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For a binary classification problem, let $\eta(x) = P[Y=1 \mid x],$ and, for a given classifier $g$, we define the asymmetric cost : $$ L(g) = P[g(X)=0, Y=1] +\lambda P[g(X)=1,Y=0]$$ For this cost, we can show that the Bayes (i.e optimal, minimizing) classifier is $$g_{\lambda}^{*}(x)=\mathbb{I}_{\{\eta(x)\ge\frac{\lambda}{1+\lambda}\}}=\mathbb{I}_{\{\lambda\le\frac{\eta(x)}{1-\eta(x)}\}} $$

Now here's the question.

Consider the decision rule $$h^{*}(x) = \int_0^{+∞}g^{*}_{\lambda}(x)dλ$$ What is the learning problem solved by $h^*$ and which criterion does $h^*$ optimize?

After calculation we find that $h^*(x) = \frac{\eta(x)}{1-\eta(x)}$ which looks a bit like the the logistic regression function but I can't quite see what's the loss function / criterion it optimizes.

Any clue ?

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