代数统计框架下的最大熵：第一步
Maximum Entropy in the framework of Algebraic Statistics: A First Step
摘要：
代数统计学是一个最近发展的领域，人们将统计模型视为代数对象，从而在统计模型的分析和计算中使用计算交换代数和代数几何的工具。在这种方法中，统计模型参数的计算相当于求解一组多变量多项式方程组，对此可以使用著名的Grobner基理论。鉴于信息论在统计学中的重要作用，本文首先探讨了在代数统计框架下描述最大熵和最小熵模型的可能性。我们表明，ME-模型是复曲面模型（一类代数统计模型）的约束函数（提供有关潜在的随机变量的信息）是整数值的功能，和一组统计模型，结果从ME-方法确实是一个仿射品种。
英文摘要：
Algebraic statistics is a recently evolving field, where one would treat statistical models as algebraic objects and thereby use tools from computational commutative algebra and algebraic geometry in the analysis and computation of statistical models. In this approach, calculation of parameters of statistical models amounts to solving set of polynomial equations in several variables, for which one can use celebrated Grobner bases theory. Owing to the important role of information theory in statistics, this paper as a first step, explores the possibility of describing maximum and minimum entropy (ME) models in the framework of algebraic statistics. We show that ME-models are toric models (a class of algebraic statistical models) when the constraint functions (that provide the information about the underlying random variable) are integer valued functions, and the set of statistical models that results from ME-methods are indeed an affine variety.
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