A new algorithm for performing classification with imperfectly labeled data is presented. The proposed approach is motivated by the insight that the average prediction of a group of sufficiently informed people is often more accurate than the prediction of any one supposed expert. This idea that the "wisdom of crowds" can outperform a single expert is implemented by drawing sets of labels as samples from a Bernoulli distribution with a specified labeling error rate. Additionally, ideas from multiple imputation are exploited to provide a principled way for determining an appropriate number of label sampling rounds to consider. The approach is demonstrated in the context of an underwater mine classification application on real synthetic aperture sonar data collected at sea, with promising results.