  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 $f_4$ 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.
