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We develop a simple and unified framework for nonlinear variable selection that incorporates uncertainty in the prediction function and is compatible with a wide range of machine learning models (e.g., tree ensembles, kernel methods, neural networks, etc). In particular, for a learned nonlinear model $f(\mathbf{x})$, we consider quantifying the importance of an input variable $\mathbf{x}^j$ using the integrated partial derivative $\Psi_j = \Vert \frac{\partial}{\partial \mathbf{x}^j} f(\mathbf{x})\Vert^2_{P_\mathcal{X}}$. We then (1) provide a principled approach for quantifying variable selection uncertainty by deriving its posterior distribution, and (2) show that the approach is generalizable even to non-differentiable models such as tree ensembles. Rigorous Bayesian nonparametric theorems are derived to guarantee the posterior consistency and asymptotic uncertainty of the proposed approach. Extensive simulations and experiments on healthcare benchmark datasets confirm that the proposed algorithm outperforms existing classic and recent variable selection methods., Comment: 50 pages, 16 figures, 11 tables

The layered In₄Se₃ based material is recognized as a state-of-the-art n-type thermoelectric material for the middle temperature range of 500 K to 900 K. Despite excellent thermoelectric properties, its inferior mechanical properties restrict its commercial possibilities. In this work, we use Quantum Mechanics (density functional theory) to investigate the ideal strength and failure mechanisms of ideal and Se deficient In₄Se₃ under pure shear and biaxial shear loads. We found that the lowest ideal shear strength of ideal In₄Se₃ is 1.25 GPa along the (100)/ slip system. Slippage between the In/Se layer dominates its deformation and failure. With Se vacancies, the ideal strength of In₄Se_(2.75) drops to 1.00 GPa while the failure mechanism remains almost the same as that of ideal In₄Se₃. Moreover, under biaxial shear loads (as in nano-indentation experiments) the ideal strength of In4₄Se₃ increases to 1.50 GPa, with compression now accounting for the failure. Even so, In₄Se₃ shows poorer mechanical properties under biaxial shear loads. These insights into the deformation and failure mechanism of In₄Se₃ compounds should help suggest designing modifications to improve mechanical properties.