How to Evaluate Uncertainty Estimates in Machine Learning for Regression?

TL;DR

This work shows that prevailing evaluation methods for regression uncertainty—density loglikelihood and PICP-based interval coverage—fail to guarantee reliable, per-point uncertainty quantification. It formalizes the uncertainty framework and demonstrates that improvements in loglikelihood or marginal coverage do not ensure accurate interval behavior for individual inputs. To remedy this, the authors propose simulation-based testing that estimates pointwise coverage via PICF and CICF, using metrics such as the Brier score and average interval width. Through synthetic experiments, out-of-distribution analyses, and a Boston Housing-inspired benchmark, they illustrate that per-point coverage is essential for trustworthy uncertainty estimates, albeit at higher computational cost. The paper thus motivates benchmark simulations to enable fair comparisons of uncertainty estimation methods in regression and guides practical evaluation toward reliable, input-specific calibration.

Abstract

As neural networks become more popular, the need for accompanying uncertainty estimates increases. There are currently two main approaches to test the quality of these estimates. Most methods output a density. They can be compared by evaluating their loglikelihood on a test set. Other methods output a prediction interval directly. These methods are often tested by examining the fraction of test points that fall inside the corresponding prediction intervals. Intuitively both approaches seem logical. However, we demonstrate through both theoretical arguments and simulations that both ways of evaluating the quality of uncertainty estimates have serious flaws. Firstly, both approaches cannot disentangle the separate components that jointly create the predictive uncertainty, making it difficult to evaluate the quality of the estimates of these components. Secondly, a better loglikelihood does not guarantee better prediction intervals, which is what the methods are often used for in practice. Moreover, the current approach to test prediction intervals directly has additional flaws. We show why it is fundamentally flawed to test a prediction or confidence interval on a single test set. At best, marginal coverage is measured, implicitly averaging out overconfident and underconfident predictions. A much more desirable property is pointwise coverage, requiring the correct coverage for each prediction. We demonstrate through practical examples that these effects can result in favoring a method, based on the predictive uncertainty, that has undesirable behaviour of the confidence or prediction intervals. Finally, we propose a simulation-based testing approach that addresses these problems while still allowing easy comparison between different methods.

Figures (10)

• Figure 1: The PICP values of 500 simulations. In each simulation, new data was generated, a new linear model was fit, and a new prediction interval was created.
• Figure 2: The dashed red line gives the true function $f(x) = ax$. The two dotted black lines give the confidence interval. In the left figure the true function falls inside our confidence interval, in the right figure it does not. It is clear that the measure CICP will either give 1 or 0, even when we have a method that gives a perfect $95\%$ confidence interval: It is impossible to test the coverage of our CI on a single test set.
• Figure 3: This figure illustrates that estimating the data noise variance correctly can result in a PICP close to the chosen confidence level. We assume that the model uncertainty is comparably very small. The true function, $f(x)$, is the constant zero function. The dashed blue line gives $\pm 1\sigma(x)$. The dotted black line gives $\pm 1 \hat{\sigma}(x)$. On average, the data noise variance estimate is correct and its corresponding PI captures the correct fraction of the data in this case. Using PICP in this example, we do not notice that our uncertainty estimate is wrong.
• Figure 4: These histograms give 100 evaluations of the PICP, at a $(1-\alpha)$ confidence level using both the bootstrap and dropout approach. The PICP captures the fraction of data points in the test set for which the observations $y$ falls inside the corresponding prediction interval. The data is simulated from $y = f(x) + \epsilon$ with $f(x) = (2x-1)^{3}$, and $\epsilon \sim {\mathcal{N}\left(0, (0.2)^{2}\right)}.$ The details of the construction of the PIs can be found in Section \ref{['dropout and bootstrap']}. From these histograms we can see that in a single simulation we would have a good performance of the PI on the test set with either method.
• Figure 5: These histograms give 100 evaluations of CICP, at a $(1-\alpha)$ confidence level. Each point in the histograms gives the fraction of datapoints in the test set of a new simulation for which the true function value $f(x)$ falls inside the corresponding confidence interval. Each simulation has its own CICP value. The same setup was used as in Figure \ref{['fig: PICPtoy']}. We observe that the good PICP values from Figure \ref{['fig: PICPtoy']} do not translate to good CICP values.

Theorems & Definitions (6)

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