# When are Bayes estimators injective as a function of sufficient statistics?

I know that Bayes estimators can be written only as a function of sufficient statistics. When are those functions injectives? That is, when can I say that, given a bayes estimator $$\delta (\cdot)$$ and two sufficient statistics $$t \neq t'$$, we'll have $$\delta(t) \neq \delta(t')$$?

Working with quadratic loss (whence the Bayes estimator is simply the mean of the posterior distribution) and conjugate priors, I've found that this seems to be the case. In particular, it seems that Bayes estimators are always affine functions of sufficient statistics (see the examples in this Wikipedia article). Is this a more general pattern/theorem? What are the conditions for it to hold? Is there a counter-example where it doesn't?

• A counterexample: Let $X_1, \dotsc, X_n$ be iid $\mathcal{N}(\mu, 1)$. And let $\theta = | \mu |$. (some prior) A bayes estimator of $\theta$ will not be an injective function of the suffisient statistic $\bar{X}$ Commented Jul 15 at 22:46

This trivially fails for non-minimal sufficient statistics: eg, $$t$$ is the whole sample and $$t'$$ is a different whole sample with the same value of the estimator.
It will fail more interestingly when the minimal sufficient statistic has dimension greater than the parameter. This could range from the Cauchy location problem (where there isn't a sufficient statistic of fixed dimension) to the problem where $$(X,Y)$$ are bivariate normal with known, equal variances and unknown correlation, where the minimal sufficient statistic is two-dimensional but the parameter is one-dimensional