  We define discounted differential privacy, as an alternative to
(conventional) differential privacy, to investigate privacy of evolving
datasets, containing time series over an unbounded horizon. We use privacy loss
as a measure of the amount of information leaked by the reports at a certain
fixed time. We observe that privacy losses are weighted equally across time in
the definition of differential privacy, and therefore the magnitude of
privacy-preserving additive noise must grow without bound to ensure
differential privacy over an infinite horizon. Motivated by the discounted
utility theory within the economics literature, we use exponential and
hyperbolic discounting of privacy losses across time to relax the definition of
differential privacy under continual observations. This implies that privacy
losses in distant past are less important than the current ones to an
individual. We use discounted differential privacy to investigate privacy of
evolving datasets using additive Laplace noise and show that the magnitude of
the additive noise can remain bounded under discounted differential privacy. We
illustrate the quality of privacy-preserving mechanisms satisfying discounted
differential privacy on smart-meter measurement time-series of real households,
made publicly available by Ausgrid (an Australian electricity distribution
company).
