## Of the Bayes’ code, the new posterior likelihood of y = step 1 will likely be indicated because the:

Of the Bayes’ code, the new posterior likelihood of y = step 1 will likely be indicated because the:

(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 ) .

Evidence. Think an out-of-distribution input x out that have M inv = [ I s ? s 0 step one ? s ] , and you may Yards e = [ 0 s ? e p ? ] , then the function symbol was ? elizabeth ( x ) = [ z away p ? z elizabeth ] , where p is the equipment-standard vector laid out 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: From inside the a more standard case, z aside will be modeled once the a random vector that is independent of the from inside the-shipment labels y = 1 and you can y = ? 1 and you will ecological has actually: z away ? ? y and z aside ? ? z elizabeth . Therefore into the Eq. 5 i’ve P ( z aside ? y = 1 ) = P ( z out ? y = ? step 1 ) = P ( z aside ) . Upcoming P ( y = step 1 ? ? away ) = ? ( 2 p ? z e ? + log ? / ( step one ? ? ) ) , identical to when you look at the Eq. seven . Hence our very own head theorem still retains less than a great deal more general situation.

## Appendix B Extension: Color Spurious Correlation

To further confirm our results past background and you may sex spurious (environmental) has, we provide extra fresh performance towards ColorMNIST dataset, just like the shown from kinkyads inside the Profile 5 .