EAS 5160 Mathematical Foundations for Machine Learning I: Probability (0.5 CU)
Short Description
This course introduces students to the mathematical foundations of the theory of probability. In addition to a host of classical domains, probability is one of the foundational elements of modern data science, machine learning, and artificial intelligence. The course begins with an exploration of combinatorial probabilities in the classical setting of games of chance, proceeds to the development of an axiomatic, fully mathematical theory of probability, and concludes with the discovery of the remarkable limit laws and the eminence grise of the classical theory, the central limit theorem. The topics covered include: discrete and continuous probability space, distributions, mass functions, densities; conditional probability; independence; the Bernoulli schema: the binomial, Poisson, and waiting time distributions; uniform, exponential, normal, and related densities; expectation, variance, moments; conditional expectation; inequalities, tail bounds, and limit laws. This material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us. Students are assessed by weekly problem set assignments and a proctored exam.
Portfolio Building Course
No
Pre-Requisites
Undergraduate calculus (as seen in a typical 2 or 3-course sequence on single and multivariable calculus): the student should be familiar with functions, sequences, derivatives, and integrals. The focus is on conceptual understanding, but some computation is needed, and the student should be comfortable with changes of variables, basic integration by parts, derivatives and integrals of polynomials, exponentials, and logarithms. The student should also be familiar with basic vector-matrix theory, which includes notions of matrix transpose, inverse, and determinant. Optional reviews of combinatorics and set theory are available during the course.
Content
This course introduces students to the mathematical foundations of the theory of probability. In addition to a host of classical domains, probability is one of the foundational elements of modern data science, machine learning, and artificial intelligence. The course begins with an exploration of combinatorial probabilities in the classical setting of games of chance, proceeds to the development of an axiomatic, fully mathematical theory of probability, and concludes with the discovery of the remarkable limit laws and the eminence grise of the classical theory, the central limit theorem. The topics covered include: discrete and continuous probability space, distributions, mass functions, densities; conditional probability; independence; the Bernoulli schema: the binomial, Poisson, and waiting time distributions; uniform, exponential, normal, and related densities; expectation, variance, moments; conditional expectation; inequalities, tail bounds, and limit laws. This material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us. Students are assessed by weekly problem set assignments and a proctored exam.
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Course Creators
- Santosh S. Venkatesh