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A Unified Definition of Mutual Information with Applications in Machine Learning

机译：互信息的统一定义及其在机器学习中的应用

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摘要

There are various definitions of mutual information. Essentially, these definitions can be divided into two classes: (1) definitions with random variables and (2) definitions with ensembles. However, there are some mathematical flaws in these definitions. For instance, Class 1 definitions either neglect the probability spaces or assume the two random variables have the same probability space. Class 2 definitions redefine marginal probabilities from the joint probabilities. In fact, the marginal probabilities are given from the ensembles and should not be redefined from the joint probabilities. Both Class 1 and Class 2 definitions assume a joint distribution exists. Yet, they all ignore an important fact that the joint or the joint probability measure is not unique. In this paper, we first present a new unified definition of mutual information to cover all the various definitions and to fix their mathematical flaws. Our idea is to define the joint distribution of two random variables by taking the marginal probabilities into consideration. Next, we establish some properties of the newly defined mutual information. We then propose a method to calculate mutual information in machine learning. Finally, we apply our newly defined mutual information to credit scoring.
机译：互信息有多种定义。从本质上讲，这些定义可以分为两类：（1）具有随机变量的定义和（2）具有合奏的定义。但是，这些定义存在一些数学缺陷。例如，类别1的定义要么忽略了概率空间，要么假设两个随机变量具有相同的概率空间。 2类定义从联合概率中重新定义了边际概率。实际上，边际概率是从集合中给出的，不应从联合概率中重新定义。 1类和2类定义均假定存在联合分布。但是，他们都忽略了一个重要事实，即联合或联合概率度量不是唯一的。在本文中，我们首先提出一个新的互信息统一定义，以涵盖所有各种定义并修复其数学缺陷。我们的想法是通过考虑边际概率来定义两个随机变量的联合分布。接下来，我们建立新定义的互信息的一些属性。然后，我们提出了一种在机器学习中计算互信息的方法。最后，我们将新定义的共同信息应用于信用评分。

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