Bayes estimator

Remarks

If $r(\pi) < \infty$, an estimator that minimizes the posterior expected loss $E[L(\theta, \delta(x)) \mid X = x]$ for almost every $x$ is a Bayes estimator. Under squared error loss $L(\theta, a) = (a - \theta)^2$, the Bayes estimator is the posterior mean $E[\theta \mid X]$. Under absolute error loss, it is a posterior median.

Example: let $X$ be binomial with parameters $n$ and $\theta$, and let $\pi$ be the beta distribution with parameters $a, b > 0$. The posterior is the beta distribution with parameters $a + X$ and $b + n - X$. So under squared error loss the Bayes estimator is its mean $(a + X)/(a + b + n)$.

Non-example: for the uniform prior, $a = b = 1$, the Bayes estimator is $(X + 1)/(n + 2)$. The proportion $X/n$ differs from it whenever $X \ne n/2$, and for $n \ge 1$ its Bayes risk $1/(6n)$ is larger, so $X/n$ is not the Bayes estimator for this prior.