  Consider a branching process with a homogeneous reproduction law. Sampling a
single cell uniformly from the population at a time $T > 0$ 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. 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, and the bias's strength grows 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.
