20592 - STATISTICS AND PROBABILITY

A solid background in Probability and Statistics is an absolute MUST for a data scientist, in whatever field she/he is willing to work. This course aims at providing such a solid methodological background. We start with a recap of fundamental notions in probabiilty theory and stochastic processes (in particular, Markov chains), presented in a friendly but rigorous way. We then go to classical statistical inference, giving the basis of maximum likelihood estimation, confidence intervals and tests, to end with an introduction to Bayesian learning. In this all, we will have in mind the "exlain or predict" big debate; simplifying a lot, "classical statistics" towards "modern statistics" and machine learning. The course is completed with a module on computational methods (stochastic integration & Monte Carlo, optimization, bootstrap, Markov chain Monte Carlo), with Python. The lectures include frontal lecturing, group work with periodic assigments, coding and simulations.

PART I : Probability recap

• Definition and basic properties
• Random variables. Multivariate distributions
• Expectation and conditional expectation.
• Convergence of random variables.
• Basic notions on stocastic processes. Random noise. Random walks. Markov chains.

Part II : Statistical inference

Models, Statistical Inference and Learning

Elements of nonparametric estimation.

The bootstrap.

Parametric Inference

MLE and asymptotics

Confidence intervals

Hypothesis testing and p-values

PART III - Bayesian learning

• Fundamentals of Bayesian learning
• Bayes rule and examples.
• Bayesian linear regression (if time permits)

ALL OVER: Computational methods

• Stochastic integration and Monte Carlo.
• Optimization. EM algorithm.
• Parametric bootstrap (if time permits)
• Markov Chain Monte Carlo.

• Define, describe and explain rigorously the main notions of probability and statistical learning in the frequentist and Bayesian approach.

* Identify computational strategies for fundamental complex problems

* Recognize the role of probability and statistics in "data science" and related fields

• Estimate and predict, and quantify uncertainty, in fundamental problems
• Write algorithms in Python for the implementation of computational statistic techniques, namely optimization and integration techniques.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)
• Individual assignments
• Group assignments

Students will be given periodic group or individual assignments,  on the theory and on the implementation of computational methods (with Python).

ASSIGNMENTS:

Students wil be given periodic assigments, on the theory and on the computational methods presented in class.

The assigments (take-home) can be done individually or in groups (up to 5  people).

They are meant to support and engage students to follow and verify their ongoing understanding along the lectures - actually, students usually find them very helpful!

As such, the assigments are not mandatory and are not formally evaluated; however, students' work on the assigments is acknowldged in the final exam:

***  students who did not deliver the assigments will have additional questions in the written proof, with up to 10 min of extra time. These questions do not contribute to the final grade, if the are reasonably well answered; but may penalize the final grade if poorly done.

*** students who did deliver the assigments will not have to answer those additional questions.

EXAM:

The exam will consist in an individual written proof  that will count 70%, and a final project on computational methods, that counts 30%.

NOTE 1: The final project is done in groups, while the written proof is individual. Therefore, the written proof may count 100% if poorly done.

NOTE 2: The exam structure might be slightly modified, in order to accomodate for unforseen (as it happened with the COVID-19 pandemic), taking into account the  students' needs. In that case, students will be promptly informed, through BBoard announcements and more.

NOTE 3: The assigments contribute to the achievement of all the learning objectices of the course; 

in particular are of support to
- *define, describe and explain rigorously the main notions of probability and statistical learning in the frequentist and Bayesian approach,

which is the necessary basis for being able to *estimate and predict, and quantify uncertainty, in fundamental problems and *recognize the role of probability and statistics in "data science" and related fields;
-  *Identify computational strategies for fundamental complex problems and implement those statistical techniques, *writing algorithms in Python.

The EXAM aims at assessing all the learning objectives. In particular,

-- the wrtiten proof aims at assessing ILOs:
 *define, describe and explain rigorously the main notions of probability and statistical learning in the frequentist and Bayesian approach and *estimate and predict,   and quantify uncertainty*, and *recognize the role of probability and statistics in "data science";
--  the final project aims at assessing ILOs
   * Identify computational strategies for fundamental complex problems, and write algorithms in Python for the implementation of computational statistic techniques,

textbook

• L. Wasserman, "All of Statistics", Springer

More teaching material, lecture notes, Python code etc will be provided on BBoard.