Artificial intelligence provides low-cost predictions for many analytics tasks, but their reliability and the value of another human label vary across tasks. We study how to allocate a shared human-validation budget when task-level uncertainty is unknown and each acquired label informs both estimation and subsequent allocation. For known difficulty levels, the benchmark follows a cost-adjusted Neyman rule that accounts for task importance and label cost. Within our Tuned Rectification-Difficulty framework, we develop an upper-confidence-bound policy that jointly learns the correction coefficients of tuned prediction-powered inference and the resulting difficulties from paired human labels and predictions, and establish an allocation-regret guarantee for the confidence-radius policy. In synthetic experiments and a real digital-twin survey with 68 tasks and over 2,000 respondents, our method closes most of the gap to the oracle on the allocation objective. It matches the estimation error of uniform allocation with up to 22.7% and 4.5% fewer human labels, respectively. The same allocation structure extends beyond scalar means to vector-valued decision models, including regression and choice/conjoint estimation, by replacing the scalar residual variance with a scalarized sandwich covariance. The value of AI depends not only on model accuracy but also on the operational policy that targets human oversight where it matters most.
Score-based variational inference (SBVI) provides an alternative to classical Kullback--Leibler (KL)-based variational inference by learning a distribution using only score queries of the target. EigenVI~\cite{cai2024eigenvi} formulates SBVI as an eigenvalue problem, but faces two limitations: the number of variational parameters grows exponentially with the data dimension, and the single-eigenvector formulation cannot represent the full minimum-eigenvalue eigenspace. We propose QuanVI, a quantum-inspired SBVI algorithm that addresses both limitations. Rather than selecting a single eigenvector, QuanVI captures a learned subspace through a density-operator formulation. To enable scalable computation, \model{} employs a matrix-product-operator (MPO) structure parameterized by a quantum tensor network (QTN) with unitary-constrained local tensors and auxiliary ancilla wires. Numerical experiments demonstrate the scalability of QuanVI beyond EigenVI and competitive performance on synthetic distributions and Bayesian posterior-approximation benchmarks, including challenging non-Gaussian targets. The code is available at https://github.com/ChaoLiAtRIKEN/quanvi.