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Routers have the ability to output statistics about packets and flows of packets that traverse them. Since, however, the generation of detailed traffic statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this paper, we study both theoretically and practically what information about the original traffic can be inferred when sampling, or "thinning", is performed at the packet level. While basic packet level characteristics such as first order statistics can be fairly directly recovered, other aspects require more attention. We focus mainly on the spectral density, a second-order statistic, and the distribution of the number of packets per flow, showing how both can be exactly recovered, in theory. We then show in detail why in practice this cannot be done using the traditional packet based sampling, even for high sampling rate. We introduce an alternative flow-based thinning, where practical inversion is possible even at arbitrarily low sampling rate. We also investigate the theory and practice of fitting the parameters of a Poisson cluster process, modeling the full packet traffic, from sampled data.