Sufficient conditions are provided to ensure that the Bayes estimator with squared error loss for the natural parameter of an exponential family distribution is a linear function of the sample value, given that various contractions of the Bayes estimator are also Bayes estimators, generalizing a result of Meeden (1976).
"On Contractions of Bayes Estimators for Exponential Family Distributions." Ann. Statist. 5 (6) 1235 - 1239, November, 1977. https://doi.org/10.1214/aos/1176344008