An Operator-Theoretic Analysis of Nonlinear Filtering under Model Misspecification
Abstract
We study nonlinear filtering with misspecified initial distributions and dynamics, but a correctly specified observation model. Observations are treated as a random process generated by the true model, rather than as fixed realizations, and both the correct filter and the misspecified filter are analyzed under this common observation law. Our main result is a Bayesian total variation balance identity for the expected change in total variation distance under a Bayesian update: a covariance term minus a non-negative remainder. The identity characterizes conditional and averaged update non-expansivity. Combined with prediction contraction, it yields finite-time bounds separating the initialization error from accumulated dynamical discrepancies, while allowing expansive updates. When the averaged updates are non-expansive and a uniform joint prediction--update coefficient is less than one, a persistent dynamical misspecification yields an asymptotic error bound proportional to the one-step discrepancy. When the conditional updates are non-expansive, we characterize an accumulated joint contraction that implies almost-sure and L1 coalescence of the true and misspecified filters. Finally, we verify update non-expansivity for scalar Gaussian predictive laws under a variance-ordering condition and, separately, derive explicit prediction contraction bounds for truncated kernels and bounded transition functions with elliptically symmetric noise.
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BibTeX
@misc{gonzalez2026operatortheoretic,
title = {{An Operator-Theoretic Analysis of Nonlinear Filtering under Model Misspecification}},
author = {Fabián González and Ömer Deniz Akyildiz and Dan Crisan and Joaquín Míguez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.11378},
eprint = {2607.11378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.11378},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - An Operator-Theoretic Analysis of Nonlinear Filtering under Model Misspecification
AU - Fabián González
AU - Ömer Deniz Akyildiz
AU - Dan Crisan
AU - Joaquín Míguez
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2607.11378
UR - https://arxiv.org/abs/2607.11378
AB - We study nonlinear filtering with misspecified initial distributions and dynamics, but a correctly specified observation model. Observations are treated as a random process generated by the true model, rather than as fixed realizations, and both the correct filter and the misspecified filter are analyzed under this common observation law. Our main result is a Bayesian total variation balance identity for the expected change in total variation distance under a Bayesian update: a covariance term minus a non-negative remainder. The identity characterizes conditional and averaged update non-expansivity. Combined with prediction contraction, it yields finite-time bounds separating the initialization error from accumulated dynamical discrepancies, while allowing expansive updates. When the averaged updates are non-expansive and a uniform joint prediction--update coefficient is less than one, a persistent dynamical misspecification yields an asymptotic error bound proportional to the one-step discrepancy. When the conditional updates are non-expansive, we characterize an accumulated joint contraction that implies almost-sure and L^1 coalescence of the true and misspecified filters. Finally, we verify update non-expansivity for scalar Gaussian predictive laws under a variance-ordering condition and, separately, derive explicit prediction contraction bounds for truncated kernels and bounded transition functions with elliptically symmetric noise.
ER -