  Importance sampling is a widely used technique to estimate properties of a
distribution. This paper investigates trading-off some bias for variance by
adaptively winsorizing the importance sampling estimator. The novel winsorizing
procedure, based on the Balancing Principle (or Lepskii's Method), chooses a
threshold level among a pre-defined set by roughly balancing the bias and
variance of the estimator when winsorized at different levels. As a
consequence, it provides a principled way to perform winsorization with
finite-sample optimality guarantees under minimal assumptions. In various
examples, the proposed estimator is shown to have smaller mean squared error
and mean absolute deviation than leading alternatives.
