与熵链式法则有关的广义多项式系数的同调刻画
A homological characterization of generalized multinomial coefficients related to the entropic chain rule
摘要：
多项式系数之间的乘法关系与香农熵的（加性）递归性质之间存在一种渐近关系，称为链式法则。我们发现，这两种类型的身份是一个独特的代数结构的表现形式：一个$1$ -上循环条件在\emph{information cohomology}，代数不变的phesheaves模块\emph{information structures}（类的观测值）。Baudot和Bennequin引入了这种上同调，并证明了当系数是概率泛函的自然预层时，Shannon熵表示唯一的非平凡的$1$度上的上同调类。作者后来得到了该预层的形变的$1$ -参数族，以这样的方式，每个Tsallis $α$ -熵表现为与参数$α$相关联的唯一的$1$ -上循环。在这篇文章中，我们介绍了一个新的预层的\emph{combinatorial functionals}，这是可测的功能有限数组的整数，这些数组表示\emph{histograms}相关的随机实验。在这种情况下，唯一的上同调类的次数$0$是由指数函数和$1$ -上循环生成的Fontené-Ward广义多项式系数。作为一个副产品，我们得到一个简单的组合模拟的基本方程的信息理论的特点广义二项式系数。将上述渐近关系推广到某些广义多项式系数与任何$α$ -熵之间的对应关系，从而对链式法则及其变形的意义有了新的认识.
英文摘要：
There is an asymptotic relationship between the multiplicative relations among multinomial coefficients and the (additive) recurrence property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a $1$-cocycle condition in \emph{information cohomology}, an algebraic invariant of phesheaves of modules on \emph{information structures} (categories of observables). Baudot and Bennequin introduced this cohomology and proved that Shannon entropy represents the only nontrivial cohomology class in degree $1$ when the coefficients are a natural presheaf of probabilistic functionals. The author obtained later a $1$-parameter family of deformations of that presheaf, in such a way that each Tsallis $α$-entropy appears as the unique $1$-cocycle associated to the parameter $α$. In this article, we introduce a new presheaf of \emph{combinatorial functionals}, which are measurable functions of finite arrays of integers; these arrays represent \emph{histograms} associated to random experiments. In this case, the only cohomology class in degree $0$ is generated by the exponential function and $1$-cocycles are Fontené-Ward generalized multinomial coefficients. As a byproduct, we get a simple combinatorial analogue of the fundamental equation of information theory that characterizes the generalized binomial coefficients. The asymptotic relationship mentioned above is extended to a correspondence between certain generalized multinomial coefficients and any $α$-entropy, that sheds new light on the meaning of the chain rule and its deformations.
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