A Bayesian approach to stochastic root finding

A stylized model of one-dimensional stochastic root-finding involves repeatedly querying an oracle as to whether the root lies to the left or right of a given point x. The oracle answers this question, but the received answer is incorrect with probability 1 - p(x). A Bayesian-style algorithm for this problem that assumes knowledge of p(·) repeatedly updates a density giving, in some sense, one's belief about the location of the root. We demonstrate how the algorithm works, and provide some results that shed light on its performance, both when p(·) is constant and when p(·) varies with x.