Delay Tolerant Networks are wireless networks where disconnections may occur frequently due to propagation phenomena, node mobility, and power outages. Propagation delays may also be long due to the operational environment (e.g. deep space, underwater). In order to achieve data delivery in such challenging networking environments, researchers have proposed the use of \emph{store-carry-and-forward} protocols: there, a node may store a message in its buffer and carry it along for long periods of time, until an appropriate forwarding opportunity arises. Additionally, multiple message replicas are often propagated to increase delivery probability. This combination of long-term storage and replication imposes a high storage overhead on untethered nodes (e.g. handhelds). Thus, efficient buffer management policies are necessary to decide which messages should be discarded, when node buffers are operated close to their capacity. In this paper, we propose efficient buffer management policies for delay tolerant networks. We show that traditional buffer management drop policies like drop-tail or drop-front fail to consider all relevant information in this context and are, thus, sub-optimal. Using the theory of encounter-based message dissemination, we propose an optimal buffer management policy based on global knowledge about the network. Our policy can be tuned either to minimize the average delivery delay or to maximize the average delivery ratio. Finally, we introduce a distributed algorithm that uses statistical learning to approximate the required global knowledge of the optimal algorithm, in practice. Using simulations based on a synthetic mobility model and real mobility traces, we show that our buffer management policy based on statistical learning successfully approximates the performance of the optimal policy in all considered scenarios. At the same time, our policies outperform existing ones in terms of both average delivery ratio and delivery delay.