<b>Bayesian Inference for Stochastic and Ordinary Differential Equations: Methodology and Applications</b>

<p dir="ltr">Stochastic differential equations (SDEs) and ordinary differential equations (ODEs) play a crucial role in both applied and mathematical sciences. However, when they are incorporated as components of probabilistic Bayesian models, posterior inference given noisy observations remain a significant challenge. In the first part of this thesis, we focus on continuous-time SDEs, where the nonlinear continuous-time dynamics result in intractable transition probabilities, making posterior simulation a so-called doubly-intractable distribution. In practice, one typically has to use approximate MCMC schemes that discretize time. Building on work in Beskos and Roberts, 2005, we develop an exact MCMC algorithm that targets the posterior distribution with no approximation error, while also being flexible, simple and scalable. At a high level, our algorithm can be understood as alternately sample a new discretization of time given a diffusion path, and then a new path given the discretization. We extend earlier work (Wang, Rao and Teh, 2022) to a larger to class of diffusions. Naively extending that work would involve expensive evaluations of probabilities and gradients of Bessel bridges. Instead, we propose a novel ’double accept’ procedure involving only standard Gaussian process techniques. In our experiments, we demonstrate that our approach has minimal impact on MCMC mixing, while significantly speeding up run-time. The second part of this thesis investigates the Bayesian application to parameter estimation in a family of ODE models for pharmacokinetic (PK) modeling in clinical trial data. Using Stan with R and Torsten, we explore Bayesian approaches for population PK modeling. These methods are particularly useful for analyzing special populations, small sample datasets, and complex model structures. Finally, the third part of this thesis applies Bayesian inference to prognostic and predictive covariate-adjusted response-adaptive randomization designs, further demonstrating the versatility of Bayesian methods in clinical and experimental settings.</p>