Prediction Intervals for Neural Networks via Nonlinear Regression

Opening the Black Box: Bootstrapping Sensitivity Measures in Neural Networks for Interpretable Machine Learning

Artificial neural networks are powerful tools for data analysis, particularly in the context of highly nonlinear regression models. However, their utility is critically limited due to the lack of interpretation of the model given its black-box nature. To partially address the problem, the paper focuses on the important problem of feature selection. It proposes and discusses a statistical test procedure for selecting a set of input variables that are relevant to the model while taking into account the multiple testing nature of the problem. The approach is within the general framework of sensitivity analysis and uses the conditional expectation of functions of the partial derivatives of the output with respect to the inputs as a sensitivity measure. The proposed procedure extensively uses the bootstrap to approximate the test statistic distribution under the null while controlling the familywise error rate to correct for data snooping arising from multiple testing. In particular, a pair bootstrap scheme was implemented in order to obtain consistent results when using misspecified statistical models, a typical characteristic of neural networks. Numerical examples and a Monte Carlo simulation were carried out to verify the ability of the proposed test procedure to correctly identify the set of relevant features.