  In this paper, we are concerned with obtaining distribution-free
concentration inequalities for mixture of independent Bernoulli variables that
incorporate a notion of variance. Missing mass is the total probability mass
associated to the outcomes that have not been seen in a given sample which is
an important quantity that connects density estimates obtained from a sample to
the population for discrete distributions. Therefore, we are specifically
motivated to apply our method to study the concentration of missing mass -
which can be expressed as a mixture of Bernoulli - in a novel way.
  We not only derive - for the first time - Bernstein-like large deviation
bounds for the missing mass whose exponents behave almost linearly with respect
to deviation size, but also sharpen McAllester and Ortiz (2003) and Berend and
Kontorovich (2013) for large sample sizes in the case of small deviations which
is the most interesting case in learning theory. In the meantime, our approach
shows that the heterogeneity issue introduced in McAllester and Ortiz (2003) is
resolvable in the case of missing mass in the sense that one can use standard
inequalities but it may not lead to strong results. Thus, we postulate that our
results are general and can be applied to provide potentially sharp
Bernstein-like bounds under some constraints.
