Note that this vignette is adapted from the paper as cited below (Kamulete 2022).

## Background

Suppose we fit a predictive model on a training set and predict on a test set. Dataset shift, also known as data or population drift, occurs when training and test distributions are not alike. This is essentially a sample mismatch problem. Some regions of the data space are either too sparse or absent during training and gain importance at test time. We want methods to alert us to the presence of unexpected inputs in the test set. To do so, a measure of divergence between training and test set is required. Can we not simply use the many modern off-the-shelf multivariate tests of equal distributions for this?

One reason for moving beyond tests of equal distributions is that they are often too strict. They require high fidelity between training and test set everywhere in the input domain. However, not all changes in distribution are a cause for concern – some changes are benign. Practitioners distrust these tests because of false alarms. Polyzotis et al. (2019) comment:

statistical tests for detecting changes in the data distribution […] are too sensitive and also uninformative for the typical scale of data in machine learning pipelines, which led us to seek alternative methods to quantify changes between data distributions.

Even when the difference is small or negligible, tests of equal distributions reject the null hypothesis of no difference. An alarm should only be raised if a shift warrants intervention. Retraining models when distribution shifts are benign is both costly and ineffective. Monitoring model performance and data quality is a critical part of deploying safe and mature models in production. To tackle these challenges, we propose D-SOS instead.

In comparing the test set to the training set, D-SOS pays more attention to the regions –- typically, the outlying regions –- where we are most vulnerable. To confront false alarms, it uses a robust test statistic, namely the weighted area under the receiver operating characteristic curve (WAUC). The weights in the WAUC discount the safe regions of the distribution. To the best of our knowledge, this is the first time that the WAUC is being used as a test statistic in this context. The goal of D-SOS is to detect non-negligible adverse shifts. This is reminiscent of noninferiority tests, widely used in healthcare to determine if a new treatment is in fact not inferior to an older one. Colloquially, the D-SOS null hypothesis holds that the new sample is not substantively worse than the old sample, and not that the two are equal.

D-SOS moves beyond tests of equal distributions and lets users specify which notions of outlyingness to probe. The choice of the score function plays a central role in formalizing what we mean by worse. The scores can come from out-of-distribution detection, two-sample classification, uncertainty quantification, residual diagnostics, density estimation, dimension reduction, and more. While some of these scores are underused and underappreciated in two-sample tests, they can be more informative in some cases. The main takeaway is that given a method to assign an outlier score to a data point, D-SOS uplifts these scores and turns them into a two-sample test for no adverse shift.

## Motivational example

For illustration, we apply D-SOS to the canonical iris dataset. The task is to classify the species of Iris flowers based on $$d=4$$ covariates (features) and $$n=50$$ observations for each species. We show how D-SOS helps diagnose false alarms. We highlight that (1) changes in distribution do not necessarily hurt predictive performance, and (2) points in the densest regions of the distribution can be the most difficult – unsafe – to predict.

We consider four tests of no adverse shift. Each test uses a different score. For two-sample classification, this score is the probability of belonging to the test set. For density-based out-of-distribution (OOD) detection, the score comes from isolation forest – this is (roughly) inversely related to the local density. For residual diagnostics, it is the out-of-sample (out-of-bag) prediction error from random forests. Finally, for confidence-based OOD detection (prediction uncertainty), it is the standard error of the mean prediction from random forests. Only the first notion of outlyingness – two-sample classification – pertains to modern tests of equal distributions; the others capture other meaningful notions of adverse shifts. For all these scores, higher is worse: higher scores indicate that the observation is diverging from the desired outcome or that it does not conform to the training set.

For the subsequent tests, we split iris into 2/3 training and 1/3 test set. The train-test pairs correspond to two partitioning strategies: (1) random sampling and (2) in-distribution (most dense) examples in the test set. How do these sample splits fare with respect to the aforementioned tests? Let $$s$$ and $$p$$ denote $$s$$−value and $$p$$−value. The results are reported on the $$s = − log2(p)$$ scale because it is intuitive. An $$s$$−value of k can be interpreted as seeing k independent coin flips with the same outcome –- all heads or all tails –- if the null is that of a fair coin. This conveys how incompatible the data is with the null.

As plotted, the case with (1) random sampling in green exemplifies the type of false alarms we want to avoid. Two-sample classification, standing in for tests of equal distributions, is incompatible with the null of no adverse shift (a $$s$$−value of around 8). But this shift does not carry over to the other tests. Residual diagnostics, density-based and confidence-based OOD detection are all fairly compatible with the view that the test set is not worse. Had we been entirely reliant on two-sample classification, we may not have realized that this shift is essentially benign. Tests of equal distributions alone give a narrow perspective on dataset shift.

Moving on to the case with (2) in-distribution test set in orange, density-based OOD detection does not flag this sample as expected. We might be tempted to conclude that the in-distribution observations are safe, and yet, the tests based on residual diagnostics and confidence-based OOD detection are fairly incompatible with this view. Some of the densest points are concentrated in a region where the classifier does not discriminate very well: the species versicolor and virginica overlap. That is, the densest observations are not necessarily safe. Density-based OOD detection glosses over this: the trouble may well come from inliers that are difficult to predict. We get a more holistic perspective of dataset shift because of these complementary notions of outlyingness.

The point of this exercise is twofold. First, we stress the limits of tests of equal distributions when testing for dataset shift. They are unable, by definition, to detect whether the shift is benign or not. Second, we propose a family of tests based on outlier scores, D-SOS, which offers a more holistic view of dataset shift. D-SOS is flexible and can be easily extended to test for other modern notions of outlyingness such as trust scores. We hope this encourages more people to test for adverse shifts.

## References

Kamulete, Vathy M. 2022. “Test for Non-Negligible Adverse Shifts.” In The 38th Conference on Uncertainty in Artificial Intelligence. https://openreview.net/forum?id=S5UG2BLi9xc.
Polyzotis, Neoklis, Martin Zinkevich, Sudip Roy, Eric Breck, and Steven Whang. 2019. “Data Validation for Machine Learning.” Proceedings of Machine Learning and Systems 1: 334–47.