A Bayesian investor learns an unknown asset drift by Kalman-Bucy filtering and trades the mean-variance optimal portfolio, but his observation model may be wrong. We make the policy robust to an adversary who distorts the law of observed prices, paying for it in relative entropy. Because wealth and beliefs are driven by the same Brownian motion, one distortion corrupts trading profits and the filter together. The robust policy and its price are then closed form. To leading order, the price of robustness is half the variance of the loss the non-robust investor would suffer. The policy pulls back from large positions by a cubic correction. With a known drift the non-robust policy is infinitely costly; under learning the loss is bounded and the cost finite. The new structure, though, comes from how the robustness penalty is scaled rather than from learning: value-scaling preserves the affine policy exactly.
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