For students seeking a single introductory course in both probability and statistics, we recommend 1.151. For students with some background in probability seeking a single introductory course on statistics, we recommend 6.434, 18.443, or 16.470. Students interested primarily in algorithms used for statistical data analysis will find 15.074, 15.077, and/or 6.867 useful. For students seeking a rigorous foundation in statistical inference we recommend 6.437 or possibly 18.466.

Below we give a list of many of the classes in statistics or areas based in statistics that are available at MIT. Clicking on a class title below will open a description of the class. In addition to all of these classes, MIT students are able to cross-register for classes at Harvard, which has a number of classes in their departments of statistics and biostatistics.

## Course 6 - Electrical Engineering and Computer Science

__6.434J Statistics for Engineers and Scientists__

Provides a rigorous introduction to fundamentals of statistics motivated by engineering applications and emphasizing the informed use of modern statistical software. Topics include sufficient statistics, exponential families, estimation, hypothesis testing, measures of performance, and notion of optimality.

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__6.435 System Identification__ (1 student review)

Mathematical models of systems from observations of their behavior. Time series, state-space, and input-output models. Model structures, parametrization, and identifiability. Non-parametric methods. Prediction error methods for parameter estimation, convergence, consistency, andasymptotic distribution. Relations to maximum likelihood estimation. Recursive estimation; relation to Kalman filters; structure determination; order estimation; Akaike criterion; and bounded but unknown noise models. Robustness and practical issues.

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__6.437 Inference and Information__ (2 student reviews)

Introduction to principles of Bayesian and non-Bayesian statistical inference. Hypothesis testing and parameter estimation, sufficient statistics; exponential families. Log-loss inference criterion, entropy and model capacity. Kullback-Leibler distance and information geometry. Asymptotic analysis and large deviations theory. Model order estimation; nonparametric statistics. Computational issues and approximation techniques; Monte Carlo methods. Selected special topics such as universal prediction and compression.

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__6.438 Algorithms for Inference__

Introduction to statistical inference with probabilistic graphical models. Covers directed and undirected graphical models, factor graphs, and Gaussian models; hidden Markov models, linear dynamical systems; sum-product and junction tree algorithms; forward-backward algorithm, Kalman filtering and smoothing; and min-sum algorithm and Viterbi algorithm. Presents variational methods, mean-field theory, and loopy belief propagation; and particle methods and filtering. Includes building graphical models from data; parameter estimation, Baum-Welch algorithm; structure learning; and selected special topics.

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__6.867 Machine Learning__ (2 student reviews)

Principles, techniques, and algorithms in machine learning from the point of view of statistical inference; representation, generalization, and model selection; and methods such as linear/additive models, active learning, boosting, support vector machines, hidden Markov models, and Bayesian networks.

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__6.869 Advances in Computer Vision__

Advanced topics in computer vision with a focus on the use of machine learning techniques and applications in graphics and human-computer interface. Topics include image representations, texture models, structure-from-motion algorithms, Bayesian techniques, object and scene recognition, tracking, shape modeling, and image databases. Applications may include face recognition, multimodal interaction, interactive systems, cinematic special effects, and photorealistic rendering. Covers topics complementary to 6.801/6.866; these subjects may be taken in sequence.

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__6.870 Advanced Topics in Computer Vision__

Seminar exploring advanced research topics in the field of computer vision; focus varies with lecturer. Typically structured around discussion of assigned research papers and presentations by students. Example research areas explored in this seminar include learning in vision, computational imaging techniques, multimodal human-computer interaction, biomedical imaging, representation and estimation methods used in modern computer vision.

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## Course 15 - Management Science

__15.034 Metrics for Managers: Big Data and Better Answers__

Enables students to understand and conduct careful empirical work using regression analysis as used in business fields such as finance, marketing and strategy, as well as in general business planning and forecasting. Emphasizes model formulation, intuition, and critical evaluation of results. Learning is primarily through empirical work done by student groups; delivered through problem sets, short write-ups, presentations and debates.

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__15.062J Data Mining: Finding the Data and Models that Create Value__

Provides an introduction to data mining and machine learning, a class of methods that that assist in recognizing patterns and making intelligent use of massive amounts of data collected via the internet, e-commerce, electronic banking, point-of-sale devices, bar-code readers, medical databases, search engines, and social networks. Topics selected from logistic regression, association rules, tree-structured classification and regression, cluster analysis, discriminant analysis, and neural network methods. Presents examples of successful applications in areas such as credit ratings, fraud detection, marketing, customer relationship management, and investments. Introduces data-mining software.

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__15.068 Statistical Consulting__

Addresses statistical issues as a consultant would face them: deciphering the client's question; finding appropriate data; performing a viable analysis; and presenting the results in compelling ways. Real-life cases and examples.

