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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Probabilistic Graphical Models

Background:

  • Loopy belief propagation (LBP) is an approximate inference algorithm for graphical models with cycles.
  • Parameter learning aims to adjust model parameters to improve LBP's accuracy.
  • Previous work suggested all locally consistent marginals are reachable via LBP.

Purpose of the Study:

  • To investigate the limitations of learning algorithms for belief propagation.
  • To identify conditions under which belief propagation cannot reach target marginals.
  • To propose a method for achieving otherwise unreachable marginals.

Main Methods:

  • Analyzing the Hessian of the Bethe free energy.
  • Identifying conditions for positive-definiteness.
  • Developing a parameter perturbation and averaging technique.

Main Results:

  • Demonstrated that many probability distributions yield 'unbelievable' marginals unreachable by LBP.
  • Showed this occurs when the Bethe free energy Hessian is not positive-definite.
  • Proved that standard learning algorithms fail in these cases, potentially worsening approximations.

Conclusions:

  • Standard learning algorithms for belief propagation are fundamentally limited.
  • Unbelievable marginals exist and cannot be reached by direct LBP parameter learning.
  • Averaging perturbed beliefs can successfully approximate unbelievable marginals.