Finite Reliability Representations: Noise-Calibrated Belief-Space Covers for Reliable Decision-Making

2026-07-04

AI Summary

This work addresses the challenge of constructing reliable and compact belief representations that support near-optimal decision-making under perceptual and actuation noise. It introduces a “reliability cell covering” approach that replaces traditional equivalence-class partitioning by defining cells in belief space wherein the optimal action-value function varies by no more than a tolerance ε. The method employs a reliability entropy measure to quantify decision-relevant belief complexity and distinguishes between representational sufficiency and performance limits imposed by noise. By leveraging a fixed observation filtering map, predictive observation laws, and a controlled belief transition kernel, the construction yields an ε-cover under Lipschitz continuity assumptions, applicable to finite POMDPs, linear-Gaussian systems, and particle-filter-based models. The resulting piecewise-constant policies achieve a suboptimality bound of 2ε/(1−γ) and admit analytical or empirical verification across diverse filtering frameworks.

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.

Problem

• belief-space representation
• decision-making under uncertainty
• sensor noise
• actuation noise
• POMDP

Innovation

• Finite Reliability Representations
• belief-space cover
• reliability entropy
• noise-calibrated decision-making
• POMDP