Inference with population genetic data usually treats the population pedigree as a nuisance parameter, the unobserved product of a past history of random mating. However, the history of genetic relationships in a given population is a fixed, unobserved object, and so an alternative approach is to treat this network of relationships as a complex object we wish to learn about, by observing how genomes have been noisily passed down through it. This paper explores this point of view, showing how to translate questions about population genetic data into calculations with a Poisson process of mutations on all ancestral genomes. This method is applied to give a robust interpretation to the f4 statistic used to identify admixture, and to design a new statistic that measures covariances in mean times to most recent common ancestor between two pairs of sequences. The method more generally interprets population genetic statistics in terms of sums of specific functions over ancestral genomes, thereby providing concrete, broadly interpretable interpretations for these statistics. This provides a method for describing demographic history without simplified demographic models. More generally, it brings into focus the population pedigree, which is averaged over in model-based demographic inference.