Differential privacy (DP) is a widely used notion for reasoning about privacy when publishing aggregate data. In this paper, we observe that certain DP mechanisms are amenable to a posteriori privacy analysis that exploits the fact that some outputs leak less information about the input database than others. To exploit this phenomenon, we introduce output differential privacy (ODP) and a new composition experiment, and leverage these new constructs to obtain significant privacy budget savings and improved privacy-utility tradeoffs under composition. All of this comes at no cost in terms of privacy; we do not weaken the privacy guarantee. To demonstrate the applicability of our a posteriori privacy analysis techniques, we analyze two well-known mechanisms: the Sparse Vector Technique and the Propose-Test-Release framework. We then show how our techniques can be used to save privacy budget in more general contexts: when a differentially private iterative mechanism terminates before its maximal number of iterations is reached, and when the output of a DP mechanism provides unsatisfactory utility. Examples of the former include iterative optimization algorithms, whereas examples of the latter include training a machine learning model with a large generalization error. Our techniques can be applied beyond the current paper to refine the analysis of existing DP mechanisms or guide the design of future mechanisms.