arXiv subjects

Fabián González

Publications and source records attributed to Fabián González.

3 recordsLinked to original sources

arXiv2026

An Operator-Theoretic Analysis of Nonlinear Filtering under Model Misspecification

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.

math.ST↗

arXiv2026

Importance sampling for Bayesian inference: polynomial-dimension dependent error bounds

Many Bayesian inference problems involve high-dimensional models where the performance of standard importance sampling (IS) methods often degrades rapidly as the dimensionality increases. Classical analyses of IS typically rely on the assumption that observations are arbitrary but fixed (i.e., deterministic), thereby neglecting the probabilistic structure that the Bayesian model induces on the data. In this paper, we adopt the perspective that observations are themselves random variables whose distribution is governed by the underlying model. Within this probabilistic framework, we identify a model-dependent function, referred to as the link function, which connects the fixed- and random-observation formulations. We provide a characterization of the $L^2$ Monte Carlo estimation error: specifically, we show that the $L^2$ error bounds are finite and converge at the standard Monte Carlo rate $O(N^{-1/2})$, for arbitrarily large dimension, if and only if the link function is Bochner integrable. This result reveals the fundamental quantity controlling the approximation error and establishes a mechanism to manage the dependence on the model state dimension. Consequently, our approach provides a principled way to alleviate the challenges of high dimensionality, offering insights that transcend worst-case analyses dominant in the existing literature. Finally, we derive explicit analytical examples of the dimensional scaling of the associated errors for several model classes, including linear-Gaussian systems and models with bounded observation functions.

math.ST↗

arXiv2025

Nested importance sampling for Bayesian inference: error bounds and the role of dimension

Many Bayesian inference problems involve high dimensional models for which only a subset of the model variables are of actual interest. All other variables are just nuisance parameters that one would ideally like to integrate out analytically. Unfortunately, such integration is often impossible. There are several computational methods that have been proposed over the past 15 years that replace intractable analytical marginalization by numerical integration, typically using different flavours of importance sampling (IS). Such methods include particle Markov chain Monte Carlo, sequential Monte Carlo squared (SMC$^2$), IS$^2$, nested particle filters, and others. In this paper, we investigate the role of the dimension of the nuisance variables in the error bounds achieved by nested IS methods in Bayesian inference. We prove that, under suitable regularity assumptions on the model, the approximation errors increase at a polynomial (rather than exponential) rate w.r.t. the dimension of the nuisance variables. Our analysis relies on tools from functional analysis and measure theory, and it includes the case of polynomials of degree zero, where the approximation error remains uniformly bounded, even as the dimension of the nuisance variables grows without bound. We also show how the general analysis can be applied to specific classes of models, including linear and Gaussian settings, models with bounded observation functions, and others. These findings improve the current understanding of how the performance of IS techniques scales with dimension in Bayesian inference problems.

stat.CO↗