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In this dissertation, we explore the effect of variable evolutionary rates on phylogenetic inference. In the first half of the thesis are introduced the biological fundamentals and the statistical framework that will be used throughout the thesis. The basic concepts in phylogenetics and an overview of Bayesian inference are presented in Chapter 1. In Chapter 2, we survey the models that are already used for rate variation. We argue that the CIR process---a diffusion process widely used in finance---is the best suited for applications in phylogenetics, for both mathematical and computational reasons. Chapter 3 shows how evolutionary rate models are incorporated to DNA substitution models. We derive the general formulae for transition probabilities of substitutions when the rate is a continuous-time Markov chain, a diffusion process or a jump process (a diffusion process with discrete jumps). / The second half of the thesis is dedicated to applications of variable evolutionary rate models in two different contexts. In Chapter 4, we use the CIR process to model heterotachy, an evolutionary hypothesis according to which positions of an alignment may evolve at rates that vary with time differently from site to site. A comparison the CIR process with the covarion---a widely-used heterotachous model---on two different data sets allows us to conclude that the CIR provides a significantly better fit. Our approach, based on a Bayesian mixture model, enables us to determine the level of heterotachy at each site. Finally, the impact of variable evolutionary rates on divergence time estimation is explored in Chapter 5. / Several models, including the CIR process are compared on three data sets. We find that autocorrelated models (including the CIR) provide the best fits.

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