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__15.074J Statistical Reasoning and Data Modeling__ (1 student review)

Designed for students who have some acquaintance with probability and/or statistics and want exposure to a wider range of topics and examples. Begins with a brief review of statistics and regression by addressing advanced topics, such as bootstrap resampling, variable selection, data and regression diagnostics, visualization, and Bayesian and robust methods. Goes on to cover data-mining and machine learning, including classification, logistic regression, and clustering. Culminates with time series analysis and forecasting, design of experiments and analysis of variance, and process control. Students use statistical computing systems based on Excel add-ins and stand-alone packages. Includes case studies involving finance, management science, consulting, risk management, and engineering systems.

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__15.077J Statistical Learning and Data Mining__ (1 student review)

Advanced introduction to the theory and application of statistics, data-mining, and machine learning, concentrating on techniques used in management science, finance, consulting, engineering systems, and bioinformatics. First half builds the statistical foundation for the second half, with topics selected from sampling, including the bootstrap, theory of estimation, testing, nonparametric statistics, analysis of variance, categorical data analysis, regression analysis, MCMC, EM, Gibbs sampling, and Bayesian methods. Second half focuses on data-mining, supervised learning, and multivariate analysis. Topics selected from logistic regression; principal components and dimension reduction; discrimination and classification analysis, including trees (CART), partial least squares, nearest neighbor and regularized methods, support vector machines, boosting and bagging, clustering, independent component analysis, and nonparametric regression. Uses statistics software packages, such as R and MATLAB for data analysis and data-mining.

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__15.097 Seminar in Operations Research and Statistics__

Group study of current topics related to operations research/statistics.

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__15.450 Analytics of Finance__

Covers several key quantitative methods of finance, including financial econometrics, Monte Carlo simulation, stochastic (Ito) calculus, and dynamic optimization. Covers these techniques, along with their computer implementation, in depth. Application areas include quantitative portfolio management, risk management, derivative pricing and hedging, and proprietary trading.

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__15.460 Analytics of Finance II__

Covers the practical aspects of the analytics of finance from the perspective of a quantitative investment manager. Develops an understanding of stochastic processes, option pricing, investment strategies, backtest simulation, data and computational architecture, portfolio construction, trading implementation, and risk management within the context of a specific quantitative equity trading strategy. Follows the natural sequence of research, development, testing, and implementation. Emphasizes financial applications, but also covers mathematical and statistical techniques in some depth, along with their computational implementation in software and the use of real-world market data.

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## Course 18 - Mathematics

__18.338 Eigenvalues of Random Matrices__

Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of MATLAB hepful, but not required.

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__18.443 Statistics for Applications__

A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.

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__18.465 Topics in Statistics__

Topics vary from term to term.

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__18.466 Mathematical Statistics__ (1 student review)

Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.

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## Course 14 - Economics

__14.381 Statistical Method in Economics__

Introduction to probability and statistics as background for advanced econometrics and introduction to the linear regression model. Covers elements of probability theory; sampling theory; asymptotic approximations; decision-theory approach to statistical estimation focusing on regression, hypothesis testing; and maximum-likelihood methods. Includes simple and multiple regression, estimation and hypothesis testing. Illustrations from economics and application of these concepts to economic problems. Enrollment limited.

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__14.382 Econometrics__

Regression analysis, focusing on departures from the standard Gauss-Markov assumptions, and simultaneous equations. Regression topics include heteroskedasticity, serial correlation, and errors in variables, generalized least squares, nonlinear regression, and limited dependent variable models. Covers identification and estimation of linear and nonlinear simultaneous equations models. Economic applications are discussed. Enrollment limited.

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__14.384 Time Series Analysis__ (1 student review)

Studies theory and application of time series methods in econometrics, including spectral analysis, estimation with stationary and non-stationary processes, VARs, factor models, unit roots, cointegration, estimation of DSGE models, and Bayesian methods.

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__14.385 Nonlinear Econometric Analysis__

Studies micro-econometric models, including large sample theory for estimation and hypothesis testing, generalized method of moments, estimation of censored and truncated specifications, quantile regression, structural estimation, nonparametric and semiparametric estimation, panel data, bootstrapping, and simulation methods. Methods illustrated with economic applications.

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__14.386 New Econometric Methods__

Focuses on recent developments in econometrics, especially structural estimation. Topics include nonseparable models, models of imperfect competition, auction models, duration models, and nonlinear panel data. Results illustrated with economic applications.

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__14.387 Topics in Applied Econometrics__

Covers core econometric ideas and widely used empirical modeling strategies. Topics vary from year to year, but course typically begins with instrumental variables, concepts, and methods; then moves on to discussion of differences-in-differences and regression discontinuity methods. Concludes with discussion of standard errors, focusing on issues such as clustering and serial correlation.

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## Course 9 - Brain and Cognitive Sciences

__9.073J Statistics for Neuroscience Research__

A survey of statistical methods for neuroscience research. Core topics include introductions to the theory of point processes, the generalized linear model, Monte Carlo methods, Bayesian methods, multivariate methods, time-series analysis, spectral analysis and state-space modeling. Emphasis on developing a firm conceptual understanding of the statistical paradigm and statistical methods primarily through analyses of actual experimental data.

