## Of the Bayes’ laws, new posterior likelihood of y = step 1 are 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 ) .

Proof. Think an aside-of-shipping enter in x-out having M inv = [ I s ? s 0 step one ? s ] , and M age = [ 0 s ? age p ? ] , then the ability image is ? age ( x ) = [ z away p ? z e ] , in which p is the device-standard vector discussed inside the Lemma dos .

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 a far more general circumstances, z out are going to be modeled while the an arbitrary vector which is in addition to the in-delivery names y = 1 and you will y = ? 1 and you will environment has: z aside ? ? y and you will z away ? ? z age . Ergo into the Eq. 5 i have P ( z out ? y = step one ) = P ( z away ? y = ? 1 ) = P ( z out ) . Upcoming P ( y = 1 ? ? out ) = ? ( 2 p ? z age ? + diary ? / ( step 1 ? ? ) ) , same as during the Eq. eight https://datingranking.net/pl/collarspace-recenzja/. Ergo our main theorem however keeps significantly less than a great deal more standard case.

## Appendix B Expansion: Colour Spurious Relationship

To help examine all of our findings beyond history and you can sex spurious (environmental) keeps, we offer more experimental show to your ColorMNIST dataset, since found for the Profile 5 .

## Evaluation Activity step three: 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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