Bayes Estimator Wikipedia has a section on the Bayes Estimator.
https://en.wikipedia.org/wiki/Bayes_estimator
Isn't Bayes Estimator simply the value of the parameter that minimizes the expected loss of a loss function under the posterior distribution?
What I am really trying to ask is, in the presence of a prior is there a reason to find this estimator in some other way without going through "the Bayesian update and finding the posterior first and then finding the optimal action"? What's the takeaway from this entire Wikipedia entry?
 A: From the first line of the Wikipedia page:

... a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss).

So yes, the Bayes estimator minimises the expected loss under the posterior; the estimator will be determined by the prior, the likelihood, and the loss function.  It can be written directly as:
$$\begin{equation} \begin{aligned}
\hat{\theta}_\text{Bayes} 
&\equiv \underset{\hat{\theta}}{\text{arg min }} \int L(\theta, \hat{\theta}) dP(\theta|\mathbf{x}) \\[10pt]
&= \underset{\hat{\theta}}{\text{arg min }} \int L(\theta, \hat{\theta}) L_\mathbf{x}(\theta) \pi(\theta) d\theta.
\end{aligned} \end{equation}$$
where the minimisation is taken over all possible estimators $\hat{\theta}: \mathscr{X} \rightarrow \Theta$.  Usually you would find this estimator by first determining the posterior distribution, and then minimising the expected loss under this distribution.  In some cases you might obtain it more directly without an intermediate step.
A: To turn a wee bit pedantic, this derivation of the Bayes estimator is the construction of the Bayes rule when the observable is $X=x$. The definition of a Bayes rule in game theory and Bayesian decision theory is rather the procedure $\hat{\theta}:\mathcal{X}\to\Theta$ that optimises the Bayes risk
$$\int_\Theta\int_\mathcal{X} \text{L}(\theta,\hat{\theta}(x))\,f(x|\theta)\,\text{d}x\,\pi(\theta)\,\text{d}\theta$$Since, by Fubini,
$$\int_\mathcal{X} \int_\Theta \text{L}(\theta,\hat{\theta}(x))\,\pi(\theta|x)\,\text{d}\theta\,m(x)\,\text{d}x=\int_\Theta\int_\mathcal{X} \text{L}(\theta,\hat{\theta}(x))\,f(x|\theta)\,\text{d}x\,\pi(\theta)\,\text{d}\theta$$
this is equivalent to minimising the posterior expected loss
$$\int_\Theta \text{L}(\theta,\hat{\theta}(x))\,\pi(\theta|x)\,\text{d}\theta$$for almost every $x$, as noted by Cowboy Trader in a comment. At least this is how it is defined in Jim Berger's book (and in mine).
