We propose a three-stage optimization model for this problem, where in the first stage some components of $\eta$ are chosen to be observed, and decisions in subsequent stages must be consistent with the obtained information. In the case that $\eta$ and $\xi$ have finite support, Goel and Grossmann gave a mixed-integer programming (MIP) formulation of this problem whose size is proportional to the square of cardinality of the sample space of the random variables.
We propose to solve the model using bounds obtained from an information-based relaxation, combined with a branching scheme that enforces the consistency of decisions with observed information. The branch-and-bound approach can naturally be combined with sampling in order to estimate both lower and upper bounds on the optimal solution value and does not require $\eta$ or $\xi$ to have finite support. We conduct a computational study of our method on instances of a stochastic facility location and sizing problem with the option to probe customers to learn about their demands before building facilities. We find that on instances with finite support, our approach scales significantly better than the MIP formulation and also demonstrate that our method can compute statistical bounds on instances with continuous distributions that improve upon the perfect information bounds.