Signal Processing
6.3000 Signal Processing
 ( ,  ) 
Prereq: 6.100A and 18.03
Units: 6-0-6
Lecture: TR2 (32-141) Lab: TR3 (32-141) +final
Fundamentals of signal processing, focusing on the use of Fourier methods to analyze and process signals such as sounds and images. Topics include Fourier series, Fourier transforms, the Discrete Fourier Transform, sampling, convolution, deconvolution, filtering, noise reduction, and compression. Applications draw broadly from areas of contemporary interest with emphasis on both analysis and design.
Fall: D. Freeman
Spring: S. You
No textbook information available
6.3010 Signals, Systems and Inference
()
Prereq: 6.3000 and (6.3700, 6.3800, or 18.05)
Units: 4-0-8
Lecture: MW3 (32-141) Recitation: TR1 (34-301) or TR2 (34-301)
Covers signals, systems and inference in communication, control and signal processing. Topics include input-output and state-space models of linear systems driven by deterministic and random signals; time- and transform-domain representations in discrete and continuous time; and group delay. State feedback and observers. Probabilistic models; stochastic processes, correlation functions, power spectra, spectral factorization. Least-mean square error estimation; Wiener filtering. Hypothesis testing; detection; matched filters.
L. Zheng
No textbook information available
6.3020[J] Fundamentals of Music Processing
(New number 6.187)
() 
(Same subject as 21M.387[J])
(Subject meets with 21M.587)
Prereq: 6.3000 and 21M.051
Units: 3-0-9
Analyzes recorded music in digital audio form using advanced signal processing and optimization techniques to understand higher-level musical meaning. Covers fundamental tools like windowing, feature extraction, discrete and short-time Fourier transforms, chromagrams, and onset detection. Addresses analysis methods including dynamic time warping, dynamic programming, self-similarity matrices, and matrix factorization. Explores a variety of applications, such as event classification, audio alignment, chord recognition, structural analysis, tempo and beat tracking, content-based audio retrieval, and audio decomposition. Students taking graduate version complete different assignments.
E. Egozy
6.7000 Discrete-Time Signal Processing
  ()
Prereq: 6.3010
Units: 4-0-8
Representation, analysis, and design of discrete time signals and systems. Decimation, interpolation, and sampling rate conversion. Noise shaping. Flowgraph structures for DT systems. IIR and FIR filter design techniques. Parametric signal modeling, linear prediction, and lattice filters. Discrete Fourier transform, DFT computation, and FFT algorithms. Spectral analysis, time-frequency analysis, relation to filter banks. Multirate signal processing, perfect reconstruction filter banks, and connection to wavelets.
A. V. Oppenheim, J. Ward
6.7010 Digital Image Processing
()
Prereq: 6.3000 and 6.3700
Units: 3-0-9
 Subject Cancelled
Introduces models, theories, and algorithms key to digital image processing. Core topics covered include models of image formation, image processing fundamentals, filtering in the spatial and frequency domains, image transforms, and feature extraction. Additional topics include image enhancement, image restoration and reconstruction, compression of images and videos, visual recognition, and the application of machine learning-based approaches to image processing. Includes student-driven term project.
Y. Rachlin, J. S. Lim
6.7020 Array Processing
 ()
Prereq: 6.7000 and (2.687 or (6.3010 and 18.06))
Units: 3-2-7
Adaptive and non-adaptive processing of signals received at arrays of sensors. Deterministic beamforming, space-time random processes, optimal and adaptive algorithms, and the sensitivity of algorithm performance to modeling errors and limited data. Methods of improving the robustness of algorithms to modeling errors and limited data are derived. Advanced topics include an introduction to matched field processing and physics-based methods of estimating signal statistics. Homework exercises providing the opportunity to implement and analyze the performance of algorithms in processing data supplied during the course.
Staff
Control
6.3100 Dynamical System Modeling and Control Design
(, ) 
(Subject meets with 6.3102)
Prereq: Physics II (GIR) and (18.06 or 18.C06)
Units: 4-4-4
Lecture: MW3 (4-163) Lab: F10-1 (38-545) or F2-5 (38-545)
A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression, and identification). Concepts are introduced with lectures and online problems, and then mastered during weekly labs. In lab, students model, design, test, and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g., optimizing thrust-driven positioners or stabilizing magnetic levitators). Students taking graduate version complete additional problems and labs.
