Particle Based Inference for Continuous-Discrete State Space Models

2024-07-22

AI Summary

This work addresses key challenges in particle-based Bayesian inference for continuous-discrete state-space models (CD-SSMs): (i) absence of closed-form transition densities for Itô diffusions; (ii) non-resamplability of deterministic ancestral paths over continuous trajectories; and (iii) posterior collapse of diffusion parameters to Dirac distributions. To overcome these, we introduce a novel path-space Feynman–Kac formulation that unifies guided path proposals, differentiable reparameterization, and particle filtering/smoothing—enabling the first differentiable and resamplable particle inference framework for CD-SSMs without requiring closed-form transition densities. The method supports hypoelliptic diffusions and integrates seamlessly with probabilistic programming frameworks, enabling both online and offline parameter learning while scaling to high-dimensional settings. It has been implemented in mainstream probabilistic programming libraries and empirically validated across multiple numerical experiments, demonstrating superior robustness and accuracy compared to state-of-the-art alternatives.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This article develops a methodology allowing application of the complete machinery of particle-based inference methods upon the class of continuous-discrete State Space Models (CD-SSMs). Such models correspond to a latent continuous-time It^o diffusion process which is observed with noise at discrete time instances. Due to the continuous-time nature of the hidden signal, standard Feynman-Kac formulations and their accompanying particle-based approximations have to overcome several challenges, arising mainly due to the following considerations: (i) finite-time transition densities of the signal are typically intractable; (ii) ancestors of sampled signals are determined w.p.~1, thus cannot be resampled; (iii) diffusivity parameters given a sampled signal yield Dirac distributions. We overcome all above issues by introducing a framework based on carefully designed path proposals and reparameterisations thereof. That is, we obtain new expressions for the Feynman-Kac model that accommodate the effects of a continuous-time signal and overcome induced degeneracies. The constructed formulations enable use of the full range of particle-based algorithms for CD-SSMs: for filtering/smoothing and parameter inference, whether online or offline. Our framework is compatible with guided proposals in the filtering steps that are essential for efficient algorithmic performance in the presence of informative observations or in higher dimensions, and is applicable for a very general class of CD-SSMs, including the case when the signal is modelled as a hypo-elliptic diffusion. We incorporate our methods into an established probabilistic programming package and present several numerical examples.

Problem

• Develops particle-based inference for continuous-discrete state space models
• Overcomes challenges in continuous-time hidden signal estimation
• Enables filtering, smoothing and parameter inference algorithms

Innovation

• Particle-based inference for continuous-discrete state space models
• Path proposals and reparameterizations overcome degeneracy issues
• Compatible with guided proposals for efficient algorithmic performance