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With the commoditization of machine learning, more and more off-the-shelf models are available as part of code libraries or cloud services. Typically, data scientists and other users apply these models as &#x201C;black boxes&#x201D;within larger projects. In the case of regressing a scalar quantity, such APIs typically offer a predict() function, which outputs the estimated target variable (often referred to as y&#x0302; or, in code, y_hat). However, many real-world problems may require some sort of deviation interval or uncertainty score rather than a single point-wise estimate. In other words, a mechanism is needed with which to answer the question &#x201C;How confident is the system about that prediction?&#x201D; Motivated by the lack of this characteristic in most predictive APIs designed for regression purposes, we propose a method that adds an uncertainty score to every black-box prediction. Since the underlying model is not accessible, and therefore standard Bayesian approaches are not applicable, we adopt an empirical approach and fit an uncertainty model using a labelled dataset (x, y) and the outputs y&#x0302; of the black box. In order to be able to use any predictive system as a black box and adapt to its complex behaviours, we propose three variants of an uncertainty model based on deep networks. The first adds a heteroscedastic noise component to the black-box output, the second predicts the residuals of the black box, and the third performs quantile regression using deep networks. Experiments using real financial data that contain an in-production black-box system and two public datasets (energy forecasting and biology responses) illustrate and quantify how uncertainty scores can be added to black-box outputs.

Building Uncertainty Models on Top of Black-Box Predictive APIs