The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

中文翻译：

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嘈杂 Ornstein-Uhlenbeck 过程的年龄与信息价值之间的关系

信息时代（AoI）已被广泛用于量化实时状态更新系统中的信息新鲜度。由于 AoI 独立于源数据和上下文的固有属性，我们为隐马尔可夫模型引入了一个基于互信息的信息价值 (VoI) 框架。在本文中，我们研究了嘈杂的 Ornstein-Uhlenbeck (OU) 过程中的 VoI 及其与 AoI 的关系。我们探索相关性和噪声对其关系的影响，并在三种不同的机制中发现两者之间的对数、指数和线性相关性。这为之前在其他作品中报告的非线性 AoI 函数的选择提供了正式的理由。此外，我们在队列模型的例子中研究了 VoI 的统计特性，推导了它的分布函数和矩。还分析了平均 VoI 的下限和上限，可用于新鲜度感知网络的设计和优化。数值结果进一步表明，与传统的线性年龄和一些基本的非线性年龄函数相比，所提出的 VoI 框架更通用，适用于各种环境。