We review the cumulant decomposition (a way of decomposing the expectation of
a product of random variables (e.g. $\mathbb{E}[XYZ]$) into a sum of terms
corresponding to partitions of these variables.) and the Wick decomposition (a
way of decomposing a product of (not necessarily random) variables into a sum
of terms corresponding to subsets of the variables). Then we generalize each
one to a new decomposition where the product function is generalized to an
arbitrary function.

我们回顾了累积分解和 Wick 分解，然后将每个分解推广到一个任意函数的新分解。

广义 Wick 分解

Generalized Wick Decompositions

Categorical logic has shown that modern logic is essentially the logic of
subsets (or "subobjects"). Partitions are dual to subsets so there is a dual
logic of partitions where a "distinction" [an ordered pair of distinct elements
(u,u') from the universe U ] is dual to an "element". An element being in a
subset is analogous to a partition p on U making a distinction, i.e., if u and
u' were in different blocks of p. Subset logic leads to finite probability
theory by taking the (Laplacian) probability as the normalized size of each
subset-event of a finite universe. The analogous step in the logic of
partitions is to assign to a partition the number of distinctions made by a
partition normalized by the total number of ordered pairs |UxU| from the finite
universe. That yields a notion of "logical entropy" for partitions and a
"logical information theory." The logical theory directly counts the
(normalized) number of distinctions in a partition while Shannon's theory gives
the average number of binary partitions needed to make those same distinctions.
Thus the logical theory is seen as providing a conceptual underpinning for
Shannon's theory based on the logical notion of "distinctions." (forthcoming in
Synthese)

通过分类逻辑从子集的角度来探讨分区逻辑，发展了逻辑熵的概念，并与 Shannon 理论的二元分区概念对应，从逻辑层面提供了 Shannon 理论的概念基础。

区分的计数：论香农信息论概念基础

Counting Distinctions: On the Conceptual Foundations of Shannon's  Information Theory

Many signal processing problems can be solved by maximizing the fitness of a
segmented model over all possible partitions of the data interval. This letter
describes a simple but powerful algorithm that searches the exponentially large
space of partitions of $N$ data points in time $O(N^2)$. The algorithm is
guaranteed to find the exact global optimum, automatically determines the model
order (the number of segments), has a convenient real-time mode, can be
extended to higher dimensional data spaces, and solves a surprising variety of
problems in signal detection and characterization, density estimation, cluster
analysis and classification.

本文提出一种搜索数据点划分的算法，可以自动确定模型阶数，适用于高维数据，可解决信号处理中的多种问题。