Fall: K. Chen, J. K. White
Spring: Y. Chen
No textbook information available
6.3102 Dynamical System Modeling and Control Design
(, )
(Subject meets with 6.3100)
Prereq: Physics II (GIR) and (18.06 or 18.C06)
Units: 4-4-4
Lecture: MW3 (4-163) Lab: F10-1 (38-545) or F2-5 (38-545)
A learn-by-design introduction to modeling and control of discrete- and continuous-time systems, from intuition-building analytical techniques to more computational and data-centric strategies. Topics include: linear difference/differential equations (natural frequencies, transfer functions); controller metrics (stability, tracking, disturbance rejection); analytical techniques (PID, root-loci, lead-lag, phase margin); computational strategies (state-space, eigen-placement, LQR); and data-centric approaches (state estimation, regression and identification). Concepts are introduced with lectures and on-line problems, and then mastered during weekly labs. In lab, students model, design, test and explain systems and controllers involving sensors, actuators, and a microcontroller (e.g. optimizing thrust-driven positioners or stabilizing magnetic levitators). Students in the graduate version complete additional problems and labs.
K. Chen, J. K. White
No textbook information available
6.7100[J] Dynamic Systems and Control
()
(Same subject as 16.338[J])
Prereq: 6.3000 and 18.06
Units: 4-0-8
Linear, discrete- and continuous-time, multi-input-output systems in control, related areas. Least squares and matrix perturbation problems. State-space models, modes, stability, controllability, observability, transfer function matrices, poles and zeros, and minimality. Internal stability of interconnected systems, feedback compensators, state feedback, optimal regulation, observers, and observer-based compensators. Measures of control performance, robustness issues using singular values of transfer functions. Introductory ideas on nonlinear systems. Recommended prerequisite: 6.3100.
Staff
6.7110 Multivariable Control Systems
()
Not offered regularly; consult department
Prereq: 6.7100 or 16.31
Units: 3-0-9
Computer-aided design methodologies for synthesis of multivariable feedback control systems. Performance and robustness trade-offs. Model-based compensators; Q-parameterization; ill-posed optimization problems; dynamic augmentation; linear-quadratic optimization of controllers; H-infinity controller design; Mu-synthesis; model and compensator simplification; nonlinear effects. Computer-aided (MATLAB) design homework using models of physical processes.
A. Megretski
6.7120 Principles of Modeling, Computing and Control for Decarbonized Electric Energy Systems
(New)
()
(Subject meets with 6.7121)
Prereq: 6.2200, (6.2000 and 6.3100), or permission of instructor
Units: 4-0-8
Introduces fundamentals of electric energy systems as complex dynamical network systems. Topics include coordinated and distributed modeling and control methods for efficient and reliable power generation, delivery, and consumption; data-enabled algorithms for integrating clean intermittent resources, storage, and flexible demand, including electric vehicles; examples of network congestion management, frequency, and voltage control in electrical grids at various scales; and design and operation of supporting markets. Students taking graduate version complete additional assignments.
M. Ilic
6.7121 Principles of Modeling, Computing and Control for Decarbonized Electric Energy Systems
(New)
()
(Subject meets with 6.7120)
Prereq: 6.2200, (6.2000 and 6.3100), or permission of instructor
Units: 4-0-8
Introduces fundamentals of electric energy systems as complex dynamical network systems. Topics include coordinated and distributed modeling and control methods for efficient and reliable power generation, delivery, and consumption; data-enabled algorithms for integrating clean intermittent resources, storage, and flexible demand, including electric vehicles; examples of network congestion management, frequency, and voltage control in electrical grids at various scales; and design and operation of supporting markets. Students taking graduate version complete additional assignments.
M. Ilic
Optimization & Engineering Mathematics
6.3260[J] Networks
() 
(Same subject as 14.15[J])
(Subject meets with 14.150)
Prereq: 6.3700 or 14.30
Units: 4-0-8
Lecture: MW2.30-4 (54-100) Recitation: F3 (E25-111) +final
Highlights common principles that permeate the functioning of diverse technological, economic and social networks. Utilizes three sets of tools for analyzing networks -- random graph models, optimization, and game theory -- to study informational and learning cascades; economic and financial networks; social influence networks; formation of social groups; communication networks and the Internet; consensus and gossiping; spread and control of epidemics; control and use of energy networks; and biological networks. Students taking graduate version complete additional assignments.
A. Wolitzky
Textbooks (Spring 2024)
6.7200[J] Optimization Methods
()
(Same subject as 15.093[J], IDS.200[J])
(Subject meets with 6.7201)
Prereq: 18.06
Units: 4-0-8
Introduces the principal algorithms for linear, network, discrete, robust, nonlinear, and dynamic optimization. Emphasizes methodology and the underlying mathematical structures. Topics include the simplex method, network flow methods, branch and bound and cutting plane methods for discrete optimization, optimality conditions for nonlinear optimization, interior point methods for convex optimization, Newton's method, heuristic methods, and dynamic programming and optimal control methods. Expectations and evaluation criteria differ for students taking graduate version; consult syllabus or instructor for specific details.
