  Accurately evaluating new policies (e.g. ad-placement models, ranking
functions, recommendation functions) is one of the key prerequisites for
improving interactive systems. While the conventional approach to evaluation
relies on online A/B tests, recent work has shown that counterfactual
estimators can provide an inexpensive and fast alternative, since they can be
applied offline using log data that was collected from a different policy
fielded in the past. In this paper, we address the question of how to estimate
the performance of a new target policy when we have log data from multiple
historic policies. This question is of great relevance in practice, since
policies get updated frequently in most online systems. We show that naively
combining data from multiple logging policies can be highly suboptimal. In
particular, we find that the standard Inverse Propensity Score (IPS) estimator
suffers especially when logging and target policies diverge -- to a point where
throwing away data improves the variance of the estimator. We therefore propose
two alternative estimators which we characterize theoretically and compare
experimentally. We find that the new estimators can provide substantially
improved estimation accuracy.
