  In this paper, we provide two views of constrained differential private (DP)
mechanisms. The first one is as belief revision. A constrained DP mechanism is
obtained by standard probabilistic conditioning, and hence can be naturally
implemented by Monte Carlo algorithms. The other is as belief update. A
constrained DP is defined according to l2-distance minimization postprocessing
or projection and hence can be naturally implemented by optimization
algorithms. The main advantage of these two perspectives is that we can make
full use of the machinery of belief revision and update to show basic
properties for constrained differential privacy especially some important new
composition properties. Within the framework established in this paper,
constrained DP algorithms in the literature can be classified either as belief
revision or belief update. At the end of the paper, we demonstrate their
differences especially in utility in a couple of scenarios.