A. Jacquillat, D. Bertsimas
6.7201 Optimization Methods
()
(Subject meets with 6.7200[J], 15.093[J], IDS.200[J])
Prereq: 18.06
Units: 4-0-8
Introduces the principal algorithms for linear, network, discrete, robust, nonlinear, and dynamic optimization. Emphasizes methodology and the underlying mathematical structures. Topics include the simplex method, network flow methods, branch and bound and cutting plane methods for discrete optimization, optimality conditions for nonlinear optimization, interior point methods for convex optimization, Newton's method, heuristic methods, and dynamic programming and optimal control methods. Expectations and evaluation criteria differ for students taking graduate version; consult syllabus or instructor for specific details.
A. Jacquillat, D. Bertsimas
6.7210[J] Introduction to Mathematical Programming
()
(Same subject as 15.081[J])
Prereq: 18.06
Units: 4-0-8
Introduction to linear optimization and its extensions emphasizing both methodology and the underlying mathematical structures and geometrical ideas. Covers classical theory of linear programming as well as some recent advances in the field. Topics: simplex method; duality theory; sensitivity analysis; network flow problems; decomposition; robust optimization; integer programming; interior point algorithms for linear programming; and introduction to combinatorial optimization and NP-completeness.
S. Gupta
6.7220[J] Nonlinear Optimization
()
(Same subject as 15.084[J])
Prereq: 18.06 and (18.100A, 18.100B, or 18.100Q)
Units: 4-0-8
Lecture: TR11-12.30 (E25-111) Recitation: F10 (E51-057) or F11 (E51-057) +final
Unified analytical and computational approach to nonlinear optimization problems. Unconstrained optimization methods include gradient, conjugate direction, Newton, sub-gradient and first-order methods. Constrained optimization methods include feasible directions, projection, interior point methods, and Lagrange multiplier methods. Convex analysis, Lagrangian relaxation, nondifferentiable optimization, and applications in integer programming. Comprehensive treatment of optimality conditions and Lagrange multipliers. Geometric approach to duality theory. Applications drawn from control, communications, machine learning, and resource allocation problems.
G. Farina
No required or recommended textbooks
6.7230[J] Algebraic Techniques and Semidefinite Optimization
()
(Same subject as 18.456[J])
Prereq: 6.7210 or 15.093
Units: 3-0-9
Lecture: WF1-2.30 (36-153)
Theory and computational techniques for optimization problems involving polynomial equations and inequalities with particular, emphasis on the connections with semidefinite optimization. Develops algebraic and numerical approaches of general applicability, with a view towards methods that simultaneously incorporate both elements, stressing convexity-based ideas, complexity results, and efficient implementations. Examples from several engineering areas, in particular systems and control applications. Topics include semidefinite programming, resultants/discriminants, hyperbolic polynomials, Groebner bases, quantifier elimination, and sum of squares.
P. Parrilo
No textbook information available
6.7240 Game Theory with Engineering Applications
()
Not offered regularly; consult department
Prereq: 6.3702
Units: 4-0-8
Introduction to fundamentals of game theory and mechanism design with motivations for each topic drawn from engineering applications (including distributed control of wireline/wireless communication networks, transportation networks, pricing). Emphasis on the foundations of the theory, mathematical tools, as well as modeling and the equilibrium notion in different environments. Topics include normal form games, supermodular games, dynamic games, repeated games, games with incomplete/imperfect information, mechanism design, cooperative game theory, and network games.
A. Ozdaglar
6.7250 Optimization for Machine Learning
()
Not offered regularly; consult department
Prereq: 6.3900 and 18.06
Units: 3-0-9
Subject Cancelled
Optimization algorithms are central to all of machine learning. Covers a variety of topics in optimization, with a focus on non-convex optimization. Focuses on both classical and cutting-edge results, including foundational topics grounded in convexity, complexity theory of first-order methods, stochastic optimization, as well as recent progress in non-Euclidean optimization, deep learning, and beyond. Prepares students to appreciate a broad spectrum of ideas in OPTML, learning to be not only informed users but also gaining exposure to research questions in the area.
Staff
6.7260 Network Science and Models
()
Prereq: 6.3702 and 18.06
Units: 3-0-9
Introduces the main mathematical models used to describe large networks and dynamical processes that evolve on networks. Static models of random graphs, preferential attachment, and other graph evolution models. Epidemic propagation, opinion dynamics, social learning, and inference in networks. Applications drawn from social, economic, natural, and infrastructure networks, as well as networked decision systems such as sensor networks.
