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Goal inference with Rao-Blackwellized Particle Filters

Yixuan Wang, Dan P. Guralnik, Warren E. Dixon

TL;DR

The paper advances intent inference for goal-directed motion by formulating a Rao-Blackwellized Particle Filter that leverages a conditionally linear–Gaussian structure to marginalize the state and sample only the intent parameters. It introduces density-based representations of intent and two estimators (complete and reduced), along with computable KL-divergence bounds that quantify information leakage to an adversary. Theoretical bounds on the difference between estimators are established, and numerical experiments in a 2D plane demonstrate rapid, robust recovery of the true goal with the reduced estimator performing nearly as well as the complete one. The work provides a principled framework for evaluating intent inference and motivates designing obfuscating controllers under KL-based leakage constraints for safer autonomy and privacy-aware systems.

Abstract

Inferring the eventual goal of a mobile agent from noisy observations of its trajectory is a fundamental estimation problem. We initiate the study of such intent inference using a variant of a Rao-Blackwellized Particle Filter (RBPF), subject to the assumption that the agent's intent manifests through closed-loop behavior with a state-of-the-art provable practical stability property. Leveraging the assumed closed-form agent dynamics, the RBPF analytically marginalizes the linear-Gaussian substructure and updates particle weights only, improving sample efficiency over a standard particle filter. Two difference estimators are introduced: a Gaussian mixture model using the RBPF weights and a reduced version confining the mixture to the effective sample. We quantify how well the adversary can recover the agent's intent using information-theoretic leakage metrics and provide computable lower bounds on the Kullback-Leibler (KL) divergence between the true intent distribution and RBPF estimates via Gaussian-mixture KL bounds. We also provide a bound on the difference in performance between the two estimators, highlighting the fact that the reduced estimator performs almost as well as the complete one. Experiments illustrate fast and accurate intent recovery for compliant agents, motivating future work on designing intent-obfuscating controllers.

Goal inference with Rao-Blackwellized Particle Filters

TL;DR

The paper advances intent inference for goal-directed motion by formulating a Rao-Blackwellized Particle Filter that leverages a conditionally linear–Gaussian structure to marginalize the state and sample only the intent parameters. It introduces density-based representations of intent and two estimators (complete and reduced), along with computable KL-divergence bounds that quantify information leakage to an adversary. Theoretical bounds on the difference between estimators are established, and numerical experiments in a 2D plane demonstrate rapid, robust recovery of the true goal with the reduced estimator performing nearly as well as the complete one. The work provides a principled framework for evaluating intent inference and motivates designing obfuscating controllers under KL-based leakage constraints for safer autonomy and privacy-aware systems.

Abstract

Inferring the eventual goal of a mobile agent from noisy observations of its trajectory is a fundamental estimation problem. We initiate the study of such intent inference using a variant of a Rao-Blackwellized Particle Filter (RBPF), subject to the assumption that the agent's intent manifests through closed-loop behavior with a state-of-the-art provable practical stability property. Leveraging the assumed closed-form agent dynamics, the RBPF analytically marginalizes the linear-Gaussian substructure and updates particle weights only, improving sample efficiency over a standard particle filter. Two difference estimators are introduced: a Gaussian mixture model using the RBPF weights and a reduced version confining the mixture to the effective sample. We quantify how well the adversary can recover the agent's intent using information-theoretic leakage metrics and provide computable lower bounds on the Kullback-Leibler (KL) divergence between the true intent distribution and RBPF estimates via Gaussian-mixture KL bounds. We also provide a bound on the difference in performance between the two estimators, highlighting the fact that the reduced estimator performs almost as well as the complete one. Experiments illustrate fast and accurate intent recovery for compliant agents, motivating future work on designing intent-obfuscating controllers.

Paper Structure

This paper contains 15 sections, 3 theorems, 38 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Intention Inference
  4. RBPF Model Development
  5. Estimation and Inference
  6. Representation of Intent by a Distribution
  7. Density-Based Estimators of Intent
  8. The Reduced Estimator
  9. Intent Estimators: Reduced vs. Complete
  10. Numerical Experiments
  11. Simulation Setup
  12. Results and Analysis
  13. Conclusion
  14. Proof of Lemma \ref{['lemma:stability of estimated controller']}
  15. Proof of Theorem \ref{['thm:bound']}

Key Result

Lemma 1

Let $\theta\in\Theta$ and let $x\in\mathds{R}^n$ evolve under Then for any $\lambda\geq\lambda_\theta$, and any $x_0\in R\mathbb{B}$, the trajectory emanating from $x(0)=x_0$ satisfies $x(t)\in x(\theta)+r(\theta)\mathbb{B}$ for all $t\geq t(\theta)$, where:

Figures (2)

  • Figure 1: Left: Hypothetical situation with three particles estimating $x(\theta{}^{{\ast}})$ with near-equal weights. The particle $A$ with weight $\tfrac{1}{3}+\varepsilon$, is a highest weight estimate of $\theta{}^{{\ast}}$ according to the PF, but it represents a false positive with high probability ($\tfrac{2}{3}-\varepsilon$). At the same time, the weighted average of the particles may not be relevant at all, since its location is not contained in any of the predicted goal regions. Right: Hypothetical situation with three effective particles with near-equal weights $\omega = 0.29$, represented by the reduced estimator (in distribution form), in Section \ref{['intention inference:estimation:effective estimator']}. Seven additional particles with negligible weights are located near the barycenter of the three effective particles. Similar to Dirac representation, the highest weight estimator remains a false positive with high probability. The position estimate from the weighted average of all particles $x(\hat{\theta}^{{\mathtt{com}}}_k)$ is distracted by the trivial weight particles, while the reduced estimator stays at the barycenter of effective particles with weight almost equal to the effective particles' weight. However, in contrast to Dirac representation, either estimator provides information about the goal distribution $q_{\theta{}^{{\ast}}}$.
  • Figure 2: RBPF intent–inference process. The upper-left panel shows the agent’s trajectory, and the upper-right panel plots the KL divergence for the complete and reduced estimators. The four lower panels display, from left to right, the effective sample size ${N_{{e\!f\!f}}}$, the position estimation error, the estimated goal radius, and the estimated arrival time.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Proposition 2: wang2025effectivemodelpruning