  This survey provides an exposition of a suite of techniques based on the
theory of polynomials, collectively referred to as polynomial methods, which
have recently been applied to address several challenging problems in
statistical inference successfully. Topics including polynomial approximation,
polynomial interpolation and majorization, moment space and positive
polynomials, orthogonal polynomials and Gaussian quadrature are discussed, with
their major probabilistic and statistical applications in property estimation
on large domains and learning mixture models. These techniques provide useful
tools not only for the design of highly practical algorithms with provable
optimality, but also for establishing the fundamental limits of the inference
problems through the method of moment matching. The effectiveness of the
polynomial method is demonstrated in concrete problems such as entropy and
support size estimation, distinct elements problem, and learning Gaussian
mixture models.