P. Jaillet, J. N. Tsitsiklis
6.7300[J] Introduction to Modeling and Simulation
()
(Same subject as 2.096[J], 16.910[J])
Prereq: 18.03 or 18.06
Units: 3-6-3
Introduction to computational techniques for modeling and simulation of a variety of large and complex engineering, science, and socio-economical systems. Prepares students for practical use and development of computational engineering in their own research and future work. Topics include mathematical formulations (e.g., automatic assembly of constitutive and conservation principles); linear system solvers (sparse and iterative); nonlinear solvers (Newton and homotopy); ordinary, time-periodic and partial differential equation solvers; and model order reduction. Students develop their own models and simulators for self-proposed applications, with an emphasis on creativity, teamwork, and communication. Prior basic linear algebra required and at least one numerical programming language (e.g., MATLAB, Julia, Python, etc.) helpful.
L. Daniel
6.7310[J] Introduction to Numerical Methods
()
(Same subject as 18.335[J])
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
Lecture: MW11-12.30 (45-230)
Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming.
J. Urschel
Textbooks (Spring 2024)
6.7320[J] Parallel Computing and Scientific Machine Learning
()
(Same subject as 18.337[J])
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
Introduction to scientific machine learning with an emphasis on developing scalable differentiable programs. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Provides direct experience with the modern realities of optimizing code performance for supercomputers, GPUs, and multicores in a high-level language.
A. Edelman
6.7330[J] Numerical Methods for Partial Differential Equations
()
(Same subject as 2.097[J], 16.920[J])
Prereq: 18.03 or 18.06
Units: 3-0-9
Covers the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential and integral equations. Topics include mathematical formulations; finite difference, finite volume, finite element, and boundary element discretization methods; and direct and iterative solution techniques. The methodologies described form the foundation for computational approaches to engineering systems involving heat transfer, solid mechanics, fluid dynamics, and electromagnetics. Computer assignments requiring programming.
J. Peraire
6.7340[J] Fast Methods for Partial Differential and Integral Equations
(, )
(Same subject as 18.336[J])
Prereq: 6.7300, 16.920, 18.085, 18.335, or permission of instructor
Units: 3-0-9
Lecture: MW9.30-11 (2-136)
Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.
Fall: A. Horning
Spring: A. Horning
No required or recommended textbooks
Communications
6.3400 Introduction to EECS via Communication Networks
()
Not offered regularly; consult department
Prereq: 6.100A
Units: 4-4-4
Studies key concepts, systems, and algorithms to reliably communicate data in settings ranging from the cellular phone network and the Internet to deep space. Weekly laboratory experiments explore these areas in depth. Topics presented in three modules - bits, signals, and packets - spanning the multiple layers of a communication system. Bits module includes information, entropy, data compression algorithms, and error correction with block and convolutional codes. Signals module includes modeling physical channels and noise, signal design, filtering and detection, modulation, and frequency-division multiplexing. Packets module includes switching and queuing principles, media access control, routing protocols, and data transport protocols.
Staff
6.7410 Principles of Digital Communication
()
(Subject meets with 6.7411)
Prereq: (6.3000 or 6.3102) and (6.3700, 6.3800, or 18.05)
Units: 3-0-9
Covers communications by progressing through signal representation, sampling, quantization, compression, modulation, coding and decoding, medium access control, and queueing and principles of protocols. By providing simplified proofs, seeks to present an integrated, systems-level view of networking and communications while laying the foundations of analysis and design. Lectures are offered online; in-class time is dedicated to recitations, exercises, and weekly group labs. Homework exercises are based on theoretical derivation and software implementation. Students taking graduate version complete additional assignments.
V. Chan
6.7411 Principles of Digital Communication
()
(Subject meets with 6.7410)
Prereq: (6.3000, 6.3100, or 6.3400) and (6.3700, 6.3800, or 18.05)
Units: 3-0-9
Covers communications by progressing through signal representation, sampling, quantization, compression, modulation, coding and decoding, medium access control, and queueing and principles of protocols. By providing simplified proofs, seeks to present an integrated, systems-level view of networking and communications while laying the foundations of analysis and design. Lectures are offered online; in-class time is dedicated to recitations, exercises, and weekly group labs. Homework exercises are based on theoretical derivation and software implementation. Students taking graduate version complete additional assignments.
V. Chan
6.7420 Heterogeneous Networks: Architecture, Transport, Proctocols, and Management
()
Prereq: 6.1200 or 6.3700
Units: 4-0-8
Introduction to modern heterogeneous networks and the provision of heterogeneous services. Architectural principles, analysis, algorithmic techniques, performance analysis, and existing designs are developed and applied to understand current problems in network design and architecture. Begins with basic principles of networking. Emphasizes development of mathematical and algorithmic tools; applies them to understanding network layer design from the performance and scalability viewpoint. Concludes with network management and control, including the architecture and performance analysis of interconnected heterogeneous networks. Provides background and insight to understand current network literature and to perform research on networks with the aid of network design projects.
