概率-不确定性与数据的科学

你将学到什么

课程概况

The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions.
Probabilistic models use the language of mathematics. But instead of relying on the traditional “theorem-proof” format, we develop the material in an intuitive — but still rigorous and mathematically-precise — manner. Furthermore, while the applications are multiple and evident, we emphasize the basic concepts and methodologies that are universally applicable.
The course covers all of the basic probability concepts, including:

multiple discrete or continuous random variables, expectations, and conditional distributions
laws of large numbers
the main tools of Bayesian inference methods
an introduction to random processes (Poisson processes and Markov chains)

The contents of this courseare heavily based upon the corresponding MIT class — Introduction to Probability — a course that has been offered and continuously refined over more than 50 years. It is a challenging class but will enable you to apply the tools of probability theory to real-world applications or to your research.
This course is part of theMITx MicroMasters Program in Statistics and Data Science. Master the skills needed to be an informed and effective practitioner of data science. You will complete this course and three others from MITx, at a similar pace and level of rigor as an on-campus course at MIT, and then take a virtually-proctored exam to earn your MicroMasters, an academic credential that will demonstrate your proficiency in data science or accelerate your path towards an MIT PhD or a Master’s at other universities. To learn more about this program, please visit https://micromasters.mit.edu/ds/.

课程大纲

Unit 1: Probability models and axioms

Probability models and axioms
Mathematical background: Sets; sequences, limits, and series; (un)countable sets.

Unit 2: Conditioning and independence

Conditioning and Bayes' rule
Independence

Unit 3: Counting

Counting

Unit 4: Discrete random variables

Probability mass functions and expectations
Variance; Conditioning on an event; Multiple random variables
Conditioning on a random variable; Independence of random variables

Unit 5: Continuous random variables

Probability density functions
Conditioning on an event; Multiple random variables
Conditioning on a random variable; Independence; Bayes' rule

Unit 6: Further topics on random variables

Derived distributions
Sums of independent random variables; Covariance and correlation
Conditional expectation and variance revisited; Sum of a random number of independent random variables

Unit 7: Bayesian inference

Introduction to Bayesian inference
Linear models with normal noise
Least mean squares (LMS) estimation
Linear least mean squares (LLMS) estimation

Unit 8: Limit theorems and classical statistics

Inequalities, convergence, and the Weak Law of Large Numbers
The Central Limit Theorem (CLT)
An introduction to classical statistics

Unit 9: Bernoulli and Poisson processes

The Bernoulli process
The Poisson process
More on the Poisson process

Unit 10 (Optional): Markov chains

Finite-state Markov chains
Steady-state behavior of Markov chains
Absorption probabilities and expected time to absorption

预备知识

College-level calculus (single-variable & multivariable). Comfort with mathematical reasoning; and familiarity with sequences, limits, infinite series, the chain rule, and ordinary or multiple integrals.

常见问题

How is this class related to 6.041x?
The material covered, and the resources (videos, etc.) are largely the same, but homeworks and exams contain revised and new problems.

What textbook do I need for the course?
None - there is no required textbook. The class follows closely the text I ntroduction to Probability, 2nd edition, by Bertsekas and Tsitsiklis, Athena Scientific, 2008. (See the publisher's website or Amazon.com for more information.) However, while this textbook is recommended as supplemental reading, the materials provided by this course are self-contained.

What is the format of the class?
The course material is organized along units, each unit containing between one and three lecture sequences. (For those who purchase the textbook, each unit corresponds to a chapter.) Each lecture sequence consists of short video clips,interwovenwith short problems to test your understanding. Each unit also contains a wealth of supplementary material, including videos that go through the solutions to various problems.

How much do I need to work for this class?
This is an ambitious class in that it covers a lot of material in substantial depth. In addition, MIT considers that the best way to master the subject is by actually solving on your own a fair number of problems. MIT students who take the corresponding residential class typically report an average of 11-12 hours spent each week, including lectures, recitations, readings, homework, and exams.