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Bayes consistency is \phi'(0)<0; originally said \phi'(0)=0 which is incorrect. See Thm 2 in Bartlett et al. 2006. |
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:<math>\phi(v)=C[f^{-1}(v)]+(1-f^{-1}(v))C'[f^{-1}(v)] \;\;\;\;\;(2)</math>,
where <math>f(\eta), (0\leq \eta \leq 1)</math> is any invertible function such that <math>f^{-1}(-v)=1-f^{-1}(v)</math> and <math>C(\eta)</math> is any differentiable strictly concave function such that <math>C(\eta)=C(1-\eta)</math>. Table-I shows the generated Bayes consistent loss functions for some example choices of <math>C(\eta)</math> and <math>f^{-1}(v)</math>. Note that the Savage and Tangent loss are not convex. Such non-convex loss functions have been shown to be useful in dealing with outliers in classification.<ref name=":0" /><ref>{{Cite journal|last1=Leistner|first1=C.|last2=Saffari|first2=A.|last3=Roth|first3=P. M.|last4=Bischof|first4=H.|date=September 2009|title=On robustness of on-line boosting - a competitive study|journal=2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops|pages=1362–1369|doi=10.1109/ICCVW.2009.5457451|isbn=978-1-4244-4442-7|s2cid=6032045}}</ref> For all loss functions generated from (2), the posterior probability <math>p(y=1|\vec{x})</math> can be found using the invertible ''link function'' as <math>p(y=1|\vec{x})=\eta=f^{-1}(v)</math>. Such loss functions where the posterior probability can be recovered using the invertible link are called ''proper loss functions''.
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