Staff
6.7430 Optical Networks
()
Not offered regularly; consult department
Prereq: 6.1200 or 6.3700
Units: 3-0-9
Introduces the fundamental and practical aspects of optical network technology, architecture, design and analysis tools and techniques. The treatment of optical networks are from the architecture and system design points of view. Optical hardware technologies are introduced and characterized as fundamental network building blocks on which optical transmission systems and network architectures are based. Beyond the Physical Layer, the higher network layers (Media Access Control, Network and Transport Layers) are treated together as integral parts of network design. Performance metrics, analysis and optimization techniques are developed to help guide the creation of high performance complex optical networks.
Staff
6.7440 Principles of Wireless Communication
()
Not offered regularly; consult department
Prereq: 6.7410
Units: 3-0-9
Introduction to design, analysis, and fundamental limits of wireless transmission systems. Wireless channel and system models; fading and diversity; resource management and power control; multiple-antenna and MIMO systems; space-time codes and decoding algorithms; multiple-access techniques and multiuser detection; broadcast codes and precoding; cellular and ad-hoc network topologies; OFDM and ultrawideband systems; architectural issues.
G. W. Wornell, L. Zheng
6.7450[J] Data-Communication Networks
()
(Same subject as 16.37[J])
Prereq: 6.3700 or 18.204
Units: 3-0-9
Provides an introduction to data networks with an analytic perspective, using wireless networks, satellite networks, optical networks, the internet and data centers as primary applications. Presents basic tools for modeling and performance analysis. Draws upon concepts from stochastic processes, queuing theory, and optimization.
Staff
6.7460 Essential Coding Theory
()
Not offered regularly; consult department
Prereq: 6.1210 and 6.1400
Units: 3-0-9
Introduces the theory of error-correcting codes. Focuses on the essential results in the area, taught from first principles. Special focus on results of asymptotic or algorithmic significance. Principal topics include construction and existence results for error-correcting codes; limitations on the combinatorial performance of error-correcting codes; decoding algorithms; and applications to other areas of mathematics and computer science.
Staff
6.7470 Information Theory
()
Prereq: 6.3700
Units: 3-0-9
Mathematical definitions of information measures, convexity, continuity, and variational properties. Lossless source coding; variable-length and block compression; Slepian-Wolf theorem; ergodic sources and Shannon-McMillan theorem. Hypothesis testing, large deviations and I-projection. Fundamental limits of block coding for noisy channels: capacity, dispersion, finite blocklength bounds. Coding with feedback. Joint source-channel problem. Rate-distortion theory, vector quantizers. Advanced topics include Gelfand-Pinsker problem, multiple access channels, broadcast channels (depending on available time).
Staff
Probability & Statistics
6.3700 Introduction to Probability
(, )
(Subject meets with 6.3702)
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 18.600
Lecture: MW2 (4-270) Recitation: TR1 (45-102) or TR2 (45-102) +final
An introduction to probability theory, the modeling and analysis of probabilistic systems, and elements of statistical inference. Probabilistic models, conditional probability. Discrete and continuous random variables. Expectation and conditional expectation, and further topics about random variables. Limit Theorems. Bayesian estimation and hypothesis testing. Elements of classical statistical inference. Bernoulli and Poisson processes. Markov chains. Students taking graduate version complete additional assignments.
Fall: L. Zheng
Spring: P. Jaillet
Textbooks (Spring 2024)
6.3702 Introduction to Probability
(, )
(Subject meets with 6.3700)
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 18.600
Lecture: MW2 (4-270) Recitation: TR1 (45-102) or TR2 (45-102) +final
An introduction to probability theory, the modeling and analysis of probabilistic systems, and elements of statistical inference. Probabilistic models, conditional probability. Discrete and continuous random variables. Expectation and conditional expectation, and further topics about random variables. Limit Theorems. Bayesian estimation and hypothesis testing. Elements of classical statistical inference. Bernoulli and Poisson processes. Markov chains. Students taking graduate version complete additional assignments.
Fall: L. Zheng
Spring: P. Jaillet
Textbooks (Spring 2024)
6.3720 Introduction to Statistical Data Analysis
()
(Subject meets with 6.3722)
Prereq: 6.100A and (6.3700, 6.3800, or 18.600)
Units: 4-0-8
Lecture: MW2.30-4 (4-370) Recitation: F12 (4-270) +final
Introduction to the central concepts and methods of data science with an emphasis on statistical grounding and modern computational capabilities. Covers principles involved in extracting information from data for the purpose of making predictions or decisions, including data exploration, feature selection, model fitting, and performance assessment. Topics include learning of distributions, hypothesis testing (including multiple comparison procedures), linear and nonlinear regression and prediction, classification, time series, uncertainty quantification, model validation, causal inference, optimization, and decisions. Computational case studies and projects drawn from applications in finance, sports, engineering, and machine learning life sciences. Students taking graduate version complete additional assignments. Recommended prerequisite: 18.06.
