Stochastic programming and robust optimization are disciplines concerned with optimal decision-making under uncertainty over time. Traditional models and solution algorithms have been tailored to problems where the order in which the uncertainties unfold is independent of the controller actions. Nevertheless, in numerous real-world decision problems, the time of information discovery can be influenced by the decision maker, and uncertainties only become observable following an (often costly) investment. Such problems can be formulated as mixed-binary multi-stage stochastic programs with decision-dependent non-anticipativity constraints. Unfortunately, these problems are severely computationally intractable. We propose an approximation scheme for multi-stage problems with decision-dependent information discovery which is based on techniques commonly used in modern robust optimization. In particular, we obtain a conservative approximation in the form of a mixed-binary linear program by restricting the spaces of measurable binary and real-valued decision rules to those that are representable as piecewise constant and linear functions of the uncertain parameters, respectively. We assess our approach on a problem of infrastructure and production planning in offshore oil fields from the literature.