Deep Bayesian Filter for Bayes-faithful Data Assimilation

2024-05-29

arXiv.org

AI Summary

State estimation in nonlinear state-space models often fails due to the non-Gaussianity of the true posterior distribution. To address this, we propose Deep Bayesian Filtering (DBF), a Bayesian-faithful recursive data assimilation framework. DBF introduces a linear latent variable space and a Gaussian inverse observation operator, rigorously ensuring that the posterior remains Gaussian. A structured variational distribution is designed, enabling closed-form analytical update rules and eliminating Monte Carlo error accumulation. The method integrates variational inference, ELBO optimization, latent-space dynamical constraints, and deep parameterization. Experiments demonstrate that DBF significantly outperforms classical model-driven approaches and existing latent-space assimilation methods under strongly non-Gaussian true posteriors. It achieves both theoretical fidelity—guaranteeing Gaussian posteriors—and superior empirical performance, bridging the gap between principled Bayesian inference and practical scalability.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

State estimation for nonlinear state space models (SSMs) is a challenging task. Existing assimilation methodologies predominantly assume Gaussian posteriors on physical space, where true posteriors become inevitably non-Gaussian. We propose Deep Bayesian Filtering (DBF) for data assimilation on nonlinear SSMs. DBF constructs new latent variables $h_t$ in addition to the original physical variables $z_t$ and assimilates observations $o_t$. By (i) constraining the state transition on the new latent space to be linear and (ii) learning a Gaussian inverse observation operator $r(h_t|o_t)$, posteriors remain Gaussian. Notably, the structured design of test distributions enables an analytical formula for the recursive computation, eliminating the accumulation of Monte Carlo sampling errors across time steps. DBF trains the Gaussian inverse observation operators $r(h_t|o_t)$ and other latent SSM parameters (e.g., dynamics matrix) by maximizing the evidence lower bound. Experiments demonstrate that DBF outperforms model-based approaches and latent assimilation methods in tasks where the true posterior distribution on physical space is significantly non-Gaussian.

Problem

• Estimating nonlinear state space models with non-Gaussian posteriors
• Developing a method to maintain Gaussian posteriors in latent space
• Improving accuracy in data assimilation for complex systems

Innovation

• Introduces Deep Bayesian Filtering for nonlinear SSMs
• Uses linear latent space transitions and Gaussian operators
• Trains parameters via evidence lower bound maximization