Y. Polyanskiy
No textbook information available
6.3722 Introduction to Statistical Data Analysis
()
(Subject meets with 6.3720)
Prereq: 6.100A and (6.3700, 6.3800, 18.600, or permission of instructor)
Units: 4-0-8
Lecture: MW2.30-4 (4-370) Recitation: F12 (4-270) +final
Introduction to the central concepts and methods of data science with an emphasis on statistical grounding and modern computational capabilities. Covers principles involved in extracting information from data for the purpose of making predictions or decisions, including data exploration, feature selection, model fitting, and performance assessment. Topics include learning of distributions, hypothesis testing (including multiple comparison procedures), linear and nonlinear regression and prediction, classification, time series, uncertainty quantification, model validation, causal inference, optimization, and decisions. Computational case studies and projects drawn from applications in finance, sports, engineering, and machine learning life sciences. Students taking graduate version complete additional assignments. Recommended prerequisite: 18.06.
Y. Polyanskiy, D. Shah, J. N. Tsitsiklis
No textbook information available
6.3730[J] Statistics, Computation and Applications
()
(Same subject as IDS.012[J])
(Subject meets with 6.3732[J], IDS.131[J])
Prereq: (6.100B, (18.03, 18.06, or 18.C06), and (6.3700, 6.3800, 14.30, 16.09, or 18.05)) or permission of instructor
Units: 3-1-8
Lecture: MW11-12.30 (2-190) Recitation: W4 (4-265) or F10 (36-144) or F11 (36-156)
Hands-on analysis of data demonstrates the interplay between statistics and computation. Includes four modules, each centered on a specific data set, and introduced by a domain expert. Provides instruction in specific, relevant analysis methods and corresponding algorithmic aspects. Potential modules may include medical data, gene regulation, social networks, finance data (time series), traffic, transportation, weather forecasting, policy, or industrial web applications. Projects address a large-scale data analysis question. Students taking graduate version complete additional assignments. Enrollment limited; priority to Statistics and Data Science minors, and to juniors and seniors.
C. Uhler, N. Azizan, M. Roozbehani
Textbooks (Spring 2024)
6.3732[J] Statistics, Computation and Applications
()
(Same subject as IDS.131[J])
(Subject meets with 6.3730[J], IDS.012[J])
Prereq: (6.100B, (18.03, 18.06, or 18.C06), and (6.3700, 6.3800, 14.30, 16.09, or 18.05)) or permission of instructor
Units: 3-1-8
Lecture: MW11-12.30 (2-190) Recitation: W4 (4-265) or F10 (36-144) or F11 (36-156)
Hands-on analysis of data demonstrates the interplay between statistics and computation. Includes four modules, each centered on a specific data set, and introduced by a domain expert. Provides instruction in specific, relevant analysis methods and corresponding algorithmic aspects. Potential modules may include medical data, gene regulation, social networks, finance data (time series), traffic, transportation, weather forecasting, policy, or industrial web applications. Projects address a large-scale data analysis question. Students taking graduate version complete additional assignments. Limited enrollment; priority to Statistics and Data Science minors and to juniors and seniors.
C. Uhler, N. Azizan, M. Roozbehani
Textbooks (Spring 2024)
6.7700[J] Fundamentals of Probability
()
(Same subject as 15.085[J])
Prereq: Calculus II (GIR)
Units: 4-0-8
Introduction to probability theory. Probability spaces and measures. Discrete and continuous random variables. Conditioning and independence. Multivariate normal distribution. Abstract integration, expectation, and related convergence results. Moment generating and characteristic functions. Bernoulli and Poisson process. Finite-state Markov chains. Convergence notions and their relations. Limit theorems. Familiarity with elementary probability and real analysis is desirable.
D. Gamarnik
6.7710 Discrete Stochastic Processes
()
Prereq: 6.3702 or 18.204
Units: 4-0-8
Review of probability and laws of large numbers; Poisson counting process and renewal processes; Markov chains (including Markov decision theory), branching processes, birth-death processes, and semi-Markov processes; continuous-time Markov chains and reversibility; random walks, martingales, and large deviations; applications from queueing, communication, control, and operations research.
