Here, we implement ancestral reconstruction of discrete states in a Bayesian statistical framework for evolutionary hypothesis testing that is geared towards rooted, time-measured phylogenies. This allows character mapping in natural time scales, calibrated under a strict or relaxed molecular clock, in combination with several models of population size change. We use this full probabilistic approach to study viral phylogeography and extend the Bayesian implementation to a mixture model in which exchange rates in the Markov model are allowed to be zero with some probability. This Bayesian stochastic search variable selection (BSSVS) enables us to construct a Bayes factor test that identifies the most parsimonious description of the phylogeographic diffusion process. We also demonstrate how the geographical distribution of the sampling locations can be incorporated as prior specifications. Through feature-rich visual summaries of the space-time process, we demonstrate how this approach can offer insights into the spatial epidemic history of Avian influenza A-H5N1 and rabies viruses in Africa.