Bayesian Inference and Uncertainty Quantification for AI: Posterior Learning, Predictive Guarantees, Calibration, and Reliable Decisions

Uncertainty is useful only when we know exactly what it is uncertainty about—and what its guarantees actually mean.

Bayesian Inference and Uncertainty Quantification for AI develops a rigorous, unified framework for reasoning about uncertainty across the modern AI pipeline. Rather than treating Bayesian posteriors, calibration, conformal prediction, and deployment risk as separate topics, the book connects them through a common question: which probability statement is being made, under what assumptions, and which parts of that statement survive approximation, distribution shift, and decision making?

Beginning with Bayesian models, priors, exchangeability, hierarchical structure, and posterior asymptotics, the book moves through Monte Carlo, gradient-based sampling, variational and amortised inference, proper scoring rules, calibration, conformal prediction, selective prediction, and distribution shift. It then brings these ideas into contemporary AI through function-space uncertainty, Bayesian deep learning, pretrained-model adaptation, retrieval and generation systems, multi-step agents, and scientific inverse problems with learned generative priors.

Throughout, theorem statements are paired with explicit assumptions, failure modes, rate interpretations, and “guarantee cards” that identify what is random, what is held fixed, and what is not claimed. Engineering lenses translate mathematical results into quantities that matter in practice—calibration data, sampling budgets, minibatch noise, abstention costs, and long-horizon reliability.

Designed for advanced master’s and early PhD students, researchers, and quantitatively trained practitioners, this volume offers a compact mathematical foundation for building, evaluating, and deploying AI systems whose uncertainty claims can be interpreted—and trusted—for the right reasons.