Ancestral reproductive bias in branching processes

Consider a branching process whose reproduction law is homogeneous. Sampling a single cell uniformly from the population at a time [Formula: see text] and looking along the sampled cell's ancestral lineage, we find that the reproduction law is heterogeneous-the expected reproductive output of ancestral cells on the lineage from time 0 to time T continuously increases with time. This 'inspection paradox' is due to sampling bias, that cells with a larger number of offspring are more likely to have one of their descendants sampled by virtue of their prolificity. The bias's strength changes with the random population size and/or the sampling time T. Our main result explicitly characterises the evolution of reproduction rates and sizes along the sampled ancestral lineage as a mixture of Poisson processes, which simplifies in special cases. The ancestral bias helps to explain recently observed variation in mutation rates along lineages of the developing human embryo.