UCLA

Melding of information from observed data, computer simulations, and scientifically-driven mechanistic models is evolving into standard practice in diverse disciplines. This dissertation presents applications and methodology for probabilistic modeling, learning, and inference in this environment. Benefits from modeling with additional scientific and domain-specific information are numerous, as methods can be tailored to answer more focused questions, test hypotheses, and make improved forecasts.

Our first application demonstrates the utility of modeling early Covid-19 dynamics in New York City with a hierarchical Bayesian structural model. Parameter calibration is achieved using multiple data streams, consisting of cell phone movement data and disease case counts over time. The parameters of the model have specific scientific importance, and this enables both improved process understanding and more accurate forecasting based on limited data. We demonstrate with out-of-sample forecasting and sensitivity analyses.

Though qualitatively and quantitatively desirable, not all scientifically-inspired models and simulations are computationally tenable for statistical inference. This is inspiration for the later chapters of the dissertation. As large scale data becomes more abundant, new methodologies and algorithms are necessary to make feasible the data fusion process of combining expensive phenomenological models and statistical or machine learning methods. This is particularly important for costly spatiotemporal simulations that arise across the physical sciences. In the spatiotemporal setting, phenomena often evolve in time and space that follow complex dynamics and require extensive computational experimentation and simulation to adequately model. These expensive spatiotemporal computer models can thus be difficult to calibrate to real-world data. This motivates our methodological development towards learning and calibrating expensive spatiotemporal computer models. We make use of state-space methodology and multiple Gaussian stochastic processes to build an efficient statistical emulator to replace the expensive computational model. The fast emulator is then used to calibrate to observed data. This model structure facilitates efficient recursive computing and sampling of parameters to provide full uncertainty quantification. We develop these inferential algorithms to make use of parallel computing and reduced rank Gaussian spatial processes for scalability to large datasets.

In our applications, we show the methodology can learn the form of complicated dynamics arising from systems of ordinary and partial nonlinear differential equations, as well as computational models with no algebraic form. This provides a black-box learning approach for the applied researcher when modeling data with expensive simulations. We hope this continues to advance research towards developing frameworks for mixing statistical, mechanistic, and computational simulations for modeling across the sciences and beyond.