Interactions between species play an important role in the function of biological communities and have important implications for human health and agriculture. For instance, interactions between pathogens and their hosts influence the stability of natural communities and shape the spread of infectious disease within human populations. Similarly, interactions between plants and pollinators are essential for the proper functioning of natural communities and also for the efficient and economical operation of agricultural systems. For these reasons, predicting how species interactions change over time is important for understanding biological communities, managing agricultural systems, and minimizing infectious disease. Unfortunately, our ability to predict how species interactions change over time is poor, in part because the forces driving evolution of interacting species is not well understood. This research will develop new mathematical and statistical methods for estimating the strength of coevolution, a force long hypothesized to play a key role in the evolution of species interactions. The research will also provide significant opportunities for graduate student training at the interface of mathematics, statistics, computation, and biology and thus contribute to human resource development in STEM. Ultimately, the mathematical and statistical tools developed by the research will be made available to the scientific community through development of free computer software; this software will allow the strength of coevolution to be easily estimated in a broad range biological systems. Two broad classes of techniques currently exist for inferring the strength of coevolutionary selection in natural populations: (1) Direct techniques that make robust inferences but are limited to specific types of systems and small numbers of populations, and (2) indirect techniques that can be applied to a broad range of systems and large numbers of populations but which provide inferences of questionable robustness. Even under the best case scenario, both techniques generally yield only qualitative results and thus cannot provide crucial quantitative information on the intensity of coevolutionary selection in natural populations. This research will capitalize on recent advances in Bayesian statistics to develop novel methods for estimating parameters of well-established coevolutionary models using data routinely collected as part of broad scale studies of trait matching in species interactions. Once these new statistical methods have been thoroughly tested using simulated data, they will be used to estimate the strength of coevolutionary selection in a textbook example of coevolution - the interaction between toxic newts and their garter snake predators.