Typo or change of interpretation in loss function in the Bayesian Choice textbook? In the proof of the following proposition, the posterior loss depends on a prior distribution $\pi$, while in the derivation it depends on the parameter $\theta$. Should the posterior loss function be 
$$L(\theta,\varphi\vert x)$$
instead, and this is just a typo,  or am I missing some fundamental change of roles? 


Robert, Christian. The Bayesian choice: from decision-theoretic foundations to computational implementation. Springer Science & Business Media, 2007.

 A: In game theory, the loss function $\mathrm{L}(\theta,\varphi)$ is a function with two arguments:


*

*the state of Nature, i.e., the parameter $\theta$ in a statistical problem;

*the decision, i.e., the value of the statistical procedure $\varphi$ (estimate, acceptance or rejection, prediction, &tc.). (Note that $\varphi$ is not an estimator but the value of the estimator at a given $x$, i.e., an estimate, e.g., $\varphi=\hat{\theta}(x)$.)
The posterior loss is the expected loss, integrated in $\theta$, wrt a prior distribution $\pi(\cdot)$ and conditional on $x$, i.e., the expectation of the loss function under the posterior distribution $\pi(\cdot|x)$. Hence the notation $\mathrm{L}(\pi,\varphi|x)$ to indicate this double dependence. Yet another notion is the Bayes risk, which integrates the loss in both the parameter and the observation, and is hence a function of two arguments, the prior $\pi(\cdot)$ and the statistical procedure $\delta$:
$$r(\pi,\delta) \stackrel{\text{def}}{=} \int_\Theta\int_\mathcal{X}
\mathrm{L}(\theta,\delta(x))\,\pi(\theta)f(x|\theta)\,\text{d}\theta\text{d}x$$
