  This work proposes an algorithmic method to verify differential privacy for
estimation mechanisms with performance guarantees. Differential privacy makes
it hard to distinguish outputs of a mechanism produced by adjacent inputs.
While obtaining theoretical conditions that guarantee differential privacy may
be possible, evaluating these conditions in practice can be hard. This is
especially true for estimation mechanisms that take values in continuous
spaces, as this requires checking for an infinite set of inequalities. Instead,
our verification approach consists of testing the differential privacy
condition for a suitably chosen finite collection of events at the expense of
some information loss. More precisely, our data-driven, test framework for
continuous range mechanisms first finds a highly-likely, compact event set, as
well as a partition of this event, and then evaluates differential privacy wrt
this partition. This results into a type of differential privacy with high
confidence, which we are able to quantify precisely. This approach is then used
to evaluate the differential-privacy properties of the recently proposed $W_2$
Moving Horizon Estimator. We confirm its properties, while comparing its
performance with alternative approaches in simulation.
