To learn the underlying parent-child influence relationships between nodes in a diffusion network, most existing approaches require timestamps that pinpoint the exact time when node infections occur in historical diffusion processes. In many real-world diffusion processes like the spread of epidemics, monitoring such infection temporal information is often expensive and difficult. In this work, we study how to carry out diffusion network inference without infection timestamps, using only the final infection statuses of nodes in each historical diffusion process, which are more readily accessible in practice. Our main result is a probabilistic model that can find for each node an appropriate number of most probable parent nodes, who are most likely to have generated the historical infection results of the node. Extensive experiments on both synthetic and real-world networks are conducted, and the results verify the effectiveness and efficiency of our approach.