraction rates for both the covariance parameter and the prediction term of the response vector. Finally, under additional assumptions, the posterior distribution is shown to contract for recovery of the unknown sparse regression vector at the same rate as observed in the linear case.

翻译：最近的研究表明，在高维线性模型中，基于稀疏性约束的贝叶斯方法的频率学派渐近性质引起了广泛兴趣。然而，现有文献中关于高维非线性模型的类似理论结果尚存在空白。本研究提供了一项新颖贡献，特别聚焦于非线性混合效应模型。在该模型中，残差方差假设已知，而随机效应的协方差矩阵和回归向量未知且需要估计。稀疏回归系数的先验分布由点质量为零的混合分布与拉普拉斯分布组成，同时采用逆威沙特先验分布处理随机效应的协方差参数。首先，以高后验概率界定了该模型的有效维度。随后，推导了协方差参数和响应向量预测项的后验收缩率。最后，在额外假设下，表明后验分布在恢复未知稀疏回归向量方面，其收缩率与线性情形下观测到的速率相同。