# Loss function : finding the criterion for which a given solution is the optimal classifier

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 ?