Robust Prediction Error Estimation with Monte-Carlo Methodology

In this paper, we aim to estimate the prediction error of machine learning models under the true distribution of the data on hand. We consider the prediction model as a data-driven black-box function and quantify its statistical properties using non-parametric methods. We propose a novel sampling technique that takes advantage of the underlying probability distribution information embedded in the data. The proposed method combines two existing frameworks for estimating the prediction inaccuracy error; $m$ out of $n$ bootstrapping and iterative bootstrapping. $m$ out of $n$ bootstrapping is to maintain the consistency, and iterative bootstrapping is often used for bias correction of the prediction error estimation. Using Monte-Carlo uncertainty quantification techniques, we disintegrate the total variance of the estimator so the user can make informed decisions regarding measures to overcome the preventable errors. In addition, via the same Monte-Carlo framework, we provide a way to estimate the bias due to using the empirical distribution. This bias captures the sensitivity of the estimator to the on hand input data and help with understanding the robustness of the estimator. The application of the proposed uncertainty quantification is tested in a model selection case study using simulated and real datasets. We evaluate the performance of the proposed estimator in two frameworks; first, directly applying is as an optimization model to find the best model; second, fixing an optimization engine and use the proposed estimator as a fitness function withing the optimizer. Furthermore, we compare the asymptotic statistical properties and numerical results in a finite dataset of the proposed estimator with the existing state-of-the-art methods.