S&DS 5000a or b, Introductory Statistics  Ethan Meyers

An introduction to statistical reasoning. Topics include numerical and graphical summaries of data, data acquisition and experimental design, probability, hypothesis testing, confidence intervals, correlation and regression. Application of statistical concepts to data; analysis of real-world problems.
HTBA

S&DS 5001a, Data Science @Yale  John Lafferty and Bhramar Mukherjee

This seminar provides early-stage graduate students with an overview of core concepts and information about resources available at Yale for research in data science. The seminar is not a typical subject-matter course; it is an orientation for students working in diverse areas of data science, providing opportunities for community building. Open only to Yale graduate students. Enrollment limited; requires permission of the instructor.  ½ Course cr
HTBA

S&DS 5170b, Applied Machine Learning and Causal Inference  P Aronow

Approaches to causal inference using machine learning. Covers randomized experiments with and without noncompliance, observational studies with and without ignorable treatment assignment, instrumental variables, and regression discontinuity. Machine-learning methods include bagging, boosting, tree-based methods such as random forests, and neural networks. Assignments provide students with hands-on experience with the methods. Applications are drawn from a variety of fields including political science, economics, public health, and medicine. Programming is central to the course and is based on the R programming language. Prerequisites: the equivalent of at least two of the following courses: S&DS 530, S&DS 538, S&DS 541, and S&DS 542; and previous programming experience (e.g., R, MATLAB, Python, C++), R preferred. Strong knowledge of OLS is assumed.
Th 4pm-5:55pm

S&DS 5200b, Intensive Introductory Statistics  Robert Wooster

An introduction to statistical reasoning designed for students with particular interest in data science and computing. Using the R language, topics include exploratory data analysis, probability, hypothesis testing, confidence intervals, regression, statistical modeling, and simulation. Computing is taught and used extensively throughout the course. Application of statistical concepts to the analysis of real-world data science problems.
TTh 9am-10:15am

S&DS 5230a or b, YData: An Introduction to Data Science  Staff

Computational, programming, and statistical skills are no longer optional in our increasingly data-driven world; they are essential for opening doors to manifold research and career opportunities. This course aims to dramatically enhance students’ knowledge and capabilities in fundamental ideas and skills in data science, especially computational and programming skills and inferential thinking. It emphasizes the development of these skills while providing opportunities for hands-on experience and practice. The course is designed to be accessible to students with little or no background in computing, programming, or statistics, but also engaging for more technically oriented students through extensive use of examples and hands-on data analysis. Python 3 is the computing language used. Enrollment is limited.
HTBA

S&DS 5300a or b / PLSC 5300a or b, Data Exploration and Analysis  Jonathan Reuning-Scherer

An expansion of introductory statistics topics that also provides an extensive introduction to the R programming language. A concise review of foundational statistical methods moves into topics such as bootstrapping, permutation tests, multiple regression, ANOVA, ANCOVA, generalized linear models, and logistic regression. All topics emphasize the use of appropriate statistical graphics. This course also builds a strong foundation in R, RStudio, and RMarkdown, including data cleaning, data types, and strategies for handling missing data. Lectures incorporate examples of how AI can support both R coding and data analysis. This is a highly applied course, with lectures and homework focused on performing data analysis and communicating results.
HTBA

S&DS 5350b, Social Algorithms  Johan Ugander

Algorithms that learn from data play increasingly central roles within modern complex social systems. In this course, we examine the design and behavior of algorithms in such contexts, including search, content recommendation, social recommendation, feed ranking, content moderation, and more. The course has a split focus on the technical design of such algorithms, as well as the literature on theoretical and empirical evaluations, particularly in the presence of strategic behavior, network effects, and algorithmic confounding. Prerequisites: S&DS 123 or S&DS 1230 YData and (S&DS 230/530 or S&DS 2300/S&DS 5300 Data Exploration or S&DS 361/S&DS 661 or S&DS 3610/S&DS 6610 Data Analysis), or equivalent.
MW 9am-10:15am

