enter image description here The previous exercise is from the book 'The Bayesian Choice', page 87. What does the author mean by uniformly optimal stat. procedure? This exercise refers to a Decision theory chapter, in a section where the only result related to some kind of 'uniformity' is one which states that to minimize the integrated risk is equivalent to minimize the expected posterior loss.

Are we supposed to prove that given $\pi(\theta|x)$, there's no $d(x)$ which minimizes $\int_{\mathbb{R}}L(\theta,d)\pi(\theta|x) \ d\theta$? If so, how do we do it?

  • $\begingroup$ The only thing I seem to get, is that the loss function is strictly convex, and so the bayes estimator is unique. I have no idea how to find a second estimator with the same minimum expected posterior loss... $\endgroup$ Jul 20, 2016 at 18:01
  • $\begingroup$ Maybe with a posterior symmetric around zero, and then consider the estimator $-d$? $\endgroup$ Jul 20, 2016 at 18:05

1 Answer 1


Since the loss function is strictly convex, the bayes estimator is unique.

For a given $\theta$, the estimator $\delta (X)=\theta$ minimizes the loss function, since $L(\theta,\theta)=0$. Then the unique bayes estimator must have $L(\theta,\delta_{bayes})= 0 \ \forall \theta$, which is impossible.

  • $\begingroup$ Proper answer to the question! Uniformly is to be understood as over the whole parameter space. $\endgroup$
    – Xi'an
    Dec 13, 2018 at 10:54

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