Why is it that it is self-defeating to use the posterior mode as the bayes estimator in this case? I am reading through an applied statistics book and in it, it makes a very luminous statement for a posterior case where the likelihood was taken from $X_1,...,X_n$ iid random variables from a continuous uniform distribution on $[0,\theta]$, for $\theta$ unknown, and for where the prior distribution is the Pareto(a,b) distribution where:
$\pi(\theta)=\frac{ba^b}{\theta^{b+1}}$ if $\theta \geq a$ and 0 if $\theta <a$,
the posterior of $\theta$ is:
$\pi(\theta |x) = Pareto(max(x_1,...,x_n,k),n+b)$.
It states that reporting the posterior mode as the Bayes estimator would defeat the purpose of using a Bayes estimator. Can anyone help me understand why?
I tried to calculate the posterior mode,
argmax $\frac{(n+b)( max(x_1,...,x_n,k))^{n+b}}{\theta^{n+b+1}}$. Why would reporting the posterior mode defeat the purpose of using a Bayes estimator?
Thanks!
 A: It defeats the purpose of a Bayes estimator because you are not defining any loss function in the first place, the expected posterior value of which is what you want to minimize in order to get a Bayes estimator. The idea of a Bayes estimator $\theta_0$ is to minimize the expected posterior value of a given loss function, $L(\theta_0,\theta)$, i.e., to minimize
$$\mathbb{E}_{\theta|x}[L(\theta_0,\theta)]=\int L(\theta_0,\theta) \pi(\theta|x)d\theta,$$
where $\pi(\theta|x)$ is the posterior distribution of $\theta$ given the data and any prior information, $x$. If you define, for example, $L(\theta_0,\theta)=(\theta_0-\theta)^2$, then your Bayes estimator would be the posterior mean. Different loss functions, then, lead to different bayes estimators; reporting the posterior mode and not defining the loss function in the first place is like reporting a value for a distance but not reporting what actually is that you measured and in what surface: was it the distance between two points on a sphere? A plane?, etc.
Edit: after I wrote this, I re-discovered this very interesting post in which I learned what I just answered in the first place some time ago: Estimation of parameters as a mode of posterior distribution. It is way better explained than my explaination here ;-).
