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Toggle## Data-Science Lectures: Probability, Statistics, Machine Learning, and Information Theory

Explore our Data Science learning resources, featuring a wide array of comprehensive lectures and practical exercises in Probability, Statistics, Machine Learning, and Information Theory. Designed to cater to all skill levels, from novice learners to seasoned experts, our materials are crafted to elevate your knowledge and proficiency in the field.

### I. Measure Theory, Lebesgue Integral, and Probability

**Measure Theory, Lebesgue Integral, and Probability** webpage provides an extensive collection of learning materials, including detailed PDFs, informative video lectures, and carefully designed exercises complete with step-by-step solutions, all aimed at deepening your comprehension of Measure Theory, the Lebesgue Integral, and Probability Theory.

**Measure Theory****:**Learn the fundamentals of Measure Theory with our thorough guide, which elucidates key concepts such as σ-algebras, measurable functions, and the intricate properties of measures. Gain insights into various measure examples, including the Dirac and weighted counting measures, and explore the pivotal existence and uniqueness of the Lebesgue measure on ℝ^n.**Lebesgue Integral**: Delve into the Lebesgue Integral with our comprehensive guide, detailing its construction and convergence properties. Explore advanced topics including Fubini’s Theorem, the Radon-Nikodym Theorem, and the intricacies of Lp spaces, all presented with clarity and depth to enhance your mathematical expertise.**Probability Theory**: Advance your understanding of Probability Theory with our detailed course, which takes you from the measure-theoretic underpinnings to complex notions such as random variables, distribution functions, and the pivotal monotone and dominated convergence theorems, ensuring a comprehensive mastery of the subject.**Discrete Random Variables and their Transform**: This course covers discrete random variables, and the use of probability generating functions to analyze their distributions and moments.**Continous Random Variables and their Transforms**: The lecture explores continuous random variables, and delve into the moment generating function, characteristic function, and the Laplace transform to understand and solve related probability problems.**Convergence of Random Variables**: This lesson covers the various modes of convergence for random variables in probability theory, starting with almost-sure convergence, then moving to convergence in probability, quadratic mean, and weak convergence, while also examining their interrelations and concluding with Prohorov’s Theorem.**Exercises with Solutions**: Reinforce your knowledge with a curated collection of exercises covering key topics in measure theory, the Lebesgue integral, and probability theory. Each exercise is accompanied by detailed solutions to aid in your learning process.

### II. Statistics: Univariate & Multivariate Data

**Multivariate Statistics Theory** webpage offers comprehensive statistics course on both univariate and multivariate analysis techniques, catering to beginners and advanced practitioners. Our resources encompass a wide range of topics to enhance your statistical expertise.

- Basic Statistics: We will explore the fundamental concepts of statistical estimation, confidence intervals, hypothesis testing, and likelihood, along with their practical applications. These crucial topics serve as the building blocks for advanced statistical methods and play a pivotal role in various domains, such as scientific research, business analytics, and engineering.
- Linear Fitting and Regression: In this lecture, we delve into the heart of statistical modeling and analysis, focusing on understanding variable relationships and predictive techniques. We explore the intricacies of linear models, covering both deterministic fitting and probabilistic regression approaches. The session encompasses analysis in both univariate and multivariate contexts, and introduces Gaussian models as a framework for regression analysis.
- Logistic Regression: In statistical modeling, logistic regression is a crucial technique for analyzing discrete outcome variables. Unlike linear regression, which is tailored for continuous outcomes, logistic regression is particularly suitable for scenarios where the output variable is categorical. This lecture provides an overview of logistic regression, discussing its applications in both binary and multiclass contexts, as well as the prediction methods employed.
**Principal Component and Factor Analysis**: Principal Component Analysis (PCA) and Factor Analysis are indispensable techniques in multivariate statistics, employed to reduce redundancy among observed variables while preserving crucial information. While both techniques share a common objective, they possess distinct characteristics and methodologies. This lecture provides a comprehensive exploration of PCA and Factor Analysis, elucidating their principles, applications, and offering comparative insights.

