Search result: Catalogue data in Autumn Semester 2020

Statistics Master Information
The following courses belong to the curriculum of the Master's Programme in Statistics. The corresponding credits do not count as external credits even for course units where an enrolment at ETH Zurich is not possible.
Master Studies (Programme Regulations 2020)
Subject Specific Electives
401-3601-00LProbability Theory Information
At most one of the three course units (Bachelor Core Courses)
401-3461-00L Functional Analysis I
401-3531-00L Differential Geometry I
401-3601-00L Probability Theory
can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office ( after having received the credits.
W10 credits4V + 1UA.‑S. Sznitman
AbstractBasics of probability theory and the theory of stochastic processes in discrete time
ObjectiveThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
ContentThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Lecture notesavailable in electronic form.
LiteratureR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
401-3627-00LHigh-Dimensional Statistics
Does not take place this semester.
W4 credits2VP. L. Bühlmann
Abstract"High-Dimensional Statistics" deals with modern methods and theory for statistical inference when the number of unknown parameters is of much larger order than sample size. Statistical estimation and algorithms for complex models and aspects of multiple testing will be discussed.
ObjectiveKnowledge of methods and basic theory for high-dimensional statistical inference
ContentLasso and Group Lasso for high-dimensional linear and generalized linear models; Additive models and many smooth univariate functions; Non-convex loss functions and l1-regularization; Stability selection, multiple testing and construction of p-values; Undirected graphical modeling
LiteraturePeter Bühlmann and Sara van de Geer (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Verlag.
ISBN 978-3-642-20191-2.
Prerequisites / NoticeKnowledge of basic concepts in probability theory, and intermediate knowledge of statistics (e.g. a course in linear models or computational statistics).
401-3612-00LStochastic SimulationW5 credits3GF. Sigrist
AbstractThis course introduces statistical Monte Carlo methods. This includes applications of stochastic simulation in various fields (statistics, statistical mechanics, operations research, financial mathematics), generating uniform and arbitrary random variables (incl. rejection and importance sampling), the accuracy of methods, variance reduction, quasi-Monte Carlo, and Markov chain Monte Carlo.
ObjectiveStudents know the stochastic simulation methods introduced in this course. Students understand and can explain these methods, show how they are related to each other, know their weaknesses and strengths, apply them in practice, and proof key results.
ContentExamples of simulations in different fields (statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (including rejection sampling and importance sampling), simulation of multivariate normal variables and stochastic differential equations. The accuracy of Monte Carlo methods. Methods for variance reduction and quasi-Monte Carlo. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). Algorithms introduced in the course are illustrated with the statistical software R.
Lecture notesA script will be available in English.
LiteratureP. Glasserman, Monte Carlo Methods in Financial Engineering.
Springer 2004.

B. D. Ripley. Stochastic Simulation. Wiley, 1987.

Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
Prerequisites / NoticeIt is assumed that students have had an introduction to probability theory and statistics (random variables, joint and conditional distributions, law of large numbers, central limit theorem, basics of measure theory).

