Browsing by Author "Xu, Jason"

The spike and slab prior offers a canonical approach to Bayesian variable selection, with caveats known to be data dimension and correlation. Motivated by the pitfalls of the spike and slab prior on high dimensional correlated data, this paper introduces Bayesian decoupling (BD, proposed by Hahn and Carvalho [HC15]) as a decision theory-based approach to inducing sparsity on posterior. We formalize the decision theoretical foundation of BD, and argue that BD conducts a sparsification over the posterior mean with a tolerable degradation of the predictive ability. Moreover, the application of BD in sparse estimation motivates the notion of decoupled model fitting and variable estimation, which is an idea rooted in Bayesian decision theory stating that variable estimation should be explicitly recovered as a decision making problem after the model fitting stage. We suggest a broader use of BD in Bayesian statistics, emphasizing that it allows multiple estimation tasks to be carried out simultaneously under a single prior by using different loss functions for different estimation purposes. Our simulation results show that BD with appropriately defined loss functions leads to a desired support recovery with low MSE and FDR and offers an accurate representation of the posterior belief.

Stochastic compartmental models provide interpretable probabilistic descriptions of many dynamic biological phenomena, such as the spread of a contagious disease through a population or the progression of cancer in an individual. Many classical inferential tools are out of reach for fitting these models in missing data settings, however, due to the intractability of the marginal likelihood. To remedy this issue, practitioners typically rely on simplifying assumptions to make inference tractable, or on intensive simulation-based methods that do not scale to modern datasets. In this thesis, I demonstrate how long-held simplifying assumptions can be relaxed, improving model realism and yielding better data fit. These contributions are developed alongside efficient sampling algorithms to enable exact Bayesian inference in many partially observed settings. I focus on two driving applications: first, I model the spread of contagious disease through a population using a continuous-time, stochastic susceptible-infectious-removed model, and provide nonparametric temporal extensions. Next, I turn attention to modeling the natural history of cancer using a semi-Markov model. In each of these studies, I tailor a data-augmented Markov chain Monte Carlo sampling algorithm that efficiently explores its discrete, high-dimensional latent space. Taken together, these advances surmount computational and methodological challenges in a notoriously difficult setting for modern Markov chain Monte Carlo samplers, and leads to new insights in several timely applications and case studies.

The research studies outlined in this thesis are all geared toward improving our understanding of infectious diseases and their impact on public health. Utilizing statistical frameworks, computational methods, and deterministic and stochastic mathematical models, these studies investigate the mean ICU stay during the COVID-19 pandemic to optimize hospital resources, the decay of viral load and distribution of lifetimes of infected cells in SHIV infected rhesus macaques, and the impact of various immune effector functions in effectively managing upper respiratory viral infections. The thesis combines methodologies to develop strategies to analyze infectious disease dynamics at both the individual and population level.

The first study establishes a statistical framework for estimating occupancy rates in intensive care units, incorporating variables such as hospital bed occupancy and SARS-CoV-2 test positivity rates. Using an immigration-death model, this research enables dynamic estimation of patient influx and efflux from ICUs, a crucial element in healthcare planning during a pandemic. This methodological approach is empirically evaluated using data from the University of California, Irvine Health and Orange County, California. The second study provides an efficient algorithm to compute transition probabilities for branching processes, which are commonly used in modeling ecological and epidemiological dynamics. Our proposal introduces a new method that uses variable splitting, leading to updates in a closed form through an efficient ADMM algorithm. It is important to note that no matrix multiplications, let alone inversions, are needed during any part of the process. This results in a significant decrease in computational costs by several orders of magnitude compared to current methods. In addition, the resulting algorithm can be easily parallelized and shows a high level of robustness to changes in tuning parameters. This method is compared to prior work by applying it to two scenarios that involve models of blood cell production and transposon evolution.

The third study introduces a modification of the Gompertz model to depict the dynamics of viral load post initiation of antiretroviral therapy. This simple model is applied to the infant rhesus macaque SHIV.C.CH505 infection data set and extended using a stochastic differential equation formulation. This model aligns well with the data and suggests that contemplating a continuous distribution of infected cell lifespans may yield a more nuanced insight into viral decay trends. In the final study, a mathematical model is proposed to improve the understanding of interactions between hosts and pathogens in upper respiratory infections. This model goes beyond traditional frameworks by encompassing an array of immune effector functions, thus offering a more thorough analysis of early infection phases. Although it concentrates on SARS-CoV-2, the model’s principles have broad relevance, highlighting the potential for diverse applications.

Collectively, these studies improve the knowledge base on infectious disease modeling and may help shape a well-informed approach to public health interventions.

Stochastic versions of proximal methods have gained much attention in statistics and machine learning. These algorithms tend to admit simple, scalable forms, and enjoy numerical stability via implicit updates. In this work, we propose and analyze a stochastic version of the recently proposed proximal distance algorithm, a class of iterative optimization methods that recover a desired constrained estimation problem as a penalty parameter $\rho \rightarrow \infty$. By uncovering connections to related stochastic proximal methods and interpreting the penalty parameter as the learning rate, we justify heuristics used in practical manifestations of the proximal distance method, establishing their convergence guarantees for the first time. Moreover, we extend recent theoretical devices to establish finite error bounds and a complete characterization of convergence rates regimes. We validate our analysis via a thorough empirical study, also showing that unsurprisingly, the proposed method outpaces batch versions on popular learning tasks.