Generative Models: Principles, Architectures, and Applications

This book provides a compact, derivation-oriented introduction to the mathematical foundations of modern generative artificial intelligence. Rather than surveying every recent architecture or implementation detail, it develops a coherent route through the ideas connecting major families of generative models, from PCA, probabilistic PCA, variational autoencoders, and diffusion models to normalising flows, autoregressive factorisations, GANs, Wasserstein GANs, and energy-based models. The aim is to make the structure of generative modelling more accessible without removing the mathematical substance needed to understand how these models are derived and related. The book is intended as a foundation-building primer for mathematically curious researchers, practitioners, and students.

The incorporation of generative models into traditional computational systems presents both enormous opportunity and tremendous peril. Although many early adopters have realized these perils at great expense, the field still requires foundational frameworks to de-risk incorporation of AI into traditional systems. This manuscript establishes this foundation through the definition of four specific primitives of AI blended architecture, designed to enable deterministic encapsulation of probabilistic models. It further establishes two overarching anti-patterns broadly represented across industry to serve as warnings for engineers in this field. This framework was designed to enable successful integration of AI into traditional systems while providing a foundation upon which generative model providers could build the next generation of generative model interfaces.

Generative AI systems have entered everyday academic, professional, and personal life with remarkable speed, yet most users encounter them as mysterious artifacts rather than intelligible systems. This chapter discusses large language models within a broader historical shift in computing paradigms and argues that many of the confusions surrounding their use arise from a mismatch between how these systems are built, how they behave, and how people expect computers to behave writ large. Rather than treating generative AI as a monolithic technology, the chapter decomposes it into interacting components, spanning data, models, product features, and user inputs, each introducing distinct affordances and tensions. Particular attention is given to the statistical and data-based foundations of these systems and to the fact that their surface behavior is explicitly human-like, a combination that places them squarely within the intellectual traditions of educational and behavioral research. From this perspective, educational researchers are unusually well positioned to study, evaluate, and productively use generative AI systems, drawing on established methods for modeling latent processes, managing uncertainty, and interpreting complex human-system interactions. The goal is to equip readers with a conceptual map that supports more informed experimentation, critical interpretation, and responsible use as these systems continue to evolve.