  Gene gain-loss-duplication models are commonly based on continuous-time
birth-death processes. Employed in a phylogenetic context, such models have
been increasingly popular in studies of gene content evolution across multiple
genomes. While the applications are becoming more varied and demanding,
bioinformatics methods for probabilistic inference on copy numbers (or
integer-valued evolutionary characters, in general) are scarce. We describe a
flexible probabilistic framework for phylogenetic gene-loss-duplication models.
The framework is based on a novel elementary representation by dependent random
variables with well-characterized conditional distributions: binomial, P\'olya
(negative binomial), and Poisson. The corresponding graphical model yields
exact numerical procedures for computing the likelihood and the posterior
distribution of ancestral copy numbers. The resulting algorithms take quadratic
time in the total number of copies. In addition, we show how the likelihood
gradient can be computed by a linear-time algorithm.
