Bayesian estimation for the exponential distribution under the Bregman divergence distance

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Suppose, I have a random variable $X$ that follows a negative exponential distribution $$ f(x;\theta)=\theta e^{-\theta x};~~x\in(0,\infty), \theta>0 $$ Then my question is, what will be Bayes estimate of $\theta$ under the Bregman divergence distance.