### III. Machine and Deep Learning

**Machine and Deep Learning** webpage includes an extensive library of machine learning and deep learning resources, featuring eight detailed lectures available in both PDF and video formats. Immerse yourself in a curated educational experience designed to enhance your grasp of essential artificial intelligence principles.

**Machine Learning Frameworks**: The foundational principles for machine learning models are presented with mathematical rigor. From basic learning frameworks to addressing noise and extending to more general learning scenarios, we explore the theoretical underpinnings essential for understanding various machine-learning algorithms.**Vapnik-Chervonenkis Theory**: We present the profound concept of establishing uniform bounds between empirical and true loss within an infinite hypothesis class. Through rigorous exploration of Vapnik-Chervonenkis (VC) theory, we unravel the intricacies of covering and packing numbers, growth functions, and VC dimension, crucial elements for comprehending the complexities of learning in machine learning and statistics.**Results from Empirical Processes Theory**: This lecture complements Vapnik-Chervonenkis theory, providing essential instruments to determine consistent limits on the divergence between empirical and actual losses across an an infinite class of hypotheses. Through rigorous derivation of tail bounds, we uncover essential principles that support the robustness and generalization capabilities of machine learning algorithms.**Learnability Characterization**: This session reveals the key attributes that determine the learnability of infinite hypothesis classes within machine learning. Employing thorough analysis and drawing upon core theories like Vapnik-Chervonenkis and empirical processes, we strive to elucidate the sample complexity necessary for learning algorithms to reach peak efficacy. Building on earlier conversations, we expand our comprehension from finite to infinite hypothesis classes, setting forth the standards for learnability across diverse conditions.**Examples of Machine-Learning Problems**: Our session explores a range of real-world situations where machine learning methods are utilized for problem-solving. We examine supervised and unsupervised learning models, featuring instances from binary classification using nearest neighbors to clustering techniques. Through analyzing these cases, we intend to offer perspectives on the wide-ranging uses of machine learning and the strategies implemented to address these challenges.**Neural Networks**: This course presents the computational models that mirror the complex architecture of the human brain. Our lecture is designed to impart an in-depth comprehension of neural networks, encompassing their precise definitions, functionalities, and refinement methods. From layered architectures to the intricacies of stochastic gradient descent, we demystify the sophisticated mechanisms of neural networks and their implementation in the fields of machine learning and artificial intelligence.**Approximation Theory in Neural Networks**: We explore the core principles that dictate how neural networks can mimic functions, an essential aspect of their versatility across different fields. Our examination of the universal approximation theorem aims to clarify the potential and constraints of neural networks in emulating functions. Additionally, we tackle critical inquiries about the nature of functions that are subject to approximation and the requisite neuron count for attaining specific precision goals.**Python Software for Machine Learning and Deep Learning – Tutorial**: Explore our in-depth guide on leveraging Python for your machine learning and deep learning endeavors. Our objective is to equip you with a clear, sequential process for setting up the required software from the ground up and crafting your initial Python scripts for data science applications. Upon completing this tutorial, you’ll possess the proficiency to build your personal neural network with PyTorch and handle data with ease for various machine learning projects.

### IV. Information Theory

**Information Theory** webpage offers an in-depth presentation of information theory and coding, providing a thorough understanding of key principles and their real-world implementations. Delve into the intricacies of data transmission and compression, the cornerstone of modern communication systems, and explore how information theory shapes our interconnected world.

- Coding Problem, Entropy and Mutual Information
**Coding Problem in Claude Shannon Information-Theory**: In Claude Shannon’s groundbreaking research on information theory, the coding problem is a pivotal challenge for the efficient transfer of information. Our lecture delves deep into this issue, examining its key elements: source coding, channel coding, and their synergy in joint source-channel coding.**Shannon’s Entropy and Conditional Entropy**: This lecture offers a clear presentation of the principles quantifying uncertainty in data. This lecture is organized into four enlightening sections, providing an in-depth analysis of Shannon’s entropy and conditional entropy, and illuminating their theoretical foundations as well as their practical applications in diverse fields.**Mutual Information and Kullback-Leibler Divergence**: This session provides a deep dive into the critical concept of mutual information, a key element in information theory. It quantifies the amount of information shared between random variables, playing a vital role in disciplines such as statistics, machine learning, and cryptography. This lecture is thoughtfully divided into three detailed sections, each thoroughly exploring different aspects of mutual information.

