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Reframing the Expected Free Energy: Four Formulations and a Unification.

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Summary

Active inference unifies expected free energy formulations by analyzing two definitions. This research clarifies mathematical justifications and constraints on prior preferences in partially observable Markov decision processes (POMDPs).

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Area of Science:

  • Computational Neuroscience
  • Machine Learning
  • Robotics
  • Psychology

Background:

  • Active inference is a unifying theory for perception, learning, and decision-making across multiple scientific disciplines.
  • It relies on expected free energy, with existing formulations like risk/ambiguity and information gain lacking a unified derivation.
  • The free energy principle provides a foundational framework for understanding these processes in biological and artificial systems.

Purpose of the Study:

  • To formalize the unification problem of deriving various expected free energy formulations from a single root definition.
  • To analyze two distinct approaches for defining expected free energy and their mathematical justifications.
  • To investigate the implications of likelihood constraints on prior preferences within partially observable Markov decision processes (POMDPs).

Main Methods:

  • Analysis of two primary definitions of expected free energy: risk over observations plus ambiguity, and risk over states plus ambiguity.
  • Mathematical examination of the conditions under which different expected free energy formulations can be recovered.
  • Demonstration of how likelihood constraints impact the selection of prior preferences over observations in POMDPs.

Main Results:

  • The first definition (risk over observations plus ambiguity) allows recovery of all formulations under a specific likelihood assumption but restricts prior preferences in POMDPs.
  • The second definition (risk over states plus ambiguity) offers a justification but only accounts for two specific formulations (risk over states plus ambiguity and entropy plus expected energy).
  • A limited class of prior preferences over observations is compatible with the generative model's likelihood mapping in POMDPs.

Conclusions:

  • Unifying expected free energy formulations requires careful consideration of the chosen definition and its underlying assumptions.
  • Likelihood constraints play a critical role in determining valid prior preferences, particularly in complex systems like POMDPs.
  • Further research is needed to fully reconcile different expected free energy formulations within the broader free energy principle, especially for systems without random fluctuations.