Adaptive Bayesian Methods for Small Sample and High-Dimensional Data: Scalable Inference, Sequential Design, and Robust Prior Modeling

Abstract: Small sample sizes and high-dimensional data present fundamental obstacles for statistical modeling, inference, and decision-making across a wide range of scientific and engineering disciplines. Traditional statistical techniques often break down under these regimes, yielding unstable or biased estimates. Adaptive Bayesian methods, with their intrinsic ability to incorporate prior information, update beliefs sequentially, and flexibly model complex structures, offer a powerful alternative. This paper synthesizes insights from cutting-edge studies on scalable inference, sequential Bayesian experimental design, and robust prior modeling for small-sample and high-dimensional settings. We provide critical evaluations of existing techniques, propose new adaptive frameworks for improved inference stability, and explore strategies to enhance both computational scalability and statistical robustness. Our findings underscore the pivotal role of adaptive Bayesian methods in navigating the challenges of complex, modern data environments. Keywords: Adaptive Bayesian Inference, Small Sample Challenges, High-Dimensional Data, Scalable Bayesian Methods, Sequential Design, Robust Priors, Hybrid Sampling, Bayesian Nonparametrics, Posterior Approximation, Experimental Design under Uncertainty