- Source Coding
**Source Coding Theorem**: We delve into the essential concept of the source coding theorem in information theory. Grasping this theorem is crucial for understanding the fundamental limits and capabilities of data compression. We start with an exploration of source coding capacity and its characterization, progressing to Shannon’s source coding theorem. The lecture then examines the principles of data compression and typical sequences, culminating in a detailed proof of the source coding theorem.**Error-Free Source Coding**: We explore the essential principles and methods for encoding information efficiently and without loss. We begin by examining variable-length source coding and coding rates, laying the foundation for error-free encoding techniques. We clarify the role of uniquely decipherable codes, including codebooks and codewords, emphasizing the need for clear decoding in information transmission. We then investigate the use of tree-based codes and explore Kraft’s inequality, a key metric for code efficiency. The lecture culminates with a thorough analysis of the error-free source coding theorem, discussing optimization issues, bounds on average codeword length, and the role of Kraft’s inequality in devising optimal error-free encoding strategies.**Optimal Source Codes**: This lecture, « Optimal Source Codes, » examines two fundamental approaches: Huffman’s optimal coding algorithm and the Shannon-Fano-Elias coding method. These strategies are essential for crafting efficient, lossless data compression codes. We will dissect the underlying principles of these methods, providing insights into the development of highly effective encoding systems for diverse data types.**Parsing-Translation and Tunstall’s Codes**: This session explores advanced methods for attaining low coding rates through optimal parsing—grouping source symbols into variable-length blocks—and encoding these blocks using straightforward, non-optimized codes. Introduced by Tunstall in 1967, Tunstall’s code is a prime example of a wider category of codes that employ parsing followed by translation. We will conduct an in-depth examination of parsing-translation codes and Tunstall’s approach, gaining an understanding of the efficient encoding strategies and their real-world applications.**Universal Source Coding**: We explore data compression and encoding, examining the concept of universal source coding and its impact. This approach aims to create compression techniques that can effectively condense any data type, regardless of prior knowledge of its statistical characteristics.**Lempel-Ziv Source Code**: We examine the Lempel-Ziv algorithm, a key technique for error-free encoding of any source sequence, achieving universal source coding. We’ll break down how the algorithm works, explore its variations, assess its efficiency using automata theory, and demonstrate why it’s considered optimal compared to other encoding methods.

- Channel Coding
**Channel Information Capacity**: This lecture explores the core ideas of how much information communication channels can transmit dependably. Starting with the basics of channel properties, we build the essential knowledge needed to grasp information capacity. We then define and discuss the theory behind information capacity, and bring these ideas to life with practical examples, demonstrating capacity in actual communication situations.**Channel Coding Theorem**: In the « Channel Coding Theorem » lecture, we investigate key information theory concepts, emphasizing the link between achievable transmission rates and a channel’s coding capacity. We start with the theorem’s basic principles, examining how to maximize transmission rates despite noisy channel limitations. Next, we lay the foundation for the theorem’s proof, introducing typical sequences for two random variables and crucial mutual information inequalities.

**Exercises with Solutions on Information Theory**: This section offers a carefully selected set of problems aimed at strengthening your grasp of information theory. For each exercise, you’ll find an in-depth solution that walks you through the process, ensuring you not only understand the theory but can also apply it practically.

## Data-Science Books: Coming Soon on this Webpage

Keep an eye on this page for the upcoming launch of our books:

- B. Błaszczyszyn, L. Darlavoix, M.K. Karray: « Primer on Measure, Integral, and Probability Theories ».
- B. Błaszczyszyn, L. Darlavoix, M.K. Karray: « Data science : From multivariate statistics to machine, deep learning ».
- B. Błaszczyszyn, M.K. Karray: « Information theory ».