We present MULTIP (Multi-instance Learning Paraphrase Model), a new model suited to identify paraphrases within the short messages on Twitter. We jointly model paraphrase relations between word and sentence pairs and assume only sentence-level annotations during learning. Using this principled latent variable model alone, we achieve the performance competitive with a state-of-the-art method which combines a latent space model with a feature-based supervised classifier. Our model also captures lexically divergent paraphrases that differ from yet complement previous methods; combining our model with previous work significantly outperforms the state-of-the-art. In addition, we present a novel annotation methodology that has allowed us to crowdsource a paraphrase corpus from Twitter. We make this new dataset available to the research community.