An Information Hierarchy of Privacy Filters and Odometers

In sequential data analysis, privacy loss accumulates over time, and privacy filters and odometers are used to control and track this cumulative cost. We study how the information used by differential privacy filters and odometers fundamentally limits what they can achieve. We formalise three levels of information: spec-level information, fixed prior to the interaction; mechanism-level information, which includes the mechanisms instantiated during the interaction; and realisation-level information, which further includes the realised interaction. We classify privacy filters and odometers by the least informative level sufficient to determine their behaviour. Our main results establish strict and arbitrarily large separations in achievable performance across this information hierarchy. For privacy filters, moving up the hierarchy can yield an unbounded utility advantage, quantified by a clipped measure of the number of allowed releases. For privacy odometers, it can yield an arbitrarily large improvement in precision, quantified by the expected peak excess of the reported bound over the realised privacy loss. Further, although the information classes are distinct, we show that moving to a richer class need incur essentially no loss in performance. Finally, we show that randomisation provides a resource orthogonal to information: at every information level, it can yield an arbitrarily large utility improvement over deterministic filters. Thus, information access and randomisation can be viewed as fundamental resources governing privacy procedures in adaptive interactions.