Bayesian Post-Model-Selection Estimation

Estimation after model selection refers to the problem where the exact observation model is unknown and is assumed to belong to a set of candidate models. Thus, a data-based model-selection stage is performed prior to the parameter estimation stage, which affects the performance of the subsequent estimation. In this letter, we investigate post-model-selection Bayesian parameter estimation of a random vector with an unknown deterministic support set, where this support set represents the model. First, we present different estimators, including the oracle minimum mean-squared-error (MMSE), the coherent MMSE, the selected MMSE, and the full model MMSE. Then, we develop the selective Bayesian Cram$\acute{\text{e}}$r-Rao bound (BCRB) and selective tighter BCRB, which are lower bounds on the mean-squared-error (MSE) for any coherent estimator.