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

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.