S&DS 5380a, Probability and Statistics  Robert Wooster

Fundamental principles and techniques of probabilistic thinking, statistical modeling, and data analysis. Essentials of probability: conditional probability, random variables, distributions, law of large numbers, central limit theorem, Markov chains. Statistical inference with emphasis on the Bayesian approach: parameter estimation, likelihood, prior and posterior distributions, Bayesian inference using Markov chain Monte Carlo. Introduction to regression and linear models. Computers are used throughout for calculations, simulations, and analysis of data. Prerequisite: after or concurrently with MATH 118 or MATH 120.
TTh 1:05pm-2:20pm

S&DS 5400b, An Introduction to Probability Theory  Ilias Zadik

Introduction to probability theory. Topics include probability spaces, random variables, expectations and probabilities, conditional probability, independence, discrete and continuous distributions, central limit theorem, Markov chains, and probabilistic modeling. This course may be appropriate for non-S&DS graduate students. Prerequisite: MATH 115 or equivalent.
MW 2:35pm-3:50pm

S&DS 5410a, Probability Theory  Sinho Chewi

A first course in probability theory: probability spaces, random variables, expectations and probabilities, conditional probability, independence, some discrete and continuous distributions, central limit theorem, Markov chains, probabilistic modeling. Prerequisite: calculus of functions of several variables.
MW 9am-10:15am

S&DS 5420b, Theory of Statistics  Zhou Fan

Principles of statistical analysis: maximum likelihood, sampling distributions, estimation, confidence intervals, tests of significance, regression, analysis of variance, and the method of least squares. Prerequisite: S&DS 541.
MW 2:35pm-3:50pm

S&DS 5510b / ECE 5021b, Stochastic Processes  Shuangping Li

Introduction to the study of random processes, including Markov chains, Markov random fields, martingales, random walks, Brownian motion, and diffusions. Techniques in probability such as coupling and large deviations. Applications chosen from image reconstruction, Bayesian statistics, finance, probabilistic analysis of algorithms, genetics, and evolution.
MW 1:05pm-2:20pm

S&DS 5540b, Bayesian Modeling and Inference  Xiang Zhou

This course offers a rigorous and comprehensive introduction to Bayesian modeling and inference, encompassing foundational theory, modern computational techniques, and advanced modeling frameworks. It is designed for advanced undergraduates, master’s students, and Ph.D. students with a strong interest in statistical modeling, inference, and applications across diverse disciplines. Prerequisites: Probability theory at the level of S&DS 2410/5410, statistical inference at the level of S&DS 2420/5420), linear algebra at the level of MATH 2250 or 2260, and experience with computing and data analysis in common programing languages such as R and Python.
TTh 1:05pm-2:20pm

S&DS 5630b, Multivariate Statistical Methods for the Social Sciences  Jonathan Reuning-Scherer

An introduction to the analysis of multivariate data. Topics include principal components analysis, factor analysis, cluster analysis (hierarchical clustering, k-means), discriminant analysis, multidimensional scaling, and structural equations modeling. Emphasis on practical application of multivariate techniques to a variety of examples in the social sciences. Students complete extensive computer work using either SAS or SPSS. Prerequisites: knowledge of basic inferential procedures, experience with linear models (regression and ANOVA). Experience with some statistical package and/or familiarity with matrix notation is helpful but not required.
TTh 1:05pm-2:20pm

S&DS 5650a or b, Introductory Machine Learning  Staff

This course covers the key ideas and techniques in machine learning without the use of advanced mathematics. Basic methodology and relevant concepts are presented in lectures, including the intuition behind the methods. Assignments give students hands-on experience with the methods on different types of data. Topics include linear regression and classification, tree-based methods, clustering, topic models, word embeddings, recurrent neural networks, dictionary learning, and deep learning. Examples come from a variety of sources including political speeches, archives of scientific articles, real estate listings, natural images, and others. Programming is central to the course and is based on the Python programming language.
HTBA

