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Let \( \displaystyle \underset{\sim}{X} =( X_{1}, \dots , X_{n} ) \) be a random sample from a density \( \displaystyle f(x\theta) \), where \(\theta\) is a value of the random variable \(\Theta\) with prior density \( \displaystyle \pi_{\Theta} \). Let \( \displaystyle d^{*} \) be a Bayes estimator of \(\theta\) and let \( \mathcal{L}(\theta,\cdot) \) be the loss function for the estimation of \(\theta\). By definition, \( \displaystyle d^{*} \) is the statistic that minimizes the Bayes risk function. Equivalently, it can be easily shown that \( \displaystyle d^{*} \) is the statistic that minimizes the posterior risk function \( \displaystyle E_{\Theta  \underset{\sim}{X} }\left( \mathcal{L}(\theta,\cdot) \right) \). We have that
\begin{align*}
\displaystyle
E_{\Theta  \underset{\sim}{X} } \left( \mathcal{L}(\theta,d) \right) &= \int_{\Theta} \mathcal{L}(\theta,d) \pi_{\Theta  \underset{\sim}{X} }(\theta  \underset{\sim}{X} ) \mathrm{d}\theta \\
&= \int_{\Theta} \mathcal{L}(\theta,d) \pi_{\Theta}(\theta) f(\underset{\sim}{X}  \theta) \frac{1}{c(\underset{\sim}{X} )} \mathrm{d}\theta \; \; \; (1)
\end{align*}
where \( \displaystyle c = c(\underset{\sim}{X} ) \) is a constant. By NeymanFischer's factorization theorem, if \( \displaystyle T=T(\underset{\sim}{X} ) \) is the sufficient statistic, then
\[ \displaystyle f(\underset{\sim}{X}  \theta) = g\left( T(\underset{\sim}{X} ) , \theta \right) h(\underset{\sim}{X} ) \; \; \; (2) \]
We now observe, due to (1) and (2), that, in order to minimize the quantity
\[ E_{\Theta  \underset{\sim}{X} } \left( \mathcal{L}(\theta,d) \right) \]with respect to \( d \), we have to minimize the quantity
\[ \int_{\Theta} \mathcal{L}(\theta,d) \pi_{\Theta}(\theta) g\left( T(\underset{\sim}{X} ) , \theta \right) \mathrm{d}\theta \]with respect to \( d \), which is a function of \( \displaystyle T=T(\underset{\sim}{X} ) \text{ and } d \). We thus conclude that the Bayes estimator \( \displaystyle d^{*} \) of \( \theta \) is a function of the sufficient statistic \( T(\underset{\sim}{X} ) \), since we already mentioned that \( \displaystyle d^{*} = \min_{d} E_{\Theta  \underset{\sim}{X} }\left( \mathcal{L}(\theta,d) \right) \).
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