Researchers extend Shapley value to uncertain data with variance control

Data valuation in machine learning often relies on the Shapley value to fairly distribute rewards based on contribution. But existing formulations assume each participant contributes a fixed, deterministic dataset. In practice, data owners typically provide samples drawn from underlying probabilistic distributions—making marginal contributions stochastic and the Shapley value itself a random variable. This paper closes that gap by defining a probabilistic Shapley value that captures both the expected contribution and the variance introduced by random sampling.

Zhuofan Jia and Jian Pei derive unbiased estimators for the expectation and variance of this probabilistic Shapley value, analyzing their statistical properties. They also introduce three Monte Carlo estimation algorithms: a baseline estimator using independent samples, a pooled estimator that reuses samples for greater efficiency, and a stratified pooled estimator that adaptively allocates sampling budget based on each player's variability. Experiments on synthetic and real datasets show the stratified pooled approach achieves strong accuracy-efficiency trade-offs, delivering substantial variance reduction at minimal added cost. This work provides both theoretical rigor and practical tools for reliable, fair data valuation in modern stochastic data-sharing environments.