Staff
6.7720[J] Discrete Probability and Stochastic Processes
()
(Same subject as 15.070[J], 18.619[J])
Prereq: 6.3702, 6.7700, 18.100A, 18.100B, or 18.100Q
Units: 3-0-9
Lecture: MW2.30-4 (E25-111)
Provides an introduction to tools used for probabilistic reasoning in the context of discrete systems and processes. Tools such as the probabilistic method, first and second moment method, martingales, concentration and correlation inequalities, theory of random graphs, weak convergence, random walks and Brownian motion, branching processes, Markov chains, Markov random fields, correlation decay method, isoperimetry, coupling, influences and other basic tools of modern research in probability will be presented. Algorithmic aspects and connections to statistics and machine learning will be emphasized.
G. Bresler
No textbook information available
Inference
6.3800 Introduction to Inference
()
Prereq: Calculus II (GIR) or permission of instructor
Units: 4-4-4
Introduces probabilistic modeling for problems of inference and machine learning from data, emphasizing analytical and computational aspects. Distributions, marginalization, conditioning, and structure, including graphical and neural network representations. Belief propagation, decision-making, classification, estimation, and prediction. Sampling methods and analysis. Introduces asymptotic analysis and information measures. Computational laboratory component explores the concepts introduced in class in the context of contemporary applications. Students design inference algorithms, investigate their behavior on real data, and discuss experimental results.
P. Golland
6.7800 Inference and Information
()
Prereq: 6.3700, 6.3800, or 6.7700
Units: 4-0-8
Lecture: TR9.30-11 (32-123) Recitation: F1 (4-257) or F2 (4-257) +final
Introduction to principles of Bayesian and non-Bayesian statistical inference. Hypothesis testing and parameter estimation, sufficient statistics; exponential families. EM agorithm. 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 topics such as universal inference and learning, and universal features and neural networks.
G. Wornell
No required or recommended textbooks
6.7810 Algorithms for Inference
()
Prereq: 18.06 and (6.3700, 6.3800, or 6.7700)
Units: 4-0-8
Introduction to statistical inference with probabilistic graphical models. Directed and undirected graphical models, and factor graphs, over discrete and Gaussian distributions; hidden Markov models, linear dynamical systems. Sum-product and junction tree algorithms; forward-backward algorithm, Kalman filtering and smoothing. Min-sum and Viterbi algorithms. Variational methods, mean-field theory, and loopy belief propagation. Particle methods and filtering. Building graphical models from data, including parameter estimation and structure learning; Baum-Welch and Chow-Liu algorithms. Selected special topics.
G. Wornell
6.7820[J] Graphical Models: A Geometric, Algebraic, and Combinatorial Perspective
()
Not offered regularly; consult department
(Same subject as IDS.136[J])
Prereq: 6.3702 and 18.06
Units: 3-0-9
Provides instruction in the geometric, algebraic and combinatorial perspective on graphical models. Presents methods for learning the underlying graph and inferring its parameters. Topics include exponential families, duality theory, conic duality, polyhedral geometry, undirected graphical models, Bayesian networks, Markov properties, total positivity of distributions, hidden variables, and tensor decompositions.
C. Uhler
6.7830 Bayesian Modeling and Inference
()
Prereq: 6.7700 and 6.7900
Units: 3-0-9
Covers Bayesian modeling and inference at an advanced graduate level. Topics include de Finetti's theorem, decision theory, approximate inference (modern approaches and analysis of Monte Carlo, variational inference, etc.), hierarchical modeling, (continuous and discrete) nonparametric Bayesian approaches, sensitivity and robustness, and evaluation.
Staff
Machine Learning
6.3900 Introduction to Machine Learning
(, )
Prereq: (6.1010 or 6.1210) and (18.06, 18.700, or 18.C06)
Units: 4-0-8
URL: https://introml.mit.edu/
Lecture: MW9.30-11 (34-501, 32-044) or MW11-12.30 (34-501, 32-044) or MW1-2.30 (34-501, 32-044) or MW2.30-4 (34-501) Recitation: F11 (10-250) +final
Introduces principles, algorithms, and applications of machine learning from the point of view of modeling and prediction; formulation of learning problems; representation, over-fitting, generalization; clustering, classification, probabilistic modeling; and methods such as support vector machines, hidden Markov models, and neural networks. Recommended prerequisites: 6.1210 and 18.06. Enrollment may be limited.
