Spring 2024 Semester PhD Courses
For the most updated information on Statistics PhD courses, please go to Vergil.
|Applied Statistics II
|This is a first-year Ph.D. course on statistical machine learning and Bayesian statistics, focusing mainly on the methodology and also covering some applications. Course contents include the following: Linear and nonlinear dimension reduction; Data-driven and model-based classification and clustering methods; Graphical models including Bayesian networks and Markov random fields; Latent variable models; Variational Bayesian inference; Introduction to deep learning and neural networks; Computational Bayesian statistics including Gibbs sampler and other MCMC algorithms; Bayesian hierarchical modeling.
|Computation plays a central role in modern statistics and machine learning. This course aims to cover topics needed to develop a broad working knowledge of modern computational statistics. We seek to develop a practical understanding of how and why existing methods work, enabling effective use of modern statistical methods. Achieving these goals requires familiarity with diverse topics in statistical computing, computational statistics, computer science, and numerical analysis. Our choice of topics reflects our view of what is central to this evolving field, and what will be interesting and useful. A key theme is scalability to problems of high dimensionality, which are of most interest to many recent applications.
|Prerequisites: STAT GR6102 or instructor permission. The Deparatments doctoral student consulting practicum. Students undertake pro bono consulting activities for Columbia community researchers under the tutelage of a faculty mentor.
|Theoretical Statistics II
|Prerequisites: STAT GR6201 Continuation of STAT G6201
|Probability Theory II
|Graduate-level introduction to stochastic processes in discrete and continuous time.Topics: Martingales: inequalities, convergence and closure properties, optimal stopping theorems, Burkholder-Gundy inequalities. Semimartingles: Doob-Meyer decomposition, stochastic integration, Ito’s formula. Brownian motion: construction, invariance principles and random walks, study of sample paths, martingale representation results, Girsanov theorem. Markov processes: semigroups and infinitesimal generators. Stochastic differential equations. Connections to partial differential equations: Feynman-Kac formula, Dirichlet problem.
|Topics in Applied Statistics
|Topics in Theoretical Statistics
|Topics in Probability Theory
|Usually when one thinks of Mathematical Finance one thinks of modeling the stock market, options, and hedging, almost invariably involving Brownian motion. A key concept is the absence of arbitrage which leads to the use of Girsanov’s Theorem and changes of measure. In this course we will of course touch on all that, more or less due to necessity, but the heart of the course will be devoted to the poorly understood subject of credit risk, taking advantage of recent advances of Coculescu and Nikeghbali. We will discuss the classification of stopping times and show how totally inaccessible stopping times arise naturally in the modeling of credit defaults. Such an analysis touches on Survival Analysis and the theory of Censored Data, especially when martingales are involved.
|Topics in Machine Learning
|Field Experiments, Machine Learning, and Causality; Spring 2024; David Blei / Don Green; This course explores the challenges of extracting unbiased and generalizable causal inferences about cause and effect in policy-relevant domains. This technical level of the course is designed for doctoral students in social science, computer science, and statistics, but it will also be open to masters students and undergraduates with sufficient preparation. The partnership between the two instructors (who are also research collaborators and co-authors) reflects a growing recognition that experimental designs deployed in field settings, although informative and influential, can only support causal generalizations with the help of supplementary assumptions; similarly, observational studies that draw on big data only provide reliable causal insights with the help of supplementary assumptions. The aim of this collaboration is to explore ways that innovative research design, modeling, and machine learning methods can advance the frontiers of knowledge in policy-relevant fields. While courses on causal inference focus on a handful of off-the-shelf techniques, the proposed course aims to innovate, offering new ways of thinking about what to study and how. With real-world experimental designs and real-world data, we will study how to evaluate the strengths and weaknesses of modeling choices and methods, and how to use model-based insights to suggest more informative design choices.
|Bianca Dumitrascu & Yuqi Gu
|Seminar in Theoretical Statistics
|Departmental colloquium in statistics.
|Seminar in Probability Theory
|This is a weekly seminar in probability theory involving mostly outside speakers who present on a variety of topics including stochastic analysis and PDEs, random matrix theory, random geometry, stochastic optimal control, statistical physics and many others.
|Chenyang Zhong & Sumit Mukherjee
|Seminar in Applied Probability and Risk
|A colloquiim in applied probability and risk.
|Marcel Nutz & Philip Protter
|Seminar in Mathematical Finance
|Research seminar on mathematical finance featuring invited speakers.