  In observational causal inference, in order to emulate a randomized
experiment, weights are used to render treatments independent of observed
covariates. This property is known as balance; in its absence, estimated causal
effects may be arbitrarily biased. In this work we introduce permutation
weighting, a method for estimating balancing weights using a standard binary
classifier (regardless of cardinality of treatment). A large class of
probabilistic classifiers may be used in this method; the choice of loss for
the classifier implies the particular definition of balance. We bound bias and
variance in terms of the excess risk of the classifier, show that these
disappear asymptotically, and demonstrate that our classification problem
directly minimizes imbalance. A wide variety of existing balancing weights may
be estimated through this regime, allowing for direct comparison between
methods based on classifier loss, as well as hyper-parameter tuning using
cross-validation. Empirical evaluations indicate that permutation weighting
provides favorable performance in comparison to existing methods.
