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.

翻译：这份调查根据多面体理论(统称为多面体方法)展示了一套技术,这些技术最近被成功地用于解决统计推断中若干具有挑战性的问题。主题包括多面体近似、多面体间插和主要化、时空和积极的多面体、正形多面体和高斯方形,这些技术在大域和学习混合模型的财产估计中的主要概率和统计应用。这些技术不仅为设计高度实用且具有可辨最佳性的算法提供了有用的工具,而且为通过瞬间匹配方法确定推断问题的基本界限提供了有用的工具。多面体法的有效性表现在具体的问题中,例如:昆虫和辅助体大小估计、不同元素问题和学习高地体混合模型。