  The sample mean is among the most well studied estimators in statistics,
having many desirable properties such as unbiasedness and consistency. However,
when analyzing data collected using a multi-armed bandit (MAB) experiment, the
sample mean is biased and much remains to be understood about its properties.
For example, when is it consistent, how large is its bias, and can we bound its
mean squared error? This paper delivers a thorough and systematic treatment of
the bias, risk and consistency of MAB sample means. Specifically, we identify
four distinct sources of selection bias (sampling, stopping, choosing and
rewinding) and analyze them both separately and together. We further
demonstrate that a new notion of \emph{effective sample size} can be used to
bound the risk of the sample mean under suitable loss functions. We present
several carefully designed examples to provide intuition on the different
sources of selection bias we study. Our treatment is nonparametric and
algorithm-agnostic, meaning that it is not tied to a specific algorithm or
goal. In a nutshell, our proofs combine variational representations of
information-theoretic divergences with new martingale concentration
inequalities.
