Masayoshi Mase

Benjamin B. Seiler, Masayoshi Mase, Art B. Owen

This paper proposes a uniqueness Shapley measure to compare the extent to which different variables are able to identify a subject. Revealing the value of a variable on subject $t$ shrinks the set of possible subjects that $t$ could be. The extent of the shrinkage depends on which other variables have also been revealed. We use Shapley value to combine all of the reductions in log cardinality due to revealing a variable after some subset of the other variables has been revealed. This uniqueness Shapley measure can be aggregated over subjects where it becomes a weighted sum of conditional entropies. Aggregation over subsets of subjects can address questions like how identifying is age for people of a given zip code. Such aggregates have a corresponding expression in terms of cross entropies. We use uniqueness Shapley to investigate the differential effects of revealing variables from the North Carolina voter registration rolls and in identifying anomalous solar flares. An enormous speedup (approaching 2000 fold in one example) is obtained by using the all dimension trees of Moore and Lee (1998) to store the cardinalities we need.

Masayoshi Mase, Art B. Owen, Benjamin B. Seiler

Cohort Shapley value is a model-free method of variable importance grounded in game theory that does not use any unobserved and potentially impossible feature combinations. We use it to evaluate algorithmic fairness, using the well known COMPAS recidivism data as our example. This approach allows one to identify for each individual in a data set the extent to which they were adversely or beneficially affected by their value of a protected attribute such as their race. The method can do this even if race was not one of the original predictors and even if it does not have access to a proprietary algorithm that has made the predictions. The grounding in game theory lets us define aggregate variable importance for a data set consistently with its per subject definitions. We can investigate variable importance for multiple quantities of interest in the fairness literature including false positive predictions.

Masayoshi Mase, Art B. Owen, Benjamin Seiler

We introduce a variable importance measure to quantify the impact of individual input variables to a black box function. Our measure is based on the Shapley value from cooperative game theory. Many measures of variable importance operate by changing some predictor values with others held fixed, potentially creating unlikely or even logically impossible combinations. Our cohort Shapley measure uses only observed data points. Instead of changing the value of a predictor we include or exclude subjects similar to the target subject on that predictor to form a similarity cohort. Then we apply Shapley value to the cohort averages. We connect variable importance measures from explainable AI to function decompositions from global sensitivity analysis. We introduce a squared cohort Shapley value that splits previously studied Shapley effects over subjects, consistent with a Shapley axiom.