Probabilistic Bisection with Spatial Metamodels
This work addresses a specific problem in stochastic root finding for applications like option pricing, representing an incremental improvement over earlier generalized PBA models.
The paper tackled the problem of root finding with noisy, location-dependent oracles by proposing a Spatial PBA algorithm that uses non-parametric surrogates and adaptive sampling policies, resulting in gains demonstrated through synthetic examples and a challenging option pricing problem.
Probabilistic Bisection Algorithm performs root finding based on knowledge acquired from noisy oracle responses. We consider the generalized PBA setting (G-PBA) where the statistical distribution of the oracle is unknown and location-dependent, so that model inference and Bayesian knowledge updating must be performed simultaneously. To this end, we propose to leverage the spatial structure of a typical oracle by constructing a statistical surrogate for the underlying logistic regression step. We investigate several non-parametric surrogates, including Binomial Gaussian Processes (B-GP), Polynomial, Kernel, and Spline Logistic Regression. In parallel, we develop sampling policies that adaptively balance learning the oracle distribution and learning the root. One of our proposals mimics active learning with B-GPs and provides a novel look-ahead predictive variance formula. The resulting gains of our Spatial PBA algorithm relative to earlier G-PBA models are illustrated with synthetic examples and a challenging stochastic root finding problem from Bermudan option pricing.