(Failure of OOD detection under invariant classifier) Consider an out-of-distribution input which contains the environmental feature: ? out ( x ) = M inv z out + M e z e , where z out ? ? inv . Given the invariant classifier (cf. Lemma 2), the posterior probability for the OOD input is p ( y = 1 ? ? out ) = ? ( 2 p ? z e ? + log ? / ( 1 ? ? ) ) , where ? is the logistic function. Thus for arbitrary confidence 0 < c : = P ( y = 1 ? ? out ) < 1 , there exists ? out ( x ) with z e such that p ? z e = 1 2 ? log c ( 1 ? ? ) ? ( 1 ? c ) .

Proof. Believe an aside-of-shipping enter in x out having Yards inv = [ We s ? s 0 1 ? s ] , and you may M elizabeth = [ 0 s ? age p ? ] , then the feature signal is actually ? e ( x ) = [ z aside p ? z elizabeth ] , in which p ‘s the product-norm vector discussed during the Lemma dos .

Then we have P ( y = 1 ? ? out ) = P ( y = 1 ? z out , p ? z e ) = ? ( 2 p cena green singles? z e ? + log ? / ( 1 ? ? ) ) , where ? is the logistic function. Thus for arbitrary confidence 0 < c : = P ( y = 1 ? ? out ) < 1 , there exists ? out ( x ) with z e such that p ? z e = 1 2 ? log c ( 1 ? ? ) ? ( 1 ? c ) . ?

Remark: Inside the a very general instance, z out should be modeled since a random vector which is in addition to the inside the-shipment brands y = step 1 and y = ? step one and you will environment possess: z away ? ? y and you may z aside ? ? z e . Therefore in the Eq. 5 you will find P ( z out ? y = 1 ) = P ( z aside ? y = ? 1 ) = P ( z away ) . Up coming P ( y = 1 ? ? aside ) = ? ( dos p ? z e ? + record ? / ( step 1 ? ? ) ) , identical to during the Eq. seven . Ergo the main theorem nevertheless retains under much more standard situation.

## Appendix B Extension: Color Spurious Relationship

To help validate all of our conclusions past background and you will sex spurious (environmental) keeps, we offer most fresh results toward ColorMNIST dataset, just like the found from inside the Figure 5 .

## Research Task step 3: ColorMNIST.

[ lecun1998gradient ] , which composes colored backgrounds on digit images. In this dataset, E = < red>denotes the background color and we use Y = < 0>as in-distribution classes. The correlation between the background color e and the digit y is explicitly controlled, with r ? < 0.25>. That is, r denotes the probability of P ( e = red ? y = 0 ) = P ( e = purple ? y = 0 ) = P ( e = green ? y = 1 ) = P ( e = pink ? y = 1 ) , while 0.5 ? r = P ( e = green ? y = 0 ) = P ( e = pink ? y = 0 ) = P ( e = red ? y = 1 ) = P ( e = purple ? y = 1 ) . Note that the maximum correlation r (reported in Table 4 ) is 0.45 . As ColorMNIST is relatively simpler compared to Waterbirds and CelebA, further increasing the correlation results in less interesting environments where the learner can easily pick up the contextual information. For spurious OOD, we use digits < 5>with background color red and green , which contain overlapping environmental features as the training data. For non-spurious OOD, following common practice [ MSP ] , we use the Textures [ cimpoi2014describing ] , LSUN [ lsun ] and iSUN [ xu2015turkergaze ] datasets. We train on ResNet-18 [ he2016deep ] , which achieves 99.9 % accuracy on the in-distribution test set. The OOD detection performance is shown in Table 4 .