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

Jul 16, 2026

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Physical sensing and actuation noise should determine how finely a decision-making system can resolve beliefs for reliable control. The framework introduces Finite Reliability Representations (FRR), covering reachable belief spaces with reliability cells so the optimal action-value Q*(b,u) varies by at most ε uniformly over actions. FRR avoids using noisy Bayesian updates as globally contractive by separating filter, predictive observation law, and controlled belief-transition kernel. The resulting cell-constant policies bound suboptimality by 2ε/(1−γ) and define reliability entropy as certified decision-relevant complexity.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages25

Reading Time1 h 3 min

Introduction

Motivation: physical noise floors vs designer-chosen tolerances

Background: POMDPs and existing approximation criteria

FRR contribution and core objective

Finite Reliability Representations: Noise-Calibrated Belief-Space Covers for Reliable Decision-Making Hyung-Jin Yoon† and Hunmin Kim‡ arXiv :2607 .04019v1 [ ee ss . SY] 4 Jul 2026 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 ε, 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 belieftransition kernel. For nonlinear continuous-state systems, FRRis 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 2ε/(1 − γ). We also introduce reliability entropy, the logarithm of the minimal number of reliability cells, as a measure of certified decisionrelevant 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. I. INTRODUCTION Autonomous systems should not reason, plan, or control ata resolution unsupported by the physical channels on which their closed-loop decisions depend. Sensing noise limits what can be inferred from observations, while actuation and process uncertainty limit what can be reliably executed. Yet many methods for making partially observable decision problems tractable introduce a representation tolerance chosen by the designer: a grid resolution, a particle budget, a point-based approximation set, a value-function error tolerance, or a behavioral similarity threshold. These choices may be computationally reasonable, but they are not directly tied to what the deployed sensing, process-noise, and actionexecution channels can actually support. A controller may therefore preserve distinctions that the hardware cannot reliably use, or discard belief distinctions that remain critical for safety and performance. The central question is not only how to approximate a belief space, but what belief resolution is physically usable for reliable decision-making. Decision-making under partial observability is classically formalized as a partially observable Markov decision process (POMDP), in which an agent maintains a belief distribution over hidden states and selects actions to maximize expected discounted reward [1], [2] . Exact POMDP planning is generally intractable for large or continuous state and observation spaces, motivating belief discretization, valuefunction approximation, particle representations, pointbased value iteration, and reachability-guided approximation methods [3], [4], [5] . In parallel, state aggregation and bisimulation metrics group states by behavioral similarity, while information-state compression and predictive-state representations reduce the dimensionality of the sufficient statistic used for prediction or control [6], [7], [8], [9], [

FAQs

What problem does Finite Reliability Representations (FRR) address?

FRR connects belief-space approximation to the physical noise floors of deployed sensing and execution channels, ensuring that belief resolution used for decision-making is reliably supported.

How does FRR guarantee reliable decision-making from a belief-space cover?

FRR partitions the reachable belief space into reliability cells where Q*(b,u) changes by at most ε uniformly over actions, allowing a cell-constant policy whose suboptimality is bounded relative to the optimal policy.

Why does FRR avoid treating noisy Bayesian updates as globally contractive?

FRR separates the fixed-observation filter map, predictive observation law, and controlled belief-transition kernel because contractivity on arbitrary beliefs does not generally hold under noisy Bayesian updates.

Jul 16, 2026

Upload by Hazel