  Many important stochastic counting models can be written as general
birth-death processes (BDPs). BDPs are continuous-time Markov chains on the
non-negative integers and can be used to easily parameterize a rich variety of
probability distributions. Although the theoretical properties of general BDPs
are well understood, traditionally statistical work on BDPs has been limited to
the simple linear (Kendall) process, which arises in ecology and evolutionary
applications. Aside from a few simple cases, it remains impossible to find
analytic expressions for the likelihood of a discretely-observed BDP, and
computational difficulties have hindered development of tools for statistical
inference. But the gap between BDP theory and practical methods for estimation
has narrowed in recent years. There are now robust methods for evaluating
likelihoods for realizations of BDPs: finite-time transition, first passage,
equilibrium probabilities, and distributions of summary statistics that arise
commonly in applications. Recent work has also exploited the connection between
continuously- and discretely-observed BDPs to derive EM algorithms for maximum
likelihood estimation. Likelihood-based inference for previously intractable
BDPs is much easier than previously thought and regression approaches analogous
to Poisson regression are straightforward to derive. In this review, we outline
the basic mathematical theory for BDPs and demonstrate new tools for
statistical inference using data from BDPs. We give six examples of BDPs and
derive EM algorithms to fit their parameters by maximum likelihood. We show how
to compute the distribution of integral summary statistics and give an example
application to the total cost of an epidemic. Finally, we suggest future
directions for innovation in this important class of stochastic processes.
