Abstract
Physical sensing and actuation noise floors should inform how much belief resolution a decision-making system can reliably use. We introduce Finite Reliability Representations (FRR), a framework for covering belief spaces by reliability cells: regions within which the optimal action-value function Q*(b,u) varies by at most a tolerance epsilon, uniformly over actions. The framework is formulated on beliefs rather than states and uses a cover rather than an equivalence quotient, because approximate decision-closeness is not transitive in general. A central technical point is that noisy Bayesian updates should not be treated as globally contractive on arbitrary beliefs. We therefore separate three objects: the fixed-observation filter map, the predictive observation law, and the controlled belief-transition kernel. For nonlinear continuous-state systems, FRR is obtained under a reachable-set Lipschitz modulus for the belief-transition kernel. For finite-state POMDPs, the same construction becomes exact on the belief simplex: prediction is linear, Bayesian correction is a normalized positive linear map, sensor noise enters through observation-distribution distinguishability, and actuation uncertainty enters through an action-execution channel. Under the corresponding action-value Lipschitz condition, an FRR cover supports a cell-constant policy whose suboptimality is bounded by 2 epsilon/(1 - gamma). We also introduce reliability entropy, the logarithm of the minimal number of reliability cells, as a measure of certified decision-relevant belief complexity. The framework distinguishes representation sufficiency from fundamental performance floors imposed by sensing, process, and actuation noise. It applies to finite POMDPs, linear-Gaussian filters, locally linearized nonlinear filters, and particle-filter implementations through analytic or empirical certification of reliability cells.
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Jul 13, 2026eess.SY
Finite Reliability Representations (FRR) certify when a cell-constant policy is sufficient for reliable decision-making in a partially observed system with a known physical noise floor. In practice, however, sensing and execution noise can be latent and context-dependent. This paper develops a certificate-aware active disambiguation framework for an unknown physical noise parameter theta = (sigma_y, sigma_u), with the sensor-only case obtained by fixing sigma_u. We define the Value of Noise Information (VoNI) as the expected excess FRR certificate gap caused by using a reliability cover calibrated to the current estimate rather than to the realized noise parameter. We bound VoNI using action-value model mismatch and FRR radius inflation, showing that noise estimation has low decision value in sub-crossover regimes where the FRR certificate is insensitive to theta, but becomes valuable when posterior uncertainty can invalidate the current cover. A bi-level decision maker uses a posterior over theta, obtained from innovation statistics, execution residuals, or another online estimator, and triggers diagnostic probing only when uncertainty threatens the FRR certificate. We also interpret VoNI as a tractable, certificate-aware approximation to a high-level finite POMDP for latent sensing-execution regime disambiguation. Under stationary, identifiable, and persistently exciting regimes, we establish posterior consistency and convergence of the induced policy loss to the FRR approximation floor. Closed-loop UGV simulations with EKF-based innovation residuals show earlier detection of abrupt sensing-noise jumps, lower drift-tracking error, and substantially fewer probing actions than posterior-entropy exploration over 50 Monte Carlo trials.
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