Bayes Risk

$$\begin{array}{rl}\mathcal{R}& \triangleq E[C]\end{array}$$

Where $C(ϵ)$ is the cost function.

This gives

$$\begin{array}{rl}\mathcal{R}& ={\pi }_0\{C_{00}P\{\delta (x)=0)|H_0\}+C_{10}P\{\delta (x)=1|H_0\}\}\\ & +{\pi }_1\{C_{01}P\{\delta (x)=0)|H_1\}+C_{11}P\{\delta (x)=1|H_1\}\}\end{array}$$

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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Idk what this is

$$\mathcal{R}=E[L(\theta ,\hat{\theta })]=\int \left[\int L(\theta ,\hat{\theta })p(\theta |x)d\theta \right]p(x)dx$$