  We introduce MCCE: Monte Carlo sampling of valid and realistic Counterfactual
Explanations for tabular data, a novel counterfactual explanation method that
generates on-manifold, actionable and valid counterfactuals by modeling the
joint distribution of the mutable features given the immutable features and the
decision. Unlike other on-manifold methods that tend to rely on variational
autoencoders and have strict prediction model and data requirements, MCCE
handles any type of prediction model and categorical features with more than
two levels. MCCE first models the joint distribution of the features and the
decision with an autoregressive generative model where the conditionals are
estimated using decision trees. Then, it samples a large set of observations
from this model, and finally, it removes the samples that do not obey certain
criteria. We compare MCCE with a range of state-of-the-art on-manifold
counterfactual methods using four well-known data sets and show that MCCE
outperforms these methods on all common performance metrics and speed. In
particular, including the decision in the modeling process improves the
efficiency of the method substantially.
