Uncertainty Quantification for Deep Learning

TL;DR

This paper critiques current uncertainty quantification methods in deep learning, showing that existing approaches often miss key uncertainty sources and misweight ensemble members. It then develops a complete probabilistic framework that incorporates input, training/testing data, weights, and model imperfections to produce the full predictive distribution for regression tasks. A practical algorithm using proposal densities is introduced to compute the exact pdf by averaging over all uncertainty sources, with demonstrations on a simple regression problem and cloud autoconversion rate prediction. The results reveal that training/testing data uncertainty typically dominates, emphasize that explicit data-uncertainty modeling improves robustness to out-of-domain inputs, and argue for a principled, Bayes-based approach to uncertainty in scientific applications.

Abstract

We present a critical survey on the consistency of uncertainty quantification used in deep learning and highlight partial uncertainty coverage and many inconsistencies. We then provide a comprehensive and statistically consistent framework for uncertainty quantification in deep learning that accounts for all major sources of uncertainty: input data, training and testing data, neural network weights, and machine-learning model imperfections, targeting regression problems. We systematically quantify each source by applying Bayes' theorem and conditional probability densities and introduce a fast, practical implementation method. We demonstrate its effectiveness on a simple regression problem and a real-world application: predicting cloud autoconversion rates using a neural network trained on aircraft measurements from the Azores and guided by a two-moment bin model of the stochastic collection equation. In this application, uncertainty from the training and testing data dominates, followed by input data, neural network model, and weight variability. Finally, we highlight the practical advantages of this methodology, showing that explicitly modeling training data uncertainty improves robustness to new inputs that fall outside the training data, and enhances model reliability in real-world scenarios.

Figures (9)

• Figure 1: Training and testing data (blue dots) and example model (orange line).
• Figure 2: Posterior pdfs derived from Bagging (black vertical line), quantile regression (black curve), and new methodology (blue line), for input values -2 (left), 2 (middle), and 5(right). The pdfs can be compared with the samples in Fig. \ref{['fig:SimpleModelSamples']}. The orange vertical line is the prediction using the true model. (Note that this not the true prediction because the input value has uncertainty.) In the left panel the Bagging vertical lines fall around -0.07, outside the plot range. Bagging shows a degenerate pdf, and the quantile regression pdf is too narrow because uncertainties in training, testing, and new input data are ignored.
• Figure 3: The normalized likelihood values (i.e., the importance weights) for the $400$ neural networks.
• Figure 4: Examples of total uncertainty pdf in the output autoconversion rate for four input vectors. The blue curves are the total uncertainty pdfs; the red bar is the autoconversion rate calculated by the baseline neural network ($w_0$) without uncertainty quantification; black bars are the Bagging output samples. Note the wide variety of shapes of the blue uncertainty pdfs and the small spread in the black bars, demonstrating the inadequacy of the Bagging approach to represent uncertainty.
• Figure 5: Same as Fig. \ref{['fig:Full_pdfs']}, showing full uncertainty pdfs (blue curves) and the contribution from input uncertainty and model uncertainty (black curves).