On the Value of Information and Mean Squared Error for Noisy Gaussian Models

The relationship between the age of information (AoI) and the mean squared error (MSE) in optimisation problems has been widely investigated for various Gaussian Markov models. Recently, we put the AoI in an information-theoretic context and proposed a mutual information-based value of information (VoI) framework to characterise how valuable the status updates are. In this letter we consider general noisy Gaussian models and investigate the connection between the VoI and MSE. We show that the VoI can be presented by a logarithmic function of the MSE. We also study the rate of change of the two with respect to the signal-to-noise ratio (SNR) and the AoI. The results hold for both discrete-time and continuous-time stationary Gaussian processes. This letter illuminates a useful relationship between information theory and signal processing theory in the presence of the AoI and additive noise, and provides a possible way to infer the information value by estimating the MSE for practical applications.