We consider the task of fine-grained sentiment analysis from the perspective of multiple instance learning (MIL). Our neural model is trained on document sentiment labels, and learns to predict the sentiment of text segments, i.e. sentences or elementary discourse units (EDUs), without segment-level supervision. We introduce an attention-based polarity scoring method for identifying positive and negative text snippets and a new dataset which we call SpoT (as shorthand for Segment-level POlariTy annotations) for evaluating MIL-style sentiment models like ours. Experimental results demonstrate superior performance against multiple baselines, whereas a judgement elicitation study shows that EDU-level opinion extraction produces more informative summaries than sentence-based alternatives.