基于特异格和歧义格的点态部分信息分解
Pointwise Partial Information Decomposition using the Specificity and Ambiguity Lattices
摘要：
一组预测变量提供目标变量信息的不同方式是什么？什么时候一个变量提供唯一的信息，什么时候变量共享冗余信息，什么时候变量协同组合提供互补信息？威廉姆斯和比尔的部分信息分解的冗余格为这些问题的答案提供了一个有希望的一瞥。然而，这种结构是使用一个多批评措施的冗余信息，尽管持续的研究，没有完全令人满意的替代措施已经提出。在本文中，我们采取了不同的方法，应用公理推导的冗余格从一组离散变量的一个单一的实现。为了克服与有符号逐点互信息相关的困难，我们将这种分解分别应用于逐点互信息的无符号熵分量，我们将其称为特异性和模糊性。这为每个组件产生单独的冗余晶格。然后基于冗余的操作解释，我们定义冗余的特异性和模糊性的措施，使我们能够评估每个格子中的部分信息原子。这些原子可以重新组合以产生所追求的多元信息分解。我们应用这个框架，从文献中的典型例子，并讨论的结果和分解的各种属性。特别是，使用特异性和模糊性的逐点分解满足目标变量的链式规则，这为所谓的两位复制示例提供了新的见解。
英文摘要：
What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine synergistically to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions. However, this structure was constructed using a much criticised measure of redundant information, and despite sustained research, no completely satisfactory replacement measure has been proposed. In this paper, we take a different approach, applying the axiomatic derivation of the redundancy lattice to a single realisation from a set of discrete variables. To overcome the difficulty associated with signed pointwise mutual information, we apply this decomposition separately to the unsigned entropic components of pointwise mutual information which we refer to as the specificity and ambiguity. This yields a separate redundancy lattice for each component. Then based upon an operational interpretation of redundancy, we define measures of redundant specificity and ambiguity enabling us to evaluate the partial information atoms in each lattice. These atoms can be recombined to yield the sought-after multivariate information decomposition. We apply this framework to canonical examples from the literature and discuss the results and the various properties of the decomposition. In particular, the pointwise decomposition using specificity and ambiguity satisfies a chain rule over target variables, which provides new insights into the so-called two-bit-copy example.
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