By far the most efficient methods for global optimization are based on starting a local optimization routine from uniformly distributed starting points. As the number of local optima is frequently unknown in advance, it is then a crucial problem when to stop the sequence of sampling and searching. By viewing a set of observed local optima as a sample from a multinomial distribution whose cells correspond to the local optima of the objective function, statistical inferences can be made about the number of local optima and the relative size of their regions of attraction. This information is used to construct optimal Bayesian stopping rules for the sequential sample.