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__9.272J Topics in Neural Signal Processing__

Presents signal processing and statistical methods used to study neural systems and analyze neurophysiological data. Topics include state-space modeling formulated using the Bayesian Chapman-Kolmogorov system, theory of point processes, EM algorithm, Bayesian and sequential Monte Carlo methods. Applications include dynamic analyses of neural encoding, neural spike train decoding, studies of neural receptive field plasticity, algorithms for neural prosthetic control, EEG and MEG source localization. Students should know introductory probability theory and statistics.

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__9.520 Statistical Learning Theory and Applications__

Focuses on the problem of supervised and unsupervised learning from the perspective of modern statistical learning theory, starting with the theory of multivariate function approximation from sparse data. Develops basic tools such as regularization, including support vector machines for regression and classification. Derives generalization bounds using stability. Discusses current research topics such as manifold regularization, sparsity, feature selection, bayesian connections and techniques. Discusses applications in areas such as computer vision, speech recognition, and bioinformatics. Also covers advances in the neuroscience of the cortex and their impact on learning theory and applications. Includes a final project.

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## Course 1 - Civil Engineering

__1.151 Probability and Statistics in Engineering__

Quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, statistics, and decision analysis. Events and their probability, Total Probability and Bayes' Theorems. Random variables and vectors. Bernoulli Trial Sequence and Poisson Point Process models. Conditional distributions and distribution of functions of random variables and vectors. Probabilistic moments, second-moment uncertainty propagation and best linear unbiased estimation theory for variables and vectors. Introduction to system reliability. Estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation) and simple and multiple linear regression. Emphasis on application to engineering problems.

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__1.202J Demand Modeling__

Theory and application of modeling and statistical methods for analysis and forecasting of demand for facilities, services, and products. Topics include: review of probability and statistics, estimation and testing of linear regression models, theory of individual choice behavior, derivation, estimation, and testing of discrete choice models (including logit, nested logit, GEV, probit, and mixture models), estimation under various sample designs and data collection methods (including revealed and stated preferences), sampling, aggregate forecasting methods, and iterative proportional fitting and related methods. Lectures reinforced with case studies, which require specification, estimation, testing, and analysis of models using data sets from actual applications.

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## Course 12 - Earth, Atmospheric and Planetary Sciences

__12.515 Data and Models__

Surveys a number of methods of inverting data to obtain model parameter estimates. Topics include review of matrix theory and statistics, random and grid-search methods, linear and non-linear least squares, maximum-likelihood estimation, ridge regression, stochastic inversion, sequential estimation, singular value decomposition, solution of large systems, genetic and simulated annealing inversion, regularization, parameter error estimates, and solution uniqueness and resolution. Computer laboratory and algorithm development.

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__12.714 Computational Data Analysis__

An introduction to the theory and practice of analyzing discrete data such as are normally encountered in geophysics and geology. Emphasizes statistical aspects of data interpretation and the nonparametric discrete-time approach to spectral analysis. Topics include: elements of probability and statistics, statistical inference, robust and nonparametric statistics, the method of least squares, univariate and multivariate spectral analysis, digital filters, and aspects of multidimensional data analysis.

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## Course 16 - Aeronautics and Astronautics

__16.470J Statistical Methods in Experimental Design__ (1 student review)

Statistically based experimental design inclusive of forming hypotheses, planning and conducting experiments, analyzing data, and interpreting and communicating results. Topics include descriptive statistics, statistical inference, hypothesis testing, parametric and nonparametric statistical analyses, factorial ANOVA, randomized block designs, MANOVA, linear regression, repeated measures models, and application of statistical software packages.

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## Course 22 - Nuclear Science and Engineering

__22.38 Probability and Its Applications To Reliability, Quality Control, and Risk Assessment__ (1 student review)

Interpretations of the concept of probability. Basic probability rules; random variables and distribution functions; functions of random variables. Applications to quality control and the reliability assessment of mechanical/electrical components, as well as simple structures and redundant systems. Elements of statistics. Bayesian methods in engineering. Methods for reliability and risk assessment of complex systems, (event-tree and fault-tree analysis, common-cause failures, human reliability models). Uncertainty propagation in complex systems (Monte Carlo methods, Latin hypercube sampling). Introduction to Markov models. Examples and applications from nuclear and other industries, waste repositories, and mechanical systems.

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## Course 7 - Biology

__7.410 Applied Statistics__

Provides an introduction to modern applied statistics. Topics include likelihood-based methods for estimation, confidence intervals, and hypothesis-testing; bootstrapping; time series modeling; linear models; nonparametric regression; and model selection. Organized around examples drawn from the recent literature.

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## Engineering Systems Division

__ESD.86J Models, Data and Inference for Socio-Technical Systems__ (1 student review)

Use data and systems knowledge to build models of complex socio-technical systems for improved system design and decision-making. Enhance model-building skills, including: review and extension of functions of random variables, Poisson processes, and Markov processes. Move from applied probability to statistics via Chi-squared t and f tests, derived as functions of random variables. Review classical statistics, hypothesis tests, regression, correlation and causation, simple data mining techniques, and Bayesian vs. classical statistics. Class project.

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