DS404BKK
Probability Theory and Applications
Faculty
Andrey Khokhlov
Chief Researcher, IEPT RAS
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
Machine Learning is a powerful tool widely used today and is specified in computer science education, while Statistics and Probability are traditionally studied in mathematical departments. Nevertheless, there is a strong relationship between these theories, and it is desirable to be aware of both general principles and differences in approaches. Indeed, machine learning uses mathematical and/or statistical models to gain an overview of the data to make predictions.
The novel contributions in data mining are mostly informal and usually linked with the Bayesian point of view on statistics. But, probability theory itself is more than just Bayesian reasoning or Kolmogorov's axiomatic approach. For instance, the frequentist approach of R. von Mises coexists with novel contributions from quantum probabilities until now. Thus, the overcoming of the formal separation in theoretical understandings may help to avoid confusion and mistakes.
This course tries to show the existing diversity of approaches while staying within classical Kolmogorov's probability theory. The necessary classical theoretical material would be explained as well. We aim to consider several paradoxical situations that arise in practice.
Most examples would be taken from natural sciences and simple situations in data analysis. The outcome is expected to be practical training in understanding the popular statistical models and critically reading a professional text. A standard undergraduate course in calculus is required; the simple basic experience in PYTHON, or R or MATLAB (or OCTAVE --- the freeware clone of the MATLAB) would also be needed.
Learning highlights
- analytically calculate the distribution law of a random variable after simple algebraic transformations
- recognize random variables that have no mathematical expectation or variance
- look for obvious differences (if any) between real data and statistical samples.
- look for obvious differences (if any) in the distribution laws by means of simple tests.
- develop a computer simulation of samples subject to a given correlation
Course outline
15 classes
Session 1
Classical finite models and the need for a rigid theory. Possible paradoxes informal reasoning and observable facts in natural science
Session 2
Computational techniques of combinatorics, cases of distinguishable and indistinguishable objects. Generating functions and other analytical computing tools
Session 3
The idea of independence in the real world and its formal implementation. Conditional probabilities, independence of events and their formal properties. Bayesian approach for finite and infinite discrete cases
Session 4
Heuristic non-finite models are derived using the symmetry arguments. Algebra of events and the corresponding mathematical theory. Probabilities in discrete sample spaces
Session 5
Different approaches to Probabilities: the model of von Mises and the classical model. General discrete model and the idea of the quantum probability model. Random variable from formal and heuristic points of view.
Session 6
Random variables from formal and heuristic points of view. Examples. Scalar and random vector variables and their properties. Mutual and group independence. Geometric models in multidimensional space, sequences and their limits.
Session 7
Moments and other characteristics of the random variables. Chebyshev inequalities. Whether the moments are always defined, whether moments define the distribution.
Session 8
The Law of Large Numbers and what Statistics can do. Sample space and several approaches in Statistics.
Session 9
Integral-valued random variables and their generating functions. Computational techniques, analytical formulas and computer modelling. Pseudorandom generators
Session 10
Applications: basic discrete models and discrete random variables. Sequences of random variables and Random Walks model.
Session 11
More about the applications: binomial distribution and asymptotics. The idea of the Central Limit Theorem and its meaning for the Natural Sciences. An experimental illustration
Session 12
Important distribution functions and classification of random variables. The main problems addressed in Classical Statistics. Nonparametric tests
Session 13
In-depth: formal theory and implementations. Lebeg integration and computational formulas in general cases. Rademacher functions and the sequence of independent random variables. Kolmogorov’s axioms and other approaches
Session 14
In-depth: limits and asymptotic properties of random variables in probability theory. Sequences of random variables and several types of convergence. Characteristic functions and their properties
Mixtures of random variables.
Session 15
Central Limit Theorem and its constraints. Gaussian and non-gaussian statistics. Real data and corresponding traditional assumptions. Simulations.
Course materials
Books
Media
Prerequisites
We do not follow the unique textbook: our course consists of several topics extracted from several classical monographs, the corresponding text fragments would be available. Also, the original lecture notes would be freely available in electronic format. Some known good classical textbooks are:
Most of the examples will be taken from the natural sciences and simple situations in data analysis. Therefore, it is assumed at least minimal familiarity with the mathematical processing of arrays of numerical data. The basic concepts and formulas from the standard bachelor's course in integration and differentiation will be used. It also requires some basic experience with PYTHON, R, or MATLAB (or OCTAVE, a free MATLAB clone).
Methodology
We follow the current practice of teaching probabilistic methods from scratch: with the deconstruction of standard routine examples along with unexpected counterexamples. Each lesson is divided into a discussion of theory and a joint analysis of a computational example or homework. In the middle of the course, an intermediate test is held, which consists of solving typical problems on the basic concepts of the theory.
Grading
After getting his Ph.D. in Algebraic Topology in 1983 Andrey worked in several scientific and/or teaching organisations, among them are the Russian Academy of Sciences, Moscow State University, and Baumann Technology University. The Scientific advising of the graduate and thesis students was part of his activities, not only in Russia, but also in France.
Andrey’s main results in science are linked with geophysical data processing, so naturally his teaching interests are now concentrated in the applied methods of Statistics and their algorithmic implementations. He currently helps his students avoid some common errors within the probabilistic inferences and support their attempts to study Probability and Statistics theory in general.
See full profileApply for this course
Probability Theory and Applications
by Andrey Khokhlov
Total hours
45 Hours
Dates
Oct 25 - Nov 12, 2021
Fee for single course
€1500
Fee for degree students
€750
How to secure your spot
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FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.