Bayesian Detection

The goal is to minimize the Bayes Risk

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Where ${\pi }_{0,1}$ are the prior probabilities, and $C_{ij}$ is the cost of deciding $i$ when $j$ is true.

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Which gives the following decision rule

$$L(\mathbf{x})\triangleq \frac{p_1(\mathbf{x})}{p_2(\mathbf{x})}>\lambda =\frac{{\pi }_0}{1-{\pi }_0}\frac{C_{10}-C_{00}}{C_{01}-C_{11}}$$

Where

$$L(x){\lessgtr }_{H_1}^{H_0}\lambda$$