The course resources (including script, slides, exercises) will be provided via the Moodle online learning platform.
401-4619-67LAdvanced Topics in Computational Statistics
Does not take place this semester.
W4 credits2Vnot available
AbstractThis lecture covers selected advanced topics in computational statistics. This year the focus will be on graphical modelling.
ObjectiveStudents learn the theoretical foundations of the selected methods, as well as practical skills to apply these methods and to interpret their outcomes.
ContentThe main focus will be on graphical models in various forms:
Markov properties of undirected graphs; Belief propagation; Hidden Markov Models; Structure estimation and parameter estimation; inference for high-dimensional data; causal graphical models
Prerequisites / NoticeWe assume a solid background in mathematics, an introductory lecture in probability and statistics, and at least one more advanced course in statistics.
401-4633-00LData Analytics in Organisations and BusinessW5 credits2V + 1UI. Flückiger
AbstractOn the end-to-end process of data analytics in organisations & business and how to transform data into insights for fact based decisions. Presentation of the process from the beginning with framing the business problem to presenting the results and making decisions by the use of data analytics. For each topic case studies from the financial service, healthcare and retail sectors will be presented.
ObjectiveThe goal of this course is to give the students the understanding of the data analytics process in the business world, with special focus on the skills and techniques used besides the technical skills. The student will become familiar with the "business language", current problems and thinking in organisations and business and tools used.
ContentFraming the Business Problem
Framing the Analytics Problem
Model Building
Model Lifecycle
Soft Skills for the Statistical/Mathematical Professional
Lecture notesLecture Notes will be available.
Prerequisites / NoticePrerequisites: Basic statistics and probability theory and regression
401-6217-00LUsing R for Data Analysis and Graphics (Part II) Restricted registration - show details W1.5 credits1GM. Mächler
AbstractThe course provides the second part an introduction to the statistical software R for scientists. Topics are data generation and selection, graphical functions, important statistical functions, types of objects, models, programming and writing functions.
Note: This part builds on "Using R... (Part I)", but can be taken independently if the basics of R are already known.
ObjectiveThe students will be able to use the software R efficiently for data analysis, graphics and simple programming
ContentThe course provides the second part of an introduction to the statistical software R ( for scientists. R is free software that contains a huge collection of functions with focus on statistics and graphics. If one wants to use R one has to learn the programming language R - on very rudimentary level. The course aims to facilitate this by providing a basic introduction to R.

Part II of the course builds on part I and covers the following additional topics:
- Elements of the R language: control structures (if, else, loops), lists, overview of R objects, attributes of R objects;
- More on R functions;
- Applying functions to elements of vectors, matrices and lists;
- Object oriented programming with R: classes and methods;
- Tayloring R: options
- Extending basic R: packages

The course focuses on practical work at the computer. We will make use of the graphical user interface RStudio:
Lecture notesAn Introduction to R.
Prerequisites / NoticeBasic knowledge of R equivalent to "Using R .. (part 1)" ( = 401-6215-00L ) is a prerequisite for this course.

The course resources will be provided via the Moodle web learning platform.
Subscribing via Mystudies should *automatically* make you
a student participant of the Moodle course of this lecture, which is at

ALL material is available on this moodle page.
401-0627-00LSmoothing and Nonparametric Regression with Examples Information W4 credits2GS. Beran-Ghosh
AbstractStarting with an overview of selected results from parametric inference, kernel smoothing will be introduced along with some asymptotic theory, optimal bandwidth selection, data driven algorithms and some special topics. Examples from environmental research will be used for motivation, but the methods will also be applicable elsewhere.
ObjectiveThe students will learn about methods of kernel smoothing and application of concepts to data. The aim will be to build sufficient interest in the topic and intuition as well as the ability to implement the methods to various different datasets.
ContentRough Outline:

- Parametric estimation methods: selection of important results
o Maximum likelihood, Method of Least squares: regression & diagnostics

- Nonparametric curve estimation
o Density estimation, Kernel regression, Local polynomials, Bandwidth selection
o Selection of special topics (as time permits, we will cover as many topics as possible) such as rapid change points, mode estimation, robust smoothing, partial linear models, etc.

- Applications: potential areas of applications will be discussed such as, change assessment, trend and surface estimation, probability and quantile curve estimation, and others.
Lecture notesBrief summaries or outlines of some of the lecture material will be posted at

NOTE: The posted notes will tend to be just sketches whereas only the in-class lessons will contain complete information.

LOG IN: In order to have access to the posted notes, you will need the course user id & the password. These will be given out on the first day of the lectures.
- Statistical Inference, by S.D. Silvey, Chapman & Hall.
- Regression Analysis: Theory, Methods and Applications, by A. Sen and M. Srivastava, Springer.
- Density Estimation, by B.W. Silverman, Chapman and Hall.
- Nonparametric Simple Regression, by J. Fox, Sage Publications.
- Applied Smoothing Techniques for Data Analysis: the Kernel Approach With S-Plus Illustrations, by A.W. Bowman, A. Azzalini, Oxford University Press.
- Kernel Smoothing: Principles, Methods and Applications, by S. Ghosh, Wiley.

