Increases in demand have driven the development of complex processing technology for separating mineral resources from exceedingly low grade multi- component resources. Low mineral concentrations and variable feedstocks can make separating signal from noise difficult, while high process complexity and the multi-component nature of a feedstock can make testwork, optimization, and process simulation difficult or infeasible. A prime example of such a scenario is the recovery and separation of rare earth elements (REEs) and other critical minerals from acid mine drainage (AMD) using a solvent extraction (SX) process. In this process the REE concentration found in an AMD source can vary site to site, and season to season. SX processes take a non-trivial amount of time to reach steady state. The separation of numerous individual elements from gangue metals is a high-dimensional problem, and SX simulators can have a prohibitive computation time. Bayesian statistical methods intrinsically quantify uncertainty of model parameters and predictions given a set of data and a prior distribution and model parameter prior distributions. The uncertainty quantification possible with Bayesian methods lend well to statistical simulation, model selection, and sensitivity analysis. Moreover, Bayesian models utilizing Gaussian Process priors can be used for active learning tasks which allow for prediction, optimization, and simulator calibration while reducing data requirements. However, literature on Bayesian methods applied to separations engineering is sparse. The goal of this dissertation is to investigate, illustrate, and test the use of a handful of Bayesian methods applied to process engineering problems. First further details for the background and motivation are provided in the introduction. The literature review provides further information regarding critical minerals, solvent extraction, Bayeisan inference, data reconciliation for separations, and Gaussian process modeling. The body of work contains four chapters containing a mixture of novel applications for Bayesian methods and a novel statistical method derived for the use with the motivating problem. Chapter topics include Bayesian data reconciliation for processes, Bayesian inference for a model intended to aid engineers in deciding if a process has reached steady state, Bayesian optimization of a process with unknown dynamics, and a novel active learning criteria for reducing the computation time required for the Bayesian calibration of simulations to real data. In closing, the utility of a handfull of Bayesian methods are displayed. However, the work presented is not intended to be complete and suggestions for further improvements to the application of Bayesian methods to separations are provided. / Doctor of Philosophy / Rare earth elements (REEs) are a set of elements used in the manufacture of supplies used in green technologies and defense. Demand for REEs has prompted the development of technology for recovering REEs from unconventional resources. One unconventional resource for REEs under investigation is acid mine drainage (AMD) produced from the exposure of certain geologic strata as part of coal mining. REE concentrations found in AMD are significant, although low compared to REE ore, and can vary from site to site and season to season. Solvent extraction (SX) processes are commonly utilized to concentrate and separate REEs from contaminants using the differing solubilities of specific elements in water and oil based liquid solutions. The complexity and variability in the processes used to concentrate REEs from AMD with SX motivates the use of modern statistical and machine learning based approaches for filtering noise, uncertainty quantification, and design of experiments for testwork, in order to find the truth and make accurate process performance comparisons. Bayesian statistical methods intrinsically quantify uncertainty. Bayesian methods can be used to quantify uncertainty for predictions as well as select which model better explains a data set. The uncertainty quantification available with Bayesian models can be used for decision making. As a particular example, the uncertainty quantification provided by Gaussian process regression lends well to finding what experiments to conduct, given an already obtained data set, to improve prediction accuracy or to find an optimum. However, literature is sparse for Bayesian statistical methods applied to separation processes. The goal of this dissertation is to investigate, illustrate, and test the use of a handful of Bayesian methods applied to process engineering problems. First further details for the background and motivation are provided in the introduction. The literature review provides further information regarding critical minerals, solvent extraction, Bayeisan inference, data reconciliation for separations, and Gaussian process modeling. The body of work contains four chapters containing a mixture of novel applications for Bayesian methods and a novel statistical method derived for the use with the motivating problem. Chapter topics include Bayesian data reconciliation for processes, Bayesian inference for a model intended to aid engineers in deciding if a process has reached steady state, Bayesian optimization of a process with unknown dynamics, and a novel active learning criteria for reducing the computation time required for the Bayesian calibration of simulations to real data. In closing, the utility of a handfull of Bayesian methods are displayed. However, the work presented is not intended to be complete and suggestions for further improvements to the application of Bayesian methods to separations are provided.

The ability to accurately characterize the soundscape, or combination of sounds, of diverse geographic areas has many practical implications. Interested parties include the United States military and the National Park Service, but applications also exist in areas such as public health, ecology, community and social justice noise analyses, and real estate. I use an ensemble of machine learning models to predict ambient sound levels throughout the contiguous United States. Our data set consists of 607 training sites, where various acoustic metrics, such as overall daytime L50 levels and one-third octave frequency band levels, have been obtained. I have data for 117 geospatial features for the entire contiguous United States, which include metrics such as distance to the nearest road or airport, and the percentage of industrialization or forest in a specific area. I discuss initial model predictions in the spatial, frequency, and temporal domains, and the statistical advantages of using an ensemble of machine learning models, particularly for limited data sets. I comment on uncertainty quantification for machine learning models originating from limited data sets.