用贝叶斯方法量化两系统间依赖性的证据强度。
An inferential measure of dependence between two systems using Bayesian model comparison
- 基于贝叶斯模型比较，计算依赖模型的后验概率
- 该度量能有效区分噪声干扰下的真实依赖关系
- 适合需要可解释依赖性分析的研究者使用
我们提出一种基于贝叶斯模型比较的依赖性度量方法，用于量化数据集 $D$ 中两个系统 $X$ 与 $Y$ 之间的依赖关系。该度量 $B(X,Y|D)$ 定义为依赖模型 $H_1$ 在观测数据 $D$ 下的后验概率 $P(H_1|D)$，或其严格递增函数。通过模拟实验，研究了噪声影响及 $H_1$ 中依赖强度参数的行为。结果表明，$B(X,Y|D)$ 能准确反映数据 $D$ 支持 $H_1$ 相较于 $H_0$（独立）的信息量。该度量具备合理性与新颖性，属于“推断型”依赖度量。最后讨论了贝叶斯框架的含义，并对比了 $B(X,Y|D)$ 与互信息的异同。
原文摘要 · Abstract (English)
We propose to quantify dependence between two systems $X$ and $Y$ in a dataset $D$ based on the Bayesian comparison of two models: one, $H_0$, of statistical independence and another one, $H_1$, of dependence. In this framework, dependence between $X$ and $Y$ in $D$, denoted $B(X,Y|D)$, is quantified as $P(H_1|D)$, the posterior probability for the model of dependence given $D$, or any strictly increasing function thereof. It is therefore a measure of the evidence for dependence between $X$ and $Y$ as modeled by $H_1$ and observed in $D$. We review several statistical models and reconsider standard results in the light of $B(X,Y|D)$ as a measure of dependence. Using simulations, we focus on two specific issues: the effect of noise and the behavior of $B(X,Y|D)$ when $H_1$ has a parameter coding for the intensity of dependence. We then derive some general properties of $B(X,Y|D)$, showing that it quantifies the information contained in $D$ in favor of $H_1$ versus $H_0$. While some of these properties are typical of what is expected from a valid measure of dependence, others are novel and naturally appear as desired features for specific measures of dependence, which we call inferential. We finally put these results in perspective; in particular, we discuss the consequences of using the Bayesian framework as well as the similarities and differences between $B(X,Y|D)$ and mutual information.
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