Planted directed polymer: Inferring a random walk from noisy images

Area of Science:

• Statistical Physics
• Machine Learning
• Information Theory

Background:

• The directed polymer model is a fundamental concept in statistical physics.
• Bayesian inference is crucial for reconstructing hidden states from noisy observations.
• Hidden Markov models provide a framework for sequential data analysis.

Purpose of the Study:

• To introduce and analyze the planted directed polymer model.
• To investigate the behavior of random walkers inferred from noisy image data.
• To explore phase transitions in Bayesian inference for hidden Markov models.

Main Methods:

• Formulation as a nonlinear Bayesian inference problem.
• Generalization of the directed polymer problem in statistical physics.
• Numerical investigations and analytical arguments for one-dimensional walkers.
• Application of directed polymer methods to Cayley tree structures.

Main Results:

• No phase transition was observed for a one-dimensional walker.
• A phase transition was identified on a Cayley tree with decreasing signal-to-noise ratio.
• Inference becomes impossible at the transition, with path overlap dropping from one to zero.

Conclusions:

• The planted directed polymer model offers new insights into hidden state inference.
• The presence or absence of phase transitions depends on the underlying structure (1D vs. Cayley tree).
• Decreasing signal-to-noise ratio critically impacts the reliability of path inference.