Bayesian Mechanistic Modeling
This thesis focuses on developing Bayesian mechanistic models that can provide a fundamental tool for investigating complex systems in science and engineering. This is based on the philosophy that to be truly successful in the scientific context, we need to shift from the usual \tabula rasa" approach of only relying on the data and instead incorporate more realistic aspects of complex phenomena to obtain interpretable and meaningful models. More specifically, we are interested in modeling approaches that incorporate relevant mechanisms provided by prior domain knowledge and domain-inspired dynamic models. To this end, we use mechanistic models embedded within a hierarchical Bayesian framework. Hierarchical Bayesian models provide a natural framework for integrating information (as well as priorknowledge) available at different scales of complex systems. Mechanistic models, on the other hand, provide a flexible framework for modeling heterogeneous and dynamic systems in ways that enable prediction and control. We apply our proposed approach to two complex systems: (1) hematopoiesis and (2) infectious diseases. Hematopoiesis is a complex biological process responsible for the production of new blood cells. To model this process and design proper experiments to investigate it, we propose a hierarchical Bayesian framework, which is then used to develop a novel Bayesian optimal experimental design approach. This work helps experimentalists to properly allocate valuable resources by finding the optimal experiment that leads to maximum information gain. The proposed approach could be computationally intensive. To alleviate this issue, we have developed a novel approach that bridges the gap between Markov Chain Monte Carlo (MCMC) and Variational Bayes (VB) to find fast, yet accurate approximation of the posterior distribution. For the second application of Bayesian mechanistic models, we develop a framework that generalizes the classic time-series Susceptible-Infectious-Removed (SIR) model, which is formulated assuming homogeneous mixing (also known as the law of mass action assumption). Our generalization provides the required rigorous statistical framework to account for nonhomogeneous mixing in a population of susceptible individuals exposed to an infectious agent. In particular, we propose the use of stochastic SIR models with time-varying infection rates for the estimation of mixing and recovery rates. Using synthetic and historic time series records, we show that our method is computationally tractable, and it provides less variable estimates about the model parameters.