S&DS 5720a, YData: Data Science for Political Campaigns  Joshua Kalla

Political campaigns have become increasingly data driven. Data science is used to inform where campaigns compete, which messages they use, how they deliver them, and among which voters. In this course, we explore how data science is being used to design winning campaigns. Students gain an understanding of what data is available to campaigns, how campaigns use this data to identify supporters, and the use of experiments in campaigns. The course provides students with an introduction to political campaigns, an introduction to data science tools necessary for studying politics, and opportunities to practice the data science skills presented in S&DS 523.
W 4pm-5:55pm

S&DS 5790a, Spatial Statistics  Xiang Zhou

This course provides a comprehensive introduction to spatial statistics, focusing on statistical modeling and inference for data exhibiting spatial dependence. The course covers the major classes of spatial data, including geostatistical (continuous-domain), areal (lattice), and spatial point pattern data and develops modeling frameworks tailored to each setting. Core topics include covariance functions and variograms, Gaussian random fields, kriging and spatial prediction, spatial regression models, conditional and simultaneous autoregressive (CAR/SAR) models for areal data, and Poisson and Cox processes for point patterns. Emphasis is placed on model construction, interpretation, and derivation of key estimators and predictors, with attention to both likelihood-based and hierarchical modeling approaches. Students learn to assess model assumptions, quantify uncertainty, and address practical challenges such as spatial confounding and computational scalability for large datasets. The course also introduces modern developments connecting spatial statistics with machine learning, including Gaussian process methods and scalable approximations. Through theoretical development, problem sets, and a data-driven project, students gain the skills necessary to formulate, implement, and evaluate spatial statistical models in scientific, environmental, biomedical, and social science applications. Prerequisites: probability theory at the level of S&DS 2410/5410, statistical inference at the level of S&DS 2420/5420, linear algebra at the level of MATH 2250 or 2260, experience with computing and data analysis in common programming languages such as R and Python, and familiarity with linear models (e.g., S&DS 3120/6120) and Bayesian methods (e.g., S&DS 3540/5540).
TTh 2:35pm-3:50pm

S&DS 5800a, Neural Data Analysis  Ethan Meyers

We discuss data analysis methods that are used in the neuroscience community. Methods include classical descriptive and inferential statistics, point process models, mutual information measures, machine learning (neural decoding) analyses, dimensionality reduction methods, and representational similarity analyses. Each week we read a research paper that uses one of these methods, and we replicate these analyses using the R or Python programming language. Emphasis is on analyzing neural spiking data, although we also discuss other imaging modalities such as magneto/electro-encephalography (EEG/MEG), two-photon imaging, and possibility functional magnetic resonance imaging data (fMRI). Data we analyze includes smaller datasets, such as single neuron recordings from songbird vocal motor system, as well as larger data sets, such as the Allen Brain observatory’s simultaneous recordings from the mouse visual system.
TTh 2:35pm-3:50pm

S&DS 6000a, Advanced Probability  Shuangping Li

Measure theoretic probability, conditioning, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, martingales. Some knowledge of real analysis is assumed.
TTh 2:35pm-3:50pm

S&DS 6030a, Advanced Stochastic Processes  Sekhar Tatikonda

Martingales, Brownian motion, stochastic integration, stochastic differential equations, general theory of Markov processes, and stochastic optimal control. Prerequisites: measure theoretic probability at the level of S&DS 6000, stochastic processes at the level of S&DS 5510, and an appreciation for proofs.
TTh 9am-10:15am

S&DS 6050b, Sampling and Optimal Transport  Sinho Chewi

MCMC sampling and variational inference have long been utilized in Bayesian statistics and machine learning; what can we say about the convergence of these methods? Recently, a modern theory has emerged which blends principles from convex optimization with a geometric perspective on the space of probability distributions based on optimal transport. This course provides an introduction to this theory, as well as to related tools used for modern algorithmic analysis: Markov semigroup theory and stochastic calculus, coupling, and functional inequalities. Much of the course focuses on the complexity of log-concave sampling, but we also discuss applications to diffusion models and variational inference. Prerequisite: Advanced Probability (S&DS 400 / S&DS 600 MATH 330). The following are helpful but not required: Optimization (S&DS 431 / S&DS 631, S&DS 432 / S&DS 632) and Stochastic Processes (S&DS 351 / S&DS 551). Enrollment is limited; requires permission of the instructor.
TTh 1:05pm-2:20pm

