# By the Bayes’ code, the brand new posterior odds of y = step 1 might be indicated given that:

By the Bayes’ code, the brand new posterior odds of y = step 1 might be indicated given that:

(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. Envision an aside-of-shipments enter in x out having M inv = [ I s ? s 0 1 ? s ] , and you can M age = [ 0 s ? elizabeth p ? ] , then the ability sign was ? elizabeth ( x ) = [ z away p ? z elizabeth ] , where p is the device-standard vector defined during the 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: From inside the a more general situation, z away is modeled since the a haphazard vector that is independent of the when you look at the-shipments labels y = step 1 and y = ? step one and you can environmental provides: z out ? ? y and you can z out ? ? z elizabeth . Therefore when you look at the Eq. 5 i have P ( z away ? y = step one ) = P ( z away ? y = ? 1 ) = P ( z away ) . Up coming P ( y = 1 ? ? away ) = ? ( 2 p ? z e blendr coupons? + diary ? / ( step one ? ? ) ) , same as in the Eq. seven . Therefore our main theorem nevertheless retains below even more general instance.

## Appendix B Expansion: Colour Spurious Correlation

To advance validate the results beyond records and you can gender spurious (environmental) keeps, you can expect even more fresh overall performance for the ColorMNIST dataset, due to the fact revealed from inside the Shape 5 .

## Investigations 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 .

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