  A fundamental problem arising in many areas of machine learning is the
evaluation of the likelihood of a given observation under different nominal
distributions. Frequently, these nominal distributions are themselves estimated
from data, which makes them susceptible to estimation errors. We thus propose
to replace each nominal distribution with an ambiguity set containing all
distributions in its vicinity and to evaluate an \emph{optimistic likelihood},
that is, the maximum of the likelihood over all distributions in the ambiguity
set. When the proximity of distributions is quantified by the Fisher-Rao
distance or the Kullback-Leibler divergence, the emerging optimistic
likelihoods can be computed efficiently using either geodesic or standard
convex optimization techniques. We showcase the advantages of working with
optimistic likelihoods on a classification problem using synthetic as well as
empirical data.
