Privacy Factor and Relative Fisher Geometry for Parametric Stochastic Processes

We develop a theory of informational privacy for parametric dynamical systems observed in real time through a Gaussian noisy channel. Taking the Kullback–Leibler divergence between the law of the observations and a fixed reference law, we define the absolute privacy factor as the exponential of the negative KL rate, and show via Stein's lemma that it is exactly the per-sample geometric rate at which the optimal adversarial test's miss probability decays. On the space–time parameter manifold we construct a relative Fisher metric, obtained by pulling back the Mahalanobis inner product through the deviation of the observation mean from the reference. The associated informational energy functional admits, at fixed kinetic budget, minimisers that we characterise as eigenfunctions of a variational eigenvalue problem, the eigenvalues measuring the leakage-per-unit-work trade-off. In the linear Gaussian setting the problem reduces to a generalised matrix eigenproblem of Galerkin type, which we solve on an adapted sine basis with provably monotone convergence from above and algebraic rate. A sharp lower bound on the privacy factor in terms of the first eigenvalue λ₁ establishes an intrinsic privacy–efficiency trade-off: the first eigenfunction is the optimal perturbation of a nominal parameter trajectory that, for a prescribed variational budget, maximises the horizon over which an observer with access to the observables alone cannot detect the deviation with statistical certainty.