S&DS 6100a, Statistical Inference  Theodor Misiakiewicz

A systematic development of the mathematical theory of statistical inference covering methods of estimation, hypothesis testing, and confidence intervals. An introduction to statistical decision theory. Knowledge of probability theory at the level of S&DS 541 is assumed.
TTh 11:35am-12:50pm

S&DS 6110b, Selected Topics in Statistical Decision Theory  Harrison Zhou

Recent developments in statistical decision theory, including nonparametric estimation, high-dimensional (non)linear estimation, low rank and sparse matrices estimation, covariance matrices estimation, graphical models, and network analysis. Prerequisite: S&DS 610.
M 9:25am-11:20am

S&DS 6120a, Linear Models  Robert Wooster

The geometry of least squares; distribution theory for normal errors; regression, analysis of variance, and designed experiments; numerical algorithms (with particular reference to the R statistical language); alternatives to least squares. Prerequisites: linear algebra and some acquaintance with statistics.
MW 2:35pm-3:50pm

S&DS 6140b / PLSC 5030b, Causal Inference  Staff

This course provides an intensive introduction to the statistical theory and practice of causal inference. Topics include: the potential outcomes framework, experimental design, selection on observables, instrumental variables, difference-in-differences, and panel data. Prerequisite: PLSC 500 or equivalent. Students are expected to come in with prior knowledge about statistical inference, regression/linear models, and statistical computing in R.
HTBA

S&DS 6150a, Introduction to Random Matrix Theory and Applications  Zhou Fan

A graduate-level introduction to random matrix theory. Wigner matrices, sample covariance matrices, spiked models. Applications to statistical principal component analysis, random graphs and networks, and landscape analysis of nonconvex statistical optimization problems. Methods applicable to non-invariant models that commonly arise in statistical applications: moment method, resolvents and Stieltjes transforms, free probability, concentration of measure, Lindeberg exchange. Prerequisite: real analysis and measure-theoretic probability.
W 4pm-6:30pm

S&DS 6165a, Topics in Causal Inference  Johan Ugander

This course explores modern topics in causal inference with a focus on randomized experiments and experimental design. The course develops the potential outcomes framework and builds toward advanced topics including covariate balancing, overlap conditions and positivity violations, causal inference under interference, adaptive experimentation, and policy evaluation. The emphasis is on both theoretical foundations and applied practice. Causal inference from observational data is de-emphasized. Students complete presentation and project work including both applied and theoretical replication studies. Prerequisite: Ph.D. standing or permission from instructor.
HTBA

S&DS 6180b, Asymptotic Statistics  Zongming Ma

The course focuses on large sample theory. Time permitting, it extends to topics in high-dimensional statistical inference. It is a natural continuation of S&DS 610, Statistical Inference, in the S&DS Ph.D.-level mathematical statistics sequence. Enrollment is limited. Prerequisites: S&DS 600, 610, and permission of the instructor.
T 9:25am-11:20am

S&DS 6190a, Sequential Decision Making: Theoretical Foundations and Modern Applications  Yuejie Chi

Sequential decision making in the face of uncertainty has garnered growing interest in recent years, with successes in a multitude of applications such as recommendation systems, robotics, and AI systems. This course aims to cover important theoretical and algorithmic foundations under the paradigms of bandits and reinforcement learning (RL) and discuss research issues arising from their modern applications in generative AI. We cover multiple important topics including stochastic bandits, adversarial bandits, planning in Markov decision processes, online and offline RL, policy optimization, and multi-agent RL, gravitating our discussions around issues such as sample complexity, computational efficiency, and function approximation. We also illustrate how RL is used to enable reasoning and alignment of foundation models, along with related (open) research questions. Prerequisite: basic fluency in linear algebra, probability, optimization, machine learning, and programming.
T 9:25am-11:20am