Additional references will be given out in the lectures.
Prerequisites / NoticePrerequisites: A background in Linear Algebra, Calculus, Probability & Statistical Inference including Estimation and Testing.
447-6289-00LSampling Surveys Restricted registration - show details
Special Students "University of Zurich (UZH)" in the Master Program in Biostatistics at UZH cannot register for this course unit electronically. Forward the lecturer's written permission to attend to the Registrar's Office. Alternatively, the lecturer may also send an email directly to The Registrar's Office will then register you for the course.
W2 credits1GB. Hulliger
AbstractThe elements of a sample survey are explained. The most important classical sample designs (simple random sampling and stratified random sampling) with their estimation procedures and the use of auxiliary information including the Horvitz-Thompson estimator are introduced. Data preparation, non-response and its treatment, variance estimation and analysis of survey data is discussed.
ObjectiveKnowledge of the Elements and the process of a sample survey. Understanding of the paradigm of random samples. Knowledge of simple random samplinig and stratified random sampling and capability to apply the corresponding methods. Knowledge of further methods of sampling and estimation as well as data preparation and analysis.
Lecture notesIntroduction to the statistical methods of survey research
401-3628-14LBayesian Statistics
Does not take place this semester.
W4 credits2V
AbstractIntroduction to the Bayesian approach to statistics: decision theory, prior distributions, hierarchical Bayes models, empirical Bayes, Bayesian tests and model selection, empirical Bayes, Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods.
ObjectiveStudents understand the conceptual ideas behind Bayesian statistics and are familiar with common techniques used in Bayesian data analysis.
ContentTopics that we will discuss are:

Difference between the frequentist and Bayesian approach (decision theory, principles), priors (conjugate priors, noninformative priors, Jeffreys prior), tests and model selection (Bayes factors, hyper-g priors for regression),hierarchical models and empirical Bayes methods, computational methods (Laplace approximation, Monte Carlo and Markov chain Monte Carlo methods)
Lecture notesA script will be available in English.
LiteratureChristian Robert, The Bayesian Choice, 2nd edition, Springer 2007.

A. Gelman et al., Bayesian Data Analysis, 3rd edition, Chapman & Hall (2013).

Additional references will be given in the course.
Prerequisites / NoticeFamiliarity with basic concepts of frequentist statistics and with basic concepts of probability theory (random variables, joint and conditional distributions, laws of large numbers and central limit theorem) will be assumed.
447-6273-00LBayes Methods Restricted registration - show details
Special Students "University of Zurich (UZH)" in the Master Program in Biostatistics at UZH cannot register for this course unit electronically. Forward the lecturer's written permission to attend to the Registrar's Office. Alternatively, the lecturer may also send an email directly to The Registrar's Office will then register you for the course.
W2 credits2GY.‑L. Grize
Abstractconditional probability; bayes inference (conjugate distributions, HPD-areas; linear and empirical bayes); determination of the a-posteriori distribution through simulation (MCMC with R2Winbugs); introduction to multilevel/hierarchical models.
ContentBayes statistics is attractive, because it allows to make decisions under uncertainty where a classical frequentist statistical approach fails. The course provides an introduction into bayesian methods. It is moderately mathematically technical, but demands a flexibility of mind, which should not underestimated.
LiteratureGelman A., Carlin J.B., Stern H.S. and D.B. Rubin, Bayesian Data Analysis, Chapman and Hall, 2nd Edition, 2004.

