James Ryan G, Emenheiser Jeffrey, Crutchfield James P
Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA.
Entropy (Basel). 2018 Dec 24;21(1):12. doi: 10.3390/e21010012.
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable X i has on a target variable , relative to the other sources. For two sources, influence breaks down into the information that both X 0 and X 1 redundantly share with , what X 0 uniquely shares with , what X 1 uniquely shares with , and finally what X 0 and X 1 synergistically share with . Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique information. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PID's meaning-interpretations not present in PID's definition but that, we argue, need to be explicit. Specifically, the use of a consistent PID quantified using a secret key agreement rate naturally induces a directional interpretation of the PID. We further reveal a surprising connection between third-order connected information, two-way secret key agreement rate, and synergy. We also consider difficulties which arise with a popular PID measure in light of the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.
部分信息分解（PID）是一个很有前景的框架，用于将一个联合随机变量分解为每个源变量(X_i)相对于其他源对目标变量的影响量。对于两个源，影响可分解为(X_0)和(X_1)与目标变量冗余共享的信息、(X_0)与目标变量唯一共享的信息、(X_1)与目标变量唯一共享的信息，以及最后(X_0)和(X_1)与目标变量协同共享的信息。不幸的是，对于如何量化这四个分量出现了相当大的分歧。借鉴密码学，我们将秘密密钥协商率视为量化唯一信息的一种操作方法。秘密密钥协商率有几种形式，这取决于允许哪些方进行通信。我们证明这四种形式中的三种与PID不一致。剩下的形式暗示了对PID含义的某些解释——这些解释在PID的定义中并不存在，但我们认为需要明确。具体而言，使用通过秘密密钥协商率量化的一致PID自然会引发对PID的定向解释。我们还揭示了三阶连通信息、双向秘密密钥协商率和协同作用之间令人惊讶的联系。我们还根据这里的结果以及最大熵观点考虑了一种流行的PID度量所产生的困难。最后，我们回顾了PID面临的挑战。
