We consider a robust version of the full information best choice

problem (Gilbert and Mosteller (1966)): there is ambiguity (represented

by a set of priors) about the measure driving the observed

process. We solve the problem under a very general class of multiple

priors in the setting of Riedel (2009). As in the classical case, it is

optimal to stop if the current observation is a running maximum that

exceeds certain thresholds. We characterize the decreasing sequence

of thresholds, as well as the (history dependent) minimizing measure.

We introduce locally constant ambiguity neighborhood (LCAn) which

has connections to coherent risk measures. Sensitivity analysis is performed

using LCAn and exponential neighborhood from Riedel (2009).