Here we provide proofs of the statements made in the main text as well as further figures of numerical experiments and a more detailed discussion of heteroskedasticity effects regarding causal discovery. Z. Testing whether the Pearson correlation between X and Y is zero is equivalent to testing whether the slope parameter β is equal to zero. Therefore, this is a homoskedastic problem. A.1.2 Discussion of Effect 2: We start by discussing the homoskedastic case to see where non-constant variance of noise leads to problems within the t-test. For homoskedastic noise the second factor is an estimator of the standard error of ˆβ, which we derive by using the mean of the squared residual as an estimator for the error variance.

Contrastive Language-Image Pre-training (CLIP) models have demonstrated remarkable generalization capabilities across multiple challenging distribution shifts. However, there is still much to be explored in terms of their robustness to the variations of specific visual factors.

We use a variety of evaluation metrics to diagnose the effect that training with instance selection has on the learned distribution. In all cases where a reference distribution is required we usethe original training distribution,and not the distribution produced after instance selection. Inception Score (IS) [24] evaluates samples by extracting class probabilities from an ImageNet pretrained Inceptionv3 classifier and measuring the distribution of outputs over all samples. Classification Accuracy Score (CAS) [23, 25] was introduced for evaluating the usefulness of conditional generativemodels for augmenting downstream tasks such as image classification. Model Params (M) Batch Size Retention Ratio(%) IS FID P R D C BigGAN 52.54 512 100 25.43 10.55 - - - FQ-BigGAN 52.54 512 100 25.96 9.67 - - - - The truncation trick isasimple and popular technique which isused toincrease thevisual fidelity of samples from a GAN at the expense of reduced diversity [2].