Mathematics for artificial Intelligence-ii

Artificial Intelligence (AI) is no longer a futuristic dream; it has become the backbone of today’s digital society, influencing everything from healthcare and finance to robotics, autonomous vehicles, and natural language processing. At the heart of every intelligent algorithm lies one common foundation: mathematics. While Volume I of this series established the fundamentals of linear algebra and probability, and your separate book on Statistics for Data Science covered the practical essentials, this Volume II is designed to take the learner deeper into the advanced statistical methods and optimization techniques that truly power state-of-the-art AI models.
If Volume I was about building the walls of the AI temple, Volume II is about reinforcing its pillars and giving it strength to stand against the complexity of modern challenges. This book is not just about formulas and theorems—it is about understanding the mathematical reasoning that makes artificial intelligence systems reliable, efficient, and robust.
Today’s deep learning models with billions of parameters, reinforcement learning agents capable of defeating world champions, and generative AI systems that create human-like text, images, and music—all owe their success to statistics and optimization. Without advanced statistical methods, we cannot understand uncertainty, reliability, or the generalization capacity of a model. Without optimization, we cannot train networks, tune hyperparameters, or find efficient solutions to real-world problems.
This book is a roadmap designed for students, researchers, data scientists, engineers, and AI enthusiasts who are eager to push beyond the basics and understand the mathematics that makes modern AI possible.


Structure of the Book
This volume is divided into five parts with sixteen chapters, carefully structured to cover advanced statistics and optimization for artificial intelligence.
Part I – Advanced Statistical Foundations for AI
Here, we explore the statistical decision theories, multivariate statistics, and resampling methods that go beyond descriptive statistics. Readers will learn about covariance structures, multivariate normal distributions, factor analysis, and the crucial bias-variance trade-off in AI systems.
Part II – Optimization for Artificial Intelligence
This part introduces optimization in its purest form. Starting with the fundamentals of convex and non-convex optimization, it explores gradient-based optimization methods such as Adam, RMSProp, and NAG. It also covers convex optimization duality and real-world applications like Support Vector Machines (SVMs).
Part III – Probabilistic and Information-Theoretic Optimization
Modern AI does not rely only on deterministic optimization. This part focuses on probabilistic approaches like Monte Carlo methods, Bayesian optimization, and Expectation-Maximization. It also dives into information theory concepts such as entropy, KL divergence, and mutual information—tools critical in deep learning and generative models.
Part IV – Advanced AI Applications
Here, the focus is on practical applications: optimization in neural networks, reinforcement learning, and hyperparameter tuning. Readers learn how to address problems like vanishing gradients, saddle points, and overfitting in real-world training.
Part V – Practical Implementations and Case Studies
This part brings everything together through coding examples and case studies. Using Python libraries like NumPy, CVXPY, TensorFlow, and PyTorch, readers can directly implement optimization techniques and apply them to computer vision, NLP, reinforcement learning, and healthcare applications.