We consider the problem of decomposing the information content of three jointly Gaussian random vectors using the partial information decomposition (PID) framework. Barrett previously characterized the Gaussian PID for a scalar "source" or "message" in closed form - we extend this to the case where the message is a vector. Specifically, we revisit a connection between the notions of Blackwell sufficiency of statistical experiments and stochastic degradedness of broadcast channels, to provide a necessary and sufficient condition for the existence of unique information in the fully multivariate Gaussian PID. The condition we identify indicates that the closed form PID for the scalar case rarely extends to the vector case. We also provide a convex optimization approach for approximating a PID in the vector case, analyze its properties, and evaluate it empirically on randomly generated Gaussian systems.
翻译:我们考虑了使用部分信息分解(PID)框架将三种高斯随机矢量的信息内容分解的问题。 巴雷特以前曾以封闭形式将高斯的 PID 描述为“源” 或“消息” 的标量“源” 或“消息”, 我们将此扩展至电文为矢量的情况。 具体地说, 我们重新审视了“ Blackwell” 概念在统计实验的充分性和广播频道的随机退化性之间的联系, 以便为完全多变化的 Gaussian PID 中存在独特信息提供了必要和充分的条件。 我们确定的条件表明, 星标的封闭形式 PID 很少延伸到矢量 案例。 我们还提供了在矢量中接近 PID 的二次曲线优化方法, 分析其属性, 并以随机生成的高斯系统进行实验性评估 。