Latent Inferred Scale Adaptation (I.S.A) for Residual Modeling Under Nonstationary Regimes

Abstract

Residual variability in estimation problems is commonly modeled under fixed-dispersion assumptions, although practical residual sequences often exhibit nonstationary behavior and locally evolving uncertainty. This work presents a hierarchical Bayesian architecture for probabilistic residual modeling in which latent uncertainty, inferred scale adaptation, and progressive statistical validation are integrated within a unified formulation. The proposed $\mathcal {L}$ -model implements the Latent component by representing the local statistical scale governing residual dispersion through a latent precision variable, allowing uncertainty to be inferred directly from observed residuals rather than prescribed a priori. The Inferred Scale Adaptation (I.S.A) component transforms the inferred statistical scale into adaptive residual weighting through posterior precision estimation while preserving the geometric structure of conventional residual-based estimation. The Bayesian Evolution Layer (BEL) consolidates the proposed formulation through complementary synthetic analyses and experimental evaluation using EuRoC MAV trajectory sequences. Synthetic investigations examine the influence of latent precision variability on residual weighting under distinct operating regimes, whereas experimental analyses reveal temporally evolving structural parameters exhibiting statistically significant nonlinear dependence on independently derived local trajectory dynamics and associated influence profiles. Collectively, these results demonstrate that latent uncertainty representation, inferred scale adaptation, and complementary validation operate as coherent components of the proposed Latent–I.S.A-BEL Framework, providing a unified probabilistic architecture for residual characterization that remains fully compatible with conventional estimation methodologies while establishing a foundation for future estimation and inference problems involving nonstationary residual behavior.