Course Descriptions

This course and its sequel MA-GY 6223 rigorously treat the basic concepts and results in real analysis. Course topics include limits of sequences, topological concepts of sets for real numbers, properties of continuous functions and differentiable functions. Important concepts and theorems include supremum and infimum, Bolzano-Weierstrass theorem, Cauchy sequences, open sets, closed sets, compact sets, topological characterization of continuity, intermediate value theorem, uniform continuity, mean value theorems and inverse function theorem. | Offered in the fall.

Prerequisite(s): MA-UY 2122 or permission of adviser.

Prerequisites

Undergraduate-level linear algebra, multivariate calculus, probability and statistics. Basic program-
ming knowledge (Python).

Description
The goal of this class is to provide students with the fundamentals underlying modern computational
statistics and to understand how these methods can be effectively implemented in practical problems
of inference and estimation. Much of the material covered in the class is also found at the basis of
machine learning and will thus be useful to students following classes in statistical learning theory,
convex optimization and kernel methods. The course will include reminders on probability and random
variables, before presenting numerical methods for root finding, function minimization, function
approximation and numerical linear algebra. We will then move to numerical integration, random
variable generation and sampling methods, notably Markov chain Monte Carlo (MCMC) methods.
The final block will be elements of classical statistics : parametric regression and classification, Gaussian processes, CDF estimation, the bootstrap and the jacknife, density fitting and non- parametric regression. If time permits, we will briefly review variational inference as a complementary approach to sampling.

This course covers basic ideas of linear algebra: Groups, rings, fields, vector spaces, basis, dependence, independence, dimension. Relation to solving systems of linear equations and matrices. Homomorphisms, duality, inner products, adjoints and similarity. | Offered in the fall.

Prerequisite(s): MA-UY 2034  and MA-UY 2114 or Graduate Standing.