  We consider stochastic optimization under distributional uncertainty, where
the unknown distributional parameter is estimated from streaming data that
arrive sequentially over time. Moreover, data may depend on the decision of the
time when they are generated. For both decision-independent and
decision-dependent uncertainties, we propose an approach to jointly estimate
the distributional parameter via Bayesian posterior distribution and update the
decision by applying stochastic gradient descent on the Bayesian average of the
objective function. Our approach converges asymptotically over time and
achieves the convergence rates of classical SGD in the decision-independent
case. We demonstrate the empirical performance of our approach on both
synthetic test problems and a classical newsvendor problem.