Fall: V. Monardo
Spring: S. Shen
No textbook information available
6.3950 AI, Decision Making, and Society
()
(Subject meets with 6.3952)
Prereq: None. Coreq: 6.1200, 6.3700, 6.3800, 18.05, or 18.600
Units: 4-0-8
Introduction to fundamentals of modern data-driven decision-making frameworks, such as causal inference and hypothesis testing in statistics as well as supervised and reinforcement learning in machine learning. Explores how these frameworks are being applied in various societal contexts, including criminal justice, healthcare, finance, and social media. Emphasis on pinpointing the non-obvious interactions, undesirable feedback loops, and unintended consequences that arise in such settings. Enables students to develop their own principled perspective on the interface of data-driven decision making and society. Students taking graduate version complete additional assignments.
A. Wilson
6.3952 AI, Decision Making, and Society
()
(Subject meets with 6.3950)
Prereq: None. Coreq: 6.1200, 6.3700, 6.3800, or 18.05
Units: 4-0-8
Introduction to fundamentals of modern data-driven decision-making frameworks, such as causal inference and hypothesis testing in statistics as well as supervised and reinforcement learning in machine learning. Explores how these frameworks are being applied in various societal contexts, including criminal justice, healthcare, finance, and social media. Emphasis on pinpointing the non-obvious interactions, undesirable feedback loops, and unintended consequences that arise in such settings. Enables students to develop their own principled perspective on the interface of data-driven decision making and society. Students taking graduate version complete additional assignments.
A. Wilson
6.7900 Machine Learning
()
Prereq: 18.06 and (6.3700, 6.3800, or 18.600)
Units: 3-0-9
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, non-parametric Bayesian methods, hidden Markov models, Bayesian networks, and convolutional and recurrent neural networks. Recommended prerequisite: 6.3900 or other previous experience in machine learning. Enrollment may be limited.
P. Agrawal
6.7910[J] Statistical Learning Theory and Applications
()
(Same subject as 9.520[J])
Prereq: 6.3700, 6.7900, 18.06, or permission of instructor
Units: 3-0-9
Covers foundations and recent advances in statistical machine learning theory, with the dual goals of providing students with the theoretical knowledge to use machine learning and preparing more advanced students to contribute to progress in the field. The content is roughly divided into three parts. The first part is about classical regularization, margin, stochastic gradient methods, overparametrization, implicit regularization, and stability. The second part is about deep networks: approximation and optimization theory plus roots of generalization. The third part is about the connections between learning theory and the brain. Occasional talks by leading researchers on advanced research topics. Emphasis on current research topics.
T. Poggio
6.7920[J] Reinforcement Learning: Foundations and Methods
(New)
()
(Same subject as 1.127[J], IDS.140[J])
Prereq: 6.3700 or permission of instructor
Units: 4-0-8
Examines reinforcement learning (RL) as a methodology for approximately solving sequential decision-making under uncertainty, with foundations in optimal control and machine learning. Provides a mathematical introduction to RL, including dynamic programming, statistical, and empirical perspectives, and special topics. Core topics include: dynamic programming, special structures, finite and infinite horizon Markov Decision Processes, value and policy iteration, Monte Carlo methods, temporal differences, Q-learning, stochastic approximation, and bandits. Also covers approximate dynamic programming, including value-based methods and policy space methods. Applications and examples drawn from diverse domains. Focus is mathematical, but is supplemented with computational exercises. An analysis prerequisite is suggested but not required; mathematical maturity is necessary.
C. Wu
6.7930[J] Machine Learning for Healthcare
()
(Same subject as HST.956[J])
Prereq: 6.3900, 6.4100, 6.7810, 6.7900, 6.8611, or 9.520
Units: 4-0-8
Lecture: TR2.30-4 (54-100) Recitation: F3 (4-270) +final
Introduces students to machine learning in healthcare, including the nature of clinical data and the use of machine learning for risk stratification, disease progression modeling, precision medicine, diagnosis, subtype discovery, and improving clinical workflows. Topics include causality, interpretability, algorithmic fairness, time-series analysis, graphical models, deep learning and transfer learning. Guest lectures by clinicians from the Boston area, and projects with real clinical data, emphasize subtleties of working with clinical data and translating machine learning into clinical practice. Limited to 55.
D. Sontag, P. Szolovits
No textbook information available
6.7940 Dynamic Programming and Reinforcement Learning
()
Not offered regularly; consult department
Prereq: 6.3700 or 18.600
Units: 4-0-8
Dynamic programming as a unifying framework for sequential decision-making under uncertainty, Markov decision problems, and stochastic control. Perfect and imperfect state information models. Finite horizon and infinite horizon problems, including discounted and average cost formulations. Value and policy iteration. Suboptimal methods. Approximate dynamic programming for large-scale problems, and reinforcement learning. Applications and examples drawn from diverse domains. While an analysis prerequisite is not required, mathematical maturity is necessary.
Staff
6.7950 Advanced Topics in Control
() 
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
Advanced study of topics in control. Specific focus varies from year to year.
Consult Department
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