A basic task in explainable AI (XAI) is to identify the most important features behind a prediction made by a black box function $f$. The insertion and deletion tests of Petsiuk et al. (2018) can be used to judge the quality of algorithms that rank pixels from most to least important for a classification. Motivated by regression problems we establish a formula for their area under the curve (AUC) criteria in terms of certain main effects and interactions in an anchored decomposition of $f$. We find an expression for the expected value of the AUC under a random ordering of inputs to $f$ and propose an alternative area above a straight line for the regression setting. We use this criterion to compare feature importances computed by integrated gradients (IG) to those computed by Kernel SHAP (KS) as well as LIME, DeepLIFT, vanilla gradient and input$\times$gradient methods. KS has the best overall performance in two datasets we consider but it is very expensive to compute. We find that IG is nearly as good as KS while being much faster. Our comparison problems include some binary inputs that pose a challenge to IG because it must use values between the possible variable levels and so we consider ways to handle binary variables in IG. We show that sorting variables by their Shapley value does not necessarily give the optimal ordering for an insertion-deletion test. It will however do that for monotone functions of additive models, such as logistic regression.

A model-agnostic variable importance method can be used with arbitrary prediction functions. Here we present some model-free methods that do not require access to the prediction function. This is useful when that function is proprietary and not available, or just extremely expensive. It is also useful when studying residuals from a model. The cohort Shapley (CS) method is model-free but has exponential cost in the dimension of the input space. A supervised on-manifold Shapley method from Frye et al. (2020) is also model free but requires as input a second black box model that has to be trained for the Shapley value problem. We introduce an integrated gradient (IG) version of cohort Shapley, called IGCS, with cost $\mathcal{O}(nd)$. We show that over the vast majority of the relevant unit cube that the IGCS value function is close to a multilinear function for which IGCS matches CS. Another benefit of IGCS is that is allows IG methods to be used with binary predictors. We use some area between curves (ABC) measures to quantify the performance of IGCS. On a problem from high energy physics we verify that IGCS has nearly the same ABCs as CS does. We also use it on a problem from computational chemistry in 1024 variables. We see there that IGCS attains much higher ABCs than we get from Monte Carlo sampling. The code is publicly available at https://github.com/cohortshapley/cohortintgrad

The most popular methods for measuring importance of the variables in a black box prediction algorithm make use of synthetic inputs that combine predictor variables from multiple subjects. These inputs can be unlikely, physically impossible, or even logically impossible. As a result, the predictions for such cases can be based on data very unlike any the black box was trained on. We think that users cannot trust an explanation of the decision of a prediction algorithm when the explanation uses such values. Instead we advocate a method called Cohort Shapley that is grounded in economic game theory and unlike most other game theoretic methods, it uses only actually observed data to quantify variable importance. Cohort Shapley works by narrowing the cohort of subjects judged to be similar to a target subject on one or more features. We illustrate it on an algorithmic fairness problem where it is essential to attribute importance to protected variables that the model was not trained on.

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.

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.