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Planning to avoid ambiguous states through Gaussian approximations to non-linear sensors in active inference agents

Wouter M. Kouw

TL;DR

It is shown that Gaussian approximations that are sensitive to the curvature of the measurement function, such as a second-order Taylor approximation, produce a state-dependent ambiguity term that induces a preference over states, based on how accurately the state can be inferred from the observation.

Abstract

In nature, active inference agents must learn how observations of the world represent the state of the agent. In engineering, the physics behind sensors is often known reasonably accurately and measurement functions can be incorporated into generative models. When a measurement function is non-linear, the transformed variable is typically approximated with a Gaussian distribution to ensure tractable inference. We show that Gaussian approximations that are sensitive to the curvature of the measurement function, such as a second-order Taylor approximation, produce a state-dependent ambiguity term. This induces a preference over states, based on how accurately the state can be inferred from the observation. We demonstrate this preference with a robot navigation experiment where agents plan trajectories.

Planning to avoid ambiguous states through Gaussian approximations to non-linear sensors in active inference agents

TL;DR

It is shown that Gaussian approximations that are sensitive to the curvature of the measurement function, such as a second-order Taylor approximation, produce a state-dependent ambiguity term that induces a preference over states, based on how accurately the state can be inferred from the observation.

Abstract

In nature, active inference agents must learn how observations of the world represent the state of the agent. In engineering, the physics behind sensors is often known reasonably accurately and measurement functions can be incorporated into generative models. When a measurement function is non-linear, the transformed variable is typically approximated with a Gaussian distribution to ensure tractable inference. We show that Gaussian approximations that are sensitive to the curvature of the measurement function, such as a second-order Taylor approximation, produce a state-dependent ambiguity term. This induces a preference over states, based on how accurately the state can be inferred from the observation. We demonstrate this preference with a robot navigation experiment where agents plan trajectories.
Paper Structure (17 sections, 4 theorems, 43 equations, 3 figures)

This paper contains 17 sections, 4 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Example
  4. Agent specification
  5. Probabilistic Model
  6. Inferring states
  7. Inferring controls
  8. Gaussian approximations
  9. Experiments
  10. Discussion
  11. Conclusion
  12. Acknowledgments.
  13. Disclosure of Interests.
  14. Appendix: proof of Lemma 1
  15. Appendix: proof of Theorem 1
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

Ambiguity, as defined in Eq. eq:EFE-ambrisk, for a generative model described in Eq. eq:pmodel and a variational distribution described in Eq. eq:qmodel, is:

Figures (3)

  • Figure 1: (Left) Planned trajectory at $k=1$, from start to goal directly over the sensor station. (Middle) Planned trajectory at $k=5$ showing a mismatch between true and estimated state resulting in a strong adjustment to the planned trajectory. (Right) Executed trajectory over a trial of $10$ steps demonstrates the agent losing track of the robot when it approaches the sensor station.
  • Figure 2: Value under three EFE functions over a plane: EFE1 is risk and ambiguity under a first-order Taylor approximation, EFER is risk only under a second-order Taylor approximation and EFE2 is both risk and ambiguity under a second-order Taylor approximation. White markers indicate minimizers. Note that each EFE function induces a different preference over states.
  • Figure 3: Trajectories of agents under three EFE functions, averaged over 100 Monte Carlo samples (ribbon is standard deviation of the mean). The robot starts at the green marker and must reach the red goal marker. All agents avoid the sensor station, with EFE2 taking the widest curve and having the smoothest average trajectory.

Theorems & Definitions (8)

  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • proof
  • proof
  • proof