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Operations managers do not typically have full information about the demand distribution. Recognizing this, data-driven approaches have been proposed in which the manager has no information beyond the evolving history of demand observations. In practice, managers often have some partial information about the demand distribution in addition to demand observations. We consider a repeated newsvendor setting, and propose a maximum-entropy based technique, termed Second Order Belief Maximum Entropy (SOBME), which allows the manager to effectively combine demand observations with distributional information in the form of bounds on the moments or tails. In the proposed approach, the decision maker forms a belief about possible demand distributions, and dynamically updates it over time using the available data and the partial distributional information. We derive a closed-form solution for the updating mechanism, and highlight that it generalizes the traditional Bayesian mechanism with an exponential modifier that accommodates partial distributional information. We prove the proposed approach is (weakly) consistent under some technical regularity conditions and we analytically characterize its rate of convergence. We provide an analytical upper bound for the newsvendor’s cost of ambiguity, i.e., the extra per-period cost incurred because of ambiguity, under SOBME, and show that it approaches zero quite quickly. Numerical experiments demonstrate that SOBME performs very well. We find that it can be very beneficial to incorporate partial distributional information when deciding stocking quantities, and that information in the form of tighter moment bounds is typically more valuable than information in the form of tighter ambiguity sets. Moreover, unlike pure data-driven approaches, SOBME is fairly robust to the newsvendor quantile. Our results also show that SOBME quickly detects and responds to hidden changes in the unknown true distribution. We also extend our analysis to consider ambiguity aversion, and develop theoretical and numerical results for the ambiguity-averse, repeated newsvendor setting.