Kruschke, J.K., Doing Bayesian Data Analysis, Elsevier2011.
Prerequisites / NoticePrerequisite:Basic knowledge of statistics; Knowledge of R.
401-3901-00LMathematical OptimizationW11 credits4V + 2UR. Zenklusen
AbstractMathematical treatment of diverse optimization techniques.
ObjectiveThe goal of this course is to get a thorough understanding of various classical mathematical optimization techniques with an emphasis on polyhedral approaches. In particular, we want students to develop a good understanding of some important problem classes in the field, of structural mathematical results linked to these problems, and of solution approaches based on this structural understanding.
ContentKey topics include:
- Linear programming and polyhedra;
- Flows and cuts;
- Combinatorial optimization problems and techniques;
- Equivalence between optimization and separation;
- Brief introduction to Integer Programming.
Literature- Bernhard Korte, Jens Vygen: Combinatorial Optimization. 6th edition, Springer, 2018.
- Alexander Schrijver: Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003. This work has 3 volumes.
- Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.
- Alexander Schrijver: Theory of Linear and Integer Programming. John Wiley, 1986.
Prerequisites / NoticeSolid background in linear algebra.
252-0535-00LAdvanced Machine Learning Information W10 credits3V + 2U + 4AJ. M. Buhmann, C. Cotrini Jimenez
AbstractMachine learning algorithms provide analytical methods to search data sets for characteristic patterns. Typical tasks include the classification of data, function fitting and clustering, with applications in image and speech analysis, bioinformatics and exploratory data analysis. This course is accompanied by practical machine learning projects.
ObjectiveStudents will be familiarized with advanced concepts and algorithms for supervised and unsupervised learning; reinforce the statistics knowledge which is indispensible to solve modeling problems under uncertainty. Key concepts are the generalization ability of algorithms and systematic approaches to modeling and regularization. Machine learning projects will provide an opportunity to test the machine learning algorithms on real world data.
ContentThe theory of fundamental machine learning concepts is presented in the lecture, and illustrated with relevant applications. Students can deepen their understanding by solving both pen-and-paper and programming exercises, where they implement and apply famous algorithms to real-world data.

Topics covered in the lecture include:

What is data?
Bayesian Learning
Computational learning theory

Supervised learning:
Ensembles: Bagging and Boosting
Max Margin methods
Neural networks

Unsupservised learning:
Dimensionality reduction techniques
Mixture Models
Non-parametric density estimation
Learning Dynamical Systems
Lecture notesNo lecture notes, but slides will be made available on the course webpage.
LiteratureC. Bishop. Pattern Recognition and Machine Learning. Springer 2007.

R. Duda, P. Hart, and D. Stork. Pattern Classification. John Wiley &
Sons, second edition, 2001.

T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical
Learning: Data Mining, Inference and Prediction. Springer, 2001.

L. Wasserman. All of Statistics: A Concise Course in Statistical
Inference. Springer, 2004.
Prerequisites / NoticeThe course requires solid basic knowledge in analysis, statistics and numerical methods for CSE as well as practical programming experience for solving assignments.
Students should have followed at least "Introduction to Machine Learning" or an equivalent course offered by another institution.

PhD students are required to obtain a passing grade in the course (4.0 or higher based on project and exam) to gain credit points.
252-3005-00LNatural Language Processing Information Restricted registration - show details
Number of participants limited to 200.
W5 credits2V + 1U + 1AR. Cotterell
AbstractThis course presents topics in natural language processing with an emphasis on modern techniques, primarily focusing on statistical and deep learning approaches. The course provides an overview of the primary areas of research in language processing as well as a detailed exploration of the models and techniques used both in research and in commercial natural language systems.
ObjectiveThe objective of the course is to learn the basic concepts in the statistical processing of natural languages. The course will be project-oriented so that the students can also gain hands-on experience with state-of-the-art tools and techniques.
ContentThis course presents an introduction to general topics and techniques used in natural language processing today, primarily focusing on statistical approaches. The course provides an overview of the primary areas of research in language processing as well as a detailed exploration of the models and techniques used both in research and in commercial natural language systems.
LiteratureJacob Eisenstein: Introduction to Natural Language Processing (Adaptive Computation and Machine Learning series)
227-0423-00LNeural Network Theory Information W4 credits2V + 1UH. Bölcskei
AbstractThe class focuses on fundamental mathematical aspects of neural networks with an emphasis on deep networks: Universal approximation theorems, basics of approximation theory, fundamental limits of deep neural network learning, geometry of decision surfaces, capacity of separating surfaces, dimension measures relevant for generalization, VC dimension of neural networks.
ObjectiveAfter attending this lecture, participating in the exercise sessions, and working on the homework problem sets, students will have acquired a working knowledge of the mathematical foundations of (deep) neural networks.
Content1. Universal approximation with single- and multi-layer networks

