  SHAP explanations are a popular feature-attribution mechanism for explainable
AI. They use game-theoretic notions to measure the influence of individual
features on the prediction of a machine learning model. Despite a lot of recent
interest from both academia and industry, it is not known whether SHAP
explanations of common machine learning models can be computed efficiently. In
this paper, we establish the complexity of computing the SHAP explanation in
three important settings. First, we consider fully-factorized data
distributions, and show that the complexity of computing the SHAP explanation
is the same as the complexity of computing the expected value of the model.
This fully-factorized setting is often used to simplify the SHAP computation,
yet our results show that the computation can be intractable for commonly used
models such as logistic regression. Going beyond fully-factorized
distributions, we show that computing SHAP explanations is already intractable
for a very simple setting: computing SHAP explanations of trivial classifiers
over naive Bayes distributions. Finally, we show that even computing SHAP over
the empirical distribution is #P-hard.
