A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$: $$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne j} c(j|k) f_k(\underline{x})p_k $$
Bayes classifiers can be used even when cost of misclassification must be considered in the problem, using some tricks to include the informatino of the cost into the priors. For instance if the cost of misclassification depends only on the true group that I'm attributing the unit to (so we have $ c(j|k)=c_k$), by defining as priors $\pi_k = c_k p_k / \sum_{s}c_sp_s$ the rule for classification becomes:
$$\sum_{k\ne t} \pi_k f_k(\underline{x}) \le \sum_{k\ne j} \pi_k f_k(\underline{x})$$
Thus obtaining a Bayes classifiers.
However, I know that bayes classifiers can still be used if the cost of misclassification depends only on the group misclassified: $c(t|k) = c_t $, or even when $c(t|k) = \alpha^t \beta^k$. I was not able to find the expression for the Bayes classifiers in those two cases, was wondering if anyone already worked on it or may have some hint.
