  Many experiments are concerned with the comparison of counts between
treatment groups. Examples include the number of successful signups in
conversion rate experiments, or the number of errors produced by software
versions in canary experiments. Observations typically arrive in data streams
and practitioners wish to continuously monitor their experiments, sequentially
testing hypotheses while maintaining Type I error probabilities under optional
stopping and continuation. These goals are frequently complicated in practice
by non-stationary time dynamics. We provide practical solutions through
sequential tests of multinomial hypotheses, hypotheses about many inhomogeneous
Bernoulli processes and hypotheses about many time-inhomogeneous Poisson
counting processes. For estimation, we further provide confidence sequences for
multinomial probability vectors, all contrasts among probabilities of
inhomogeneous Bernoulli processes and all contrasts among intensities of
time-inhomogeneous Poisson counting processes. Together, these provide an
"anytime-valid" inference framework for a wide variety of experiments dealing
with count outcomes, which we illustrate with a number of industry
applications.
