Advances in Bayesian Approaches for Stochastic Process Modeling and Uncertainty Quantification

Stochastic processes serve as foundational models for systems characterized by random evolution across time or space, making them essential tools in disciplines such as finance, physics, epidemiology, and environmental science. Traditional statistical methods often yield only point estimates of model parameters, limiting their capacity to capture the full scope of uncertainty inherent in such systems. In contrast, Bayesian inference offers a rigorous and comprehensive probabilistic framework by treating both parameters and stochastic processes as random variables. This approach enables the integration of prior knowledge and yields posterior distributions that encapsulate uncertainty more fully. This paper presents a comprehensive survey of Bayesian inference as applied to stochastic processes. It begins by outlining the theoretical foundations of Bayes' Theorem in this context, emphasizing the importance of prior specification for infinite-dimensional function spaces. The discussion then turns to key classes of stochastic processes—including Gaussian Processes, Markov Models, and State-Space Models—highlighting how Bayesian methods enhance their interpretability and predictive capacity. Given the complexity of posterior distributions in these models, the paper also reviews modern computational techniques such as Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) that enable practical implementation. Applications across multiple domains are explored to demonstrate the flexibility and power of the Bayesian approach. The study concludes by identifying emerging challenges and outlining promising directions for future research in Bayesian inference for stochastic systems.