Definition
r(\theta, \delta) &= \int_{\Theta} R(\theta, \delta) \pi(\theta) d\theta = E_\theta[R(\theta, \delta)] = E_\theta[E_{x}[L(\theta, \delta(x))]]\\
&= \int_{\Theta} \int_{X} L(\theta, \delta(x))p(x|\theta) \pi(\theta) dx d\theta = \int_{X} \int_{\Theta} L(\theta, \delta(x))p(\theta|x)p(x) d\theta dx\\
&= \int_{X}E_\theta[L(\theta, \delta(x))|X=x]p(x)dx = \int_{X}\rho(x, \pi)p(x)dx
\end{aligned}$$
where $L(\theta, \delta)$ is [[Loss Function]], $R(\theta, \delta)$ is [[Risk Function]], and $\rho(x, \pi)$ is a [[Posterior Risk]].