Of the Bayes’ laws, the posterior odds of y = step 1 should be indicated since:
(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. Imagine an away-of-shipments type in x out with Yards inv = [ We s ? s 0 step one ? s ] , and you wyszukiwanie profilu swoop can Yards e = [ 0 s ? e p ? ] , then your ability expression try ? age ( x ) = [ z away p ? z elizabeth ] , where p is the tool-norm vector outlined in 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: For the a far more standard instance, z out will be modeled since a haphazard vector that is independent of the within the-shipments brands y = step one and y = ? step one and environment enjoys: z aside ? ? y and you will z away ? ? z e . For this reason within the Eq. 5 i have P ( z away ? y = step 1 ) = P ( z out ? y = ? 1 ) = P ( z aside ) . Then P ( y = step 1 ? ? aside ) = ? ( dos p ? z elizabeth ? + record ? / ( 1 ? ? ) ) , same as inside the Eq. 7 . For this reason the fundamental theorem however keeps not as much as alot more general instance.
Appendix B Expansion: Color Spurious Relationship
To further validate the findings past history and you will intercourse spurious (environmental) provides, we offer extra experimental abilities toward ColorMNIST dataset, due to the fact revealed during the Figure 5 .
Analysis Activity step three: ColorMNIST.
[ lecun1998gradient ] , which composes colored backgrounds on digit images. In this dataset, E = denotes the background color and we use Y = as in-distribution classes. The correlation between the background color e and the digit y is explicitly controlled, with r ? . 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 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 .