S&DS 6250a or b, Statistical Case Studies  Staff

Statistical analysis of a variety of statistical problems using real data. Emphasis on methods of choosing data, acquiring data, assessing data quality, and the issues posed by extremely large data sets. Extensive computations using R. Enrollment limited; requires permission of the instructor.
HTBA

S&DS 6260a or b, Practical Work  Jay Emerson

Individual one-term projects, with students working on studies outside the department, under the guidance of a statistician.
HTBA

S&DS 6270a and S&DS 6280a or b, Statistical Consulting  Jay Emerson

Statistical consulting and collaborative research projects often require statisticians to explore new topics outside their area of expertise. This course exposes students to real problems, requiring them to draw on their expertise in probability, statistics, and data analysis. Students complete the course with individual projects supervised jointly by faculty outside the department and by one of the instructors. Students enroll for both terms (S&DS 627 and 628) and receive one credit at the end of the year. Enrollment limited; requires permission of the instructor.  ½ Course cr per term
F 2:30pm-4:30pm

S&DS 6320b, Advanced Optimization Techniques  Anna Gilbert

This course covers fundamental theory and algorithms in optimization, emphasizing convex optimization. Topics covered include convex analysis; duality and KKT conditions; subgradient methods; interior point methods; semidefinite programming; distributed methods; stochastic gradient methods; robust optimization; and an introduction to nonconvex optimization. Applications from statistics and data science, economics, engineering, and the sciences. Prerequisites: knowledge of linear algebra, such as MATH 222 or MATH 225; multivariate calculus, such as MATH 120; probability, such as S&DS 541; optimization, such as S&DS 631; and comfort with proof-based exposition and problem sets.
TTh 1:05pm-2:20pm

S&DS 6530a, Advanced Markov Chains  Ilias Zadik

In this course, we cover some advanced topics on the theory of Markov chains, with applications in algorithmic counting and sampling. The topics include more standard techniques, like canonical paths and the correlation decay algorithm, to very recent research topics such as spectral independence and stochastic localization. Prerequisites: undergraduate-level understanding of discrete mathematics, linear algebra, and (discrete) stochastic processes. Some background on real analysis and algorithms/computational complexity is beneficial.
W 1:05pm-3:35pm

S&DS 6610b, Data Analysis  Brian Macdonald

By analyzing data sets using the R statistical computing language, a selection of statistical topics are studied: linear and nonlinear models, maximum likelihood, resampling methods, curve estimation, model selection, classification, and clustering. Prerequisite: after or concurrent with S&DS 542.
TTh 2:35pm-3:50pm

S&DS 6630b, Computational Mathematics Situational Awareness and Survival Skills  Roy Lederman

Are you using a computer to analyze data? Will the computer ever finish processing the data? Will the result be junk? Will you recognize that it is junk? We discuss the difference between math on paper and math on a computer and the difference between general programming and implementing mathematics on a computer. We experience benign mathematical operations failing catastrophically without any error message. We experience mathematically equivalent operations taking anywhere between a fraction of a second and a lifetime. We develop situational awareness and survival skills for this harsh environment. We discuss algorithms, complexity, numerical computation, linear algebra, data analysis, programming, and prototyping. Assignments include theory, programming, data analysis, individual work, and collaborative work. We use C (optionally, FORTRAN) and Python. Making mistakes on assignments and respectful class discussions of insights from such mistakes are integral parts of the course. Prerequisites: linear algebra, multivariate calculus, and programming experience (any language). Prior experience with C, FORTRAN, or Python is recommended but not required; students unfamiliar with these languages must be comfortable independently learning them during the course. Limited size. Instructor permission is required.
HTBA

S&DS 6640b, Information Theory  Yihong Wu

Foundations of information theory in communications, statistical inference, statistical mechanics, probability, and algorithmic complexity. Quantities of information and their properties: entropy, conditional entropy, divergence, redundancy, mutual information, channel capacity. Basic theorems of data compression, data summarization, and channel coding. Applications in statistics.
TTh 11:35am-12:50pm

