The planted directed polymer: inferring a random walk from noisy images

Abstract

We introduce and study the planted directed polymer, in which the path of a random walker is inferred from noisy 'images' accumulated at each timestep. Formulated as a nonlinear problem of Bayesian inference for a hidden Markov model, this problem is a generalization of the directed polymer problem of statistical physics, coinciding with it in the limit of zero signal to noise. For a 1D walker we present numerical investigations and analytical arguments that no phase transition is present. When formulated on a Cayley tree, methods developed for the directed polymer are used to show that there is a transition with decreasing signal to noise where effective inference becomes impossible, meaning that the average fractional overlap between the inferred and true paths falls from one to zero.

13 pages, 13 figures

References in corpus (5)

• Measurement-Induced Phase Transitions in the Dynamics of Entanglement
• Measurement-driven entanglement transition in hybrid quantum circuits
• Statistical physics of inference: Thresholds and algorithms
• An exact solution for the KPZ equation with flat initial conditions
• The KPZ equation with flat initial condition and the directed polymer with one free end

Cited by in corpus (1)

• Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer