  Distributional data Shapley value (DShapley) has recently been proposed as a
principled framework to quantify the contribution of individual datum in
machine learning. DShapley develops the foundational game theory concept of
Shapley values into a statistical framework and can be applied to identify data
points that are useful (or harmful) to a learning algorithm. Estimating
DShapley is computationally expensive, however, and this can be a major
challenge to using it in practice. Moreover, there has been little mathematical
analyses of how this value depends on data characteristics. In this paper, we
derive the first analytic expressions for DShapley for the canonical problems
of linear regression, binary classification, and non-parametric density
estimation. These analytic forms provide new algorithms to estimate DShapley
that are several orders of magnitude faster than previous state-of-the-art
methods. Furthermore, our formulas are directly interpretable and provide
quantitative insights into how the value varies for different types of data. We
demonstrate the practical efficacy of our approach on multiple real and
synthetic datasets.