2. Introduction to approximation theory: Fundamental limits on compressibility of signal classes, Kolmogorov epsilon-entropy of signal classes, non-linear approximation theory

3. Fundamental limits of deep neural network learning

4. Geometry of decision surfaces

5. Separating capacity of nonlinear decision surfaces

6. Dimension measures: Pseudo-dimension, fat-shattering dimension, Vapnik-Chervonenkis (VC) dimension

7. Dimensions of neural networks

8. Generalization error in neural network learning
Lecture notesDetailed lecture notes will be provided.
Prerequisites / NoticeThis course is aimed at students with a strong mathematical background in general, and in linear algebra, analysis, and probability theory in particular.
401-4521-70LGeometric Tomography - Uniqueness, Statistical Reconstruction and Algorithms Information Restricted registration - show details W4 credits2VJ. Hörrmann
AbstractSelf-contained course on the theoretical aspects of the reconstruction of geometric objects from tomographic projection and section data.
ObjectiveIntroduction to geometric tomography and understanding of various theoretical aspects of reconstruction problems.
ContentThe problem of reconstruction of an object from geometric information like X-ray data is a classical inverse problem on the overlap between applied mathematics, statistics, computer science and electrical engineering. We focus on various aspects of the problem in the case of prior shape information on the reconstruction object. We will answer questions on uniqueness of the reconstruction and also cover statistical and algorithmic aspects.
LiteratureR. Gardner: Geometric Tomography
F. Natterer: The Mathematics of Computerized Tomography
A. Rieder: Keine Probleme mit inversen Problemen
Prerequisites / NoticeA sound mathematical background in geometry, analysis and probability is required though a repetition of relevant material will be included. The ability to understand and write mathematical proofs is mandatory.
401-6282-00LStatistical Analysis of High-Throughput Genomic and Transcriptomic Data (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: STA426

Mind the enrolment deadlines at UZH:
W5 credits3GH. Rehrauer, M. Robinson
AbstractA range of topics will be covered, including basic molecular biology, genomics technologies and in particular, a wide range of statistical and computational methods that have been used in the analysis of DNA microarray and high throughput sequencing experiments.
Objective-Understand the fundamental "scientific process" in the field of Statistical Bioinformatics
-Be equipped with the skills/tools to preprocess genomic data (Unix, Bioconductor, mapping, etc.) and ensure reproducible research (Sweave)
-Have a general knowledge of the types of data and biological applications encountered with microarray and sequencing data
-Have the general knowledge of the range of statistical methods that get used with microarray and sequencing data
-Gain the ability to apply statistical methods/knowledge/software to a collaborative biological project
-Gain the ability to critical assess the statistical bioinformatics literature
-Write a coherent summary of a bioinformatics problem and its solution in statistical terms
ContentLectures will include: microarray preprocessing; normalization; exploratory data analysis techniques such as clustering, PCA and multidimensional scaling; Controlling error rates of statistical tests (FPR versus FDR versus FWER); limma (linear models for microarray analysis); mapping algorithms (for RNA/ChIP-seq); RNA-seq quantification; statistical analyses for differential count data; isoform switching; epigenomics data including DNA methylation; gene set analyses; classification
Lecture notesLecture notes, published manuscripts
Prerequisites / NoticePrerequisites: Basic knowlegde of the programming language R, sufficient knowledge in statistics

