How much information per joule? Measuring the energy efficiency of inferential wireless sensing

Wireless sensor networks are increasingly seen not just as data collection instruments, but as part of an infrastructure designed to construct models of their embedding environments. Their energy supplies are limited, in turn constraining the amount of data that can be acquired and reported. This motivates consideration of the inferential value and energy cost of reporting the sensed data in the design of coding algorithms and/or reporting strategies, rather than the information content of the data itself. There is a need to evaluate the efficiency of these strategies and parametric variations of them in the context of model inference. This paper describes an information-theoretic measure, the relative divergence distribution (RDD), of their efficacy that relates inferential performance and the energy cost of reporting the data used in inference. It is non-asymptotic, and, as part of a Bayesian inference framework, does not require a prior distribution on the data model but can accommodate prior information on process model parameters. The inferential energy efficiency, based on the RDD, is a measure of the "fuel economy" of inferential sensing. These measures are applied to inferential sensing of a Bernoulli process, where it is demonstrated that the entropy of a data stream is not necessarily useful in understanding its value in inference.