## Of the Bayes’ rule, this new rear probability of y = step one shall be indicated due to the fact:

(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 ) .

Research. Thought an out-of-shipping enter in x-out having M inv = [ We s ? s 0 step one ? s ] , and you can Meters elizabeth = [ 0 s ? e p ? ] , then the element symbolization was ? age ( x ) = [ z aside p ? z age ] , where p ‘s the unit-standard vector defined inside Lemma 2 .

Then we have P ( y = 1 ? ? out ) = P ( y = 1 ? z out , p ? z e ) = ? ( 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 ) . ?

Remark: In an even more standard case, z away is going to be modeled given that a random vector which is in addition to the when you look at the-distribution names y = step one and you will y = ? 1 and you may environment have: z out ? ? y and you will z away ? ? z age . Thus in Eq. 5 we have P ( z aside ? y = step one ) = P ( z aside ? y = ? step one ) = P ( z out ) . After that P ( y = step one ? ? out ) = ? ( 2 p ? z elizabeth ? + log ? / ( step 1 ? ? ) ) , same as inside Eq. 7 . Hence our very own main theorem nevertheless retains not as much as more general instance.

## Appendix B Extension: Colour Spurious Correlation

To advance verify the conclusions beyond background and you will sex spurious (environmental) features, we offer more experimental performance on ColorMNIST dataset, just like the revealed when you look at the Shape 5 .

## Testing 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 .