In exponential families where the target parameters equal the mean parameters, optimal control variate estimators and maximum likelihood estimators have identical asymptotic variance, giving a fixed-point algorithm for the MLE.
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It's all In the (Exponential) Family: An Equivalence between Maximum Likelihood Estimation and Control Variates for Sketching Algorithms
In exponential families where the target parameters equal the mean parameters, optimal control variate estimators and maximum likelihood estimators have identical asymptotic variance, giving a fixed-point algorithm for the MLE.