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

arXiv: https://arxiv.org/abs/2607.04019 · HTML: https://mlsanity.com/paper/2607.04019/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 22 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Builds on (most cited first, 1 of 1)

- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)

## Abstract (arXiv)

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.