  Rating systems are ubiquitous, with applications ranging from product
recommendation to teaching evaluations. Confidence intervals for functionals of
rating data such as empirical means or quantiles are critical to
decision-making in various applications including recommendation/ranking
algorithms. Confidence intervals derived from standard Hoeffding and Bernstein
bounds can be quite loose, especially in small sample regimes, since these
bounds do not exploit the geometric structure of the probability simplex. We
propose a new approach to deriving confidence intervals that are tailored to
the geometry associated with multi-star/value rating systems using a
combination of techniques from information theory, including Kullback-Leibler,
Sanov, and Csisz{\'a}r inequalities.
  The new confidence intervals are almost always as good or better than all
standard methods and are significantly tighter in many situations. The standard
bounds can require several times more samples than our new bounds to achieve
specified confidence interval widths.