S&DS 6650a or b, Intermediate Machine Learning  Staff

S&DS 365 is a second course in machine learning at the advanced undergraduate or beginning graduate level. The course assumes familiarity with the basic ideas and techniques in machine learning, for example as covered in S&DS 265. The course treats methods together with mathematical frameworks that provide intuition and justifications for how and when the methods work. Assignments give students hands-on experience with machine learning techniques, to build the skills needed to adapt approaches to new problems. Topics include nonparametric regression and classification, kernel methods, risk bounds, nonparametric Bayesian approaches, graphical models, attention and language models, generative models, sparsity and manifolds, and reinforcement learning. Programming is central to the course, and is based on the Python programming language and Jupyter notebooks.
HTBA

S&DS 6690a, Statistical Learning Theory  Omar Montasser

This course covers classical topics and recent advances in statistical learning theory. This includes topics such as PAC learning, VC theory, boosting, and online learning. We explore statistical and computational aspects, with an emphasis on developing a rigorous quantitative understanding of key machine learning concepts. A second aim is to introduce technical tools that help with designing learning algorithms and proving learning guarantees. Prerequisites: Mathematical maturity and comfort with proof-oriented courses. Background in probability (e.g., S&DS 241), machine learning (e.g., S&DS 265), and algorithms (e.g., CPSC 365). Familiarity with basic concepts in computational complexity (e.g., NP-hardness) is helpful but not required.
T 4pm-5:55pm

S&DS 6705a, Bayes Theory, Information Theory, and Neural Networks  Andrew Barron

This course provides the tools with which to demonstrate statistical accuracy and computational feasibility of the training of artificial neural networks. The construction and properties of Bayes procedures are developed; their frequentist accuracy is revealed by information theory and arbitrary sequence regret analysis; Markov chain Monte Carlo techniques and their analysis are reviewed; and a log-concave coupling is demonstrated for the intrinsically multimodal artificial neural net models which establishes rapid computation of Bayes estimates. The approximation capabilities are presented for flexibly trained neural networks that are not available with classical linear models and not available with more rigidly trained gradient methods or neural tangent kernel methods.  The course equips students for research on the statistical and computational foundations of deep learning, to recognize the important open problems in the field, and to appraise neural network training techniques with an appropriate critical perspective. Prerequisite: Prior exposure to mathematical probability and statistics at levels comparable to S&DS 2410 and 2420. This course is suitable for graduate students in Statistics & Data Science and related programs. Enrollment by advanced undergraduates is possible though subject to instructor evaluation for approval.
MW 11:35am-12:50pm

S&DS 6890a, Scientific Machine Learning  Lu Lu

There are two main branches of technical computing: scientific computing and machine learning. Recently, there has been a convergence of the two disciplines in the emerging scientific machine learning (SciML) field. The main objective of this course is to teach theory, algorithms, and implementation of SciML techniques to graduate students. This course entails various methods to solve a broad range of computational problems frequently encountered in solid mechanics, fluid mechanics, nondestructive evaluation of materials, systems biology, chemistry, and non-linear dynamics. The topics in this course cover multi-fidelity learning, physics-informed neural networks, deep neural operators, uncertainty quantification, and parallel computing. Certain materials are discussed through student presentations of selected publications in the area. Students should have prior coursework in advanced calculus, linear algebra, and probability. Having a background in scientific computing, Python, and/or machine learning is helpful but not mandatory. Enrollment is limited; requires permission of the instructor.
Th 4pm-5:55pm

S&DS 6900a or b, Independent Study  Jay Emerson

By arrangement with faculty. Approval of DGS required.
HTBA

S&DS 6980a, Research Seminar in Mathematical Statistics  Harrison Zhou

This course equips students with the skills to critically understand mathematical statistics papers and deliver professional seminar presentations. The instructor selects high-quality papers from The Annals of Statistics, covering diverse topics such as LLMs, diffusion, reinforcement learning, deep learning, and causal inference. Enrollment is limited; requires permission of the instructor.
F 10am-11:50am

S&DS 7000a or b, Departmental Seminar  Staff

Presentations of recent breakthroughs in statistics and data science.  0 Course cr
HTBA