Entropy Regularization under Bayesian Drift Uncertainty

We study entropy-regularized mean-variance portfolio optimization under Bayesian drift uncertainty. Gaussian policies remain optimal under partial information, the value function is quadratic in wealth, and belief-dependent coefficients admit closed-form solutions. The mean control is identical to deterministic Bayesian Markowitz feedback; entropy regularization affects only the policy variance. Additionally, this variance does not affect information gain, and instead provides belief-dependent robustness. Notably, optimal policy variance increases with posterior conviction $|m_t|$, forcing greater action randomization when mean position is most aggressive.