# Beyond Invariance: Test-Time Label-Shift Adaptation for Distributions with "Spurious" Correlations

Changes in the data distribution at test time can have deleterious effects on the performance of predictive models $p(y|x)$. We consider situations where there are additional meta-data labels (such as group labels), denoted by $z$, that can account for such changes in the distribution. In particular, we assume that the prior distribution $p(y, z)$, which models the dependence between the class label $y$ and the "nuisance" factors $z$, may change across domains, either due to a change in the correlation between these terms, or a change in one of their marginals. However, we assume that the generative model for features $p(x|y,z)$ is invariant across domains. We note that this corresponds to an expanded version of the widely used "label shift" assumption, where the labels now also include the nuisance factors $z$. Based on this observation, we propose a test-time label shift correction that adapts to changes in the joint distribution $p(y, z)$ using EM applied to unlabeled samples from the target domain distribution, $p_t(x)$. Importantly, we are able to avoid fitting a generative model $p(x|y, z)$, and merely need to reweight the outputs of a discriminative model $p_s(y, z|x)$ trained on the source distribution. We evaluate our method, which we call "Test-Time Label-Shift Adaptation" (TTLSA), on several standard image and text datasets, as well as the CheXpert chest X-ray dataset, and show that it improves performance over methods that target invariance to changes in the distribution, as well as baseline empirical risk minimization methods. Code for reproducing experiments is available at https://github.com/nalzok/test-time-label-shift .

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