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Gene gain-loss-duplication models are commonly based on continuous-timebirth-death processes. Employed in a phylogenetic context, such models havebeen increasingly popular in studies of gene content evolution across multiplegenomes. While the applications are becoming more varied and demanding,bioinformatics methods for probabilistic inference on copy numbers (orinteger-valued evolutionary characters, in general) are scarce. We describe aflexible probabilistic framework for phylogenetic gene-loss-duplication models.The framework is based on a novel elementary representation by dependent randomvariables with well-characterized conditional distributions: binomial, P\'olya(negative binomial), and Poisson. The corresponding graphical model yieldsexact numerical procedures for computing the likelihood and the posteriordistribution of ancestral copy numbers. The resulting algorithms take quadratictime in the total number of copies. In addition, we show how the likelihoodgradient can be computed by a linear-time algorithm.

Gain-loss-duplication models on a phylogeny: exact algorithms for computing the likelihood and its gradient