When Does Uncertainty Quantification Matter in Predict-then-Optimize?

Predict-then-optimize (PtO) pipelines first predict uncertain problem parameters and then solve an optimization problem using the resulting predictive distribution. A common intuition is that better uncertainty quantification (UQ) should improve downstream decisions, but the extent to which this is true depends on how distributional errors interact with the geometry of the optimization problem. This paper formalizes that dependence and characterizes when predictive uncertainty materially affects end-to-end decision quality. We model a PtO pipeline in which a predictor outputs a predictive law that is consumed directly by a downstream stochastic, risk-averse, chance-constrained, or precision-weighted optimization problem. We derive bounds showing that excess decision risk is controlled by the discrepancy between the predictive and true laws, scaled by the sensitivity of the downstream optimization problem. When the predictive law is restricted to a structured low-dimensional family, this error further decomposes into estimation error and structural bias. Moment-based coupling bounds relate decision degradation to errors in predictive means and covariances, while structured uncertainty models can substantially reduce estimation complexity. For scale-invariant decision rules, only relative uncertainty accuracy matters. In precision-weighted quadratic problems, the resulting efficiency loss admits a fully explicit characterization. Numerical experiments on newsvendor, strongly convex, chance-constrained, and precision-weighted problems confirm the theoretical bounds and sensitivity predictions. Overall, the results show that the importance of UQ is not intrinsic to the prediction problem alone, but is jointly determined by the difficulty of estimating the predictive distribution and the sensitivity of the downstream optimizer to decision-relevant distributional errors.