Partial information decompositions (PIDs), which quantify information interactions between three or more variables in terms of uniqueness, redundancy and synergy, are gaining traction in many application domains. However, our understanding of the operational interpretations of PIDs is still incomplete for many popular PID definitions. In this paper, we discuss the operational interpretations of unique information through the lens of two well-known PID definitions. We reexamine an interpretation from statistical decision theory showing how unique information upper bounds the risk in a decision problem. We then explore a new connection between the two PIDs, which allows us to develop an informal but appealing interpretation, and generalize the PID definitions using a common Lagrangian formulation. Finally, we provide a new PID definition that is able to capture the information that is unique. We also show that it has a straightforward interpretation and examine its properties.

翻译：部分信息分解（PID）通过独特性、冗余性和协同性来量化三个或更多变量之间的信息交互，这在许多应用领域中日益受到关注。然而，对于许多流行的PID定义，我们对它们操作解释的理解仍不完整。本文通过两种著名的PID定义，探讨了独特信息的操作解释。我们重新审视了统计决策理论中的一种解释，揭示了独特信息如何作为决策问题中风险的上界。随后，我们探索了这两种PID之间的新联系，从而发展出一种非正式但引人入胜的解释，并通过共同的拉格朗日公式推广了PID定义。最后，我们提出了一种新的PID定义，能够捕捉独特的信息。我们还证明了该定义具有直观的解释，并分析了其性质。