Title: Bayesian Parameter Estimation and Inference Across Scales
Student: Margaret Callahan
Abstract: In the analysis of complex phenomena arising in biology, medicine, physics, economics or the social sciences, it is not uncommon to employ mathematical models at different scales. Microscopic models tend to be better suited to capture the fine-scale details of a complex system and are typically stochastic in nature, while macroscopic models, sometimes arising as mean-field approximations to microscopic models, are typically used to describe the overall, population-level dynamics of the system. Typically, both coarse- and fine-scale models depend on parameters whose values are unknown or poorly known, hence need to be estimated. The estimation of the parameters of larger scale models tends to be more straightforward and approachable with standard optimization-based tools. In several important applications, the microscopic model parameters are of greater interest and importance, as they may be interpreted as characterizing the generative process underlying the large-scale model. Due to the stochastic nature of these types of models, the unknown parameters are difficult, if not impossible, to estimate directly. Ideally, we would like to estimate the parameters of the coarse-scale model and relate them to those of the fine-scale model. This connection, however, may be all but trivial to establish, due to the fact that models at different scales may depend on parameters that do not exhibit a one-to-one correspondence. In this work, we propose a Bayesian approach for inferring on the values of the microscopic model parameters, based on estimates of the parameters of the associated macroscopic model. We illustrate the viability of our technique with several computed examples, ranging from simple to fairly complex, with a focus on applications in the life sciences.