  Let $\mathfrak{C}$ be a class of probability distributions over the discrete
domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather
general condition -- essentially, that each distribution in $\mathfrak{C}$ can
be well-approximated by a variable-width histogram with few bins -- then there
is a highly efficient (both in terms of running time and sample complexity)
algorithm that can learn any mixture of $k$ unknown distributions from
$\mathfrak{C}.$
  We analyze several natural types of distributions over $[n]$, including
log-concave, monotone hazard rate and unimodal distributions, and show that
they have the required structural property of being well-approximated by a
histogram with few bins. Applying our general algorithm, we obtain
near-optimally efficient algorithms for all these mixture learning problems.