Former course title: Statistical Methods for the Analysis of Microarray and Short-Read Sequencing Data
401-8625-00LClinical Biostatistics (University of Zurich)
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: STA404

Mind the enrolment deadlines at UZH:
W6 credits4GUniversity lecturers
AbstractDiscussion of the different statistical methods that are used in clinical research.
ContentDiscussion of the different statistical methods that are used in
clinical research. Among other subjects the following will be
introduced: sample size calculation, randomization and blinding,
analysis of clinical trials (parallel groups design, analysis of
covariance, crossover design, equivalence studies), intention-to-treat
analysis, multiple testing, group sequential methods, adaptive designs,
diagnostic studies, and agreement studies.
Literature- Matthews, J. N. S. (2006). Introduction to Randomized Controlled
Clinical Trials. Chapman & Hall/CRC Texts in Statistical Science.
- Cook, T. D. and DeMets, L. D. (2008). Introduction to Statistical
Methods for Clinical Trials. Chapman & Hall/CRC Texts in Statistical
- Pepe, M. (2003). The Statistical Evaluation of Medical Tests for
Classification and Prediction. Oxford University Press.
- Schumacher, M. and Schulgen, G. (2008). Methodik klinischer
Studien. Springer, Berlin.
Prerequisites / NoticeBasic knowlegde of the programming language R, sufficient knowledge in
calculus, linear algebra, probability, statistics
263-3210-00LDeep Learning Information Restricted registration - show details W8 credits3V + 2U + 2AT. Hofmann
AbstractDeep learning is an area within machine learning that deals with algorithms and models that automatically induce multi-level data representations.
ObjectiveIn recent years, deep learning and deep networks have significantly improved the state-of-the-art in many application domains such as computer vision, speech recognition, and natural language processing. This class will cover the mathematical foundations of deep learning and provide insights into model design, training, and validation. The main objective is a profound understanding of why these methods work and how. There will also be a rich set of hands-on tasks and practical projects to familiarize students with this emerging technology.
Prerequisites / NoticeThis is an advanced level course that requires some basic background in machine learning. More importantly, students are expected to have a very solid mathematical foundation, including linear algebra, multivariate calculus, and probability. The course will make heavy use of mathematics and is not (!) meant to be an extended tutorial of how to train deep networks with tools like Torch or Tensorflow, although that may be a side benefit.

The participation in the course is subject to the following condition:
- Students must have taken the exam in Advanced Machine Learning (252-0535-00) or have acquired equivalent knowledge, see exhaustive list below:

Advanced Machine Learning

Computational Intelligence Lab

Introduction to Machine Learning

Statistical Learning Theory

Computational Statistics

Probabilistic Artificial Intelligence
263-5210-00LProbabilistic Artificial Intelligence Information Restricted registration - show details W8 credits3V + 2U + 2AA. Krause
AbstractThis course introduces core modeling techniques and algorithms from machine learning, optimization and control for reasoning and decision making under uncertainty, and study applications in areas such as robotics and the Internet.
ObjectiveHow can we build systems that perform well in uncertain environments and unforeseen situations? How can we develop systems that exhibit "intelligent" behavior, without prescribing explicit rules? How can we build systems that learn from experience in order to improve their performance? We will study core modeling techniques and algorithms from statistics, optimization, planning, and control and study applications in areas such as sensor networks, robotics, and the Internet. The course is designed for graduate students.
ContentTopics covered:
- Probability
- Probabilistic inference (variational inference, MCMC)
- Bayesian learning (Gaussian processes, Bayesian deep learning)
- Probabilistic planning (MDPs, POMPDPs)
- Multi-armed bandits and Bayesian optimization
- Reinforcement learning
Prerequisites / NoticeSolid basic knowledge in statistics, algorithms and programming.
The material covered in the course "Introduction to Machine Learning" is considered as a prerequisite.
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