Advanced Statistics

3 EC

Semester 1, period 3

5354ADST3Y

Owner Master Physics and Astronomy (joint degree)
Coordinator dr. C. Weniger
Part of Master Astronomy and Astrophysics, year 1Master Physics and Astronomy, track Theoretical Physics, year 1Master Physics and Astronomy, track GRAPPA, year 1

Course manual 2017/2018

Course content

This course is about the mathematical foundations of the two big schools of statistics: Bayesian and Frequentist inference. We will study applications, differences and limitations of these approaches that are at the foundation of all of modern science.

  1. Probability crash course
  2. Frequentist inference
    1. Hypothesis testing
    2. Estimators
    3. Maximum likelihood ratio test
    4. Confidence regions
  3. Bayesian inference
    1. Bayes theorem
    2. Credible intervals & priors
    3. Sampling techniques

Study materials

Literature

  • G. Cowan, 'Statistical Data Analysis'. (optional)

  • R.J. Barlow, 'Statistics, A guide to the Use of Statistical Methods in the Physical Sciences'. (optional)

  • P. Gregory, 'Bayesian Logical Data Analysis for the Physical Sciences'. (optional)

Other

  • Lecture slides, course notes.

Objectives

At the end of the course, the student is able to

  • Give examples for and explain difference between Frequentist and Bayesian probabilities
  • Recall definitions of basic probability distributions (Binomial, Poisson, Normal, Chi-squared) and their properties
  • Apply the central limit theorem and specify its limitations
  • Apply hypothesis testing and define confidence levels and various types of inference errors
  • Apply the chi-squared goodness-of-fit test to simple and composite hypothesis
  • Apply the Delta-chi-squared method for signal detection
  • Know the basic definitions related to general estimators
  • Know the definition of Fisher information and Cramer-Rao, and be able to apply these definitions in simple examples
  • Know and apply the maximum likelihood estimator (MLE) to given parametric probability distribution function
  • Understand and apply the maximum-likelihood-ratio test to various problems
  • Sketch the basic steps of the derivation of Wilk's theorem, know what the theorem is about and be able to apply it to simple examples; know the limitations of Wilk's theorem
  • Know how to estimate global significance from local significance, if number of trials is given
  • Understand general definition of confidence region
  • Be able to construct confidence regions using the likelihood ratio construction (if Wilks' theorem applies)
  • Apply Neyman belt construction to simple examples
  • Know the components of Bayes theorem, and apply them to simple scenarios
  • Apply Bayesian model comparison and interpret the Bayes factor in terms of Jeffreys' scale
  • Exemplify the mechanism of Ockham's razor in Bayesian inference
  • Understand general definition of various credible intervals
  • Give examples for non-informative and informative priors
  • Derive priors using the maximum entropy principle, using Lagrange multipliers
  • Understand various sampling techniques (inverse transform, reject/accept sampling)
  • Understand and apply Metropolis-Hastings MCMC

Teaching methods

  • Lecture
  • Seminar
  • Self-study

The main course material will be presented in the lectures, and can be read in the course notes. The three homework exercises give the student the opportunity to test the material in practice. The homework include a significant amount of writing statistical programs in PYTHON.

Learning activities

Activity

Number of hours

Zelfstudie

50

Lectures

14

Exercise sessions

14

Attendance

Requirements concerning attendance (OER-B).

  • In addition to, or instead of, classes in the form of lectures, the elements of the master’s examination programme often include a practical component as defined in article 1.2 of part A. The course catalogue contains information on the types of classes in each part of the programme. Attendance during practical components is mandatory.

    • Additional requirements for this course:

    The full attendance of both the lectures and homework sessions is strongly encouraged.  In the case of absence, the course coordinator should be notified.

    Assessment

    Item and weight Details

    Final grade

    0.7 (70%)

    Tentamen

    0.1 (10%)

    Homework 1

    0.1 (10%)

    Homework 2

    0.1 (10%)

    Homework 3

    The final grade is given by 70% exam and 30% HW grade.

    Inspection of assessed work

    Contact the course coordinator to make an appointment for inspection.

    Assignments

    Homework 1

    Homework 2

    Homework 3

    Homework exercises can be discussed in groups, but must be handed in individually.

    Fraud and plagiarism

    The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

    Course structure

    Weeknummer Onderwerpen  
    1 Introduction  
    2 Frequentist due HW1
    3 Bayesian due HW2
    4 Exam due HW3, final exam

    Timetable

    The schedule for this course is published on DataNose.

    Honours information

    Recommendend prior knowledge: Good knowledge of the material from 'Statistical Methods for the Physical Sciences' (Uttley) or 'Statistical Data Analysis' (Decowski) is highly recommended. Many of the exercises will require the use of the programming language python, which is also used in Uttley's course.

    Additional information

    Recommendend prior knowledge: Good knowledge of the material from 'Statistical Methods for the Physical Sciences' (Uttley) or 'Statistical Data Analysis' (Decowski) is highly recommended. Many of the exercises will require the use of the programming language python, which is also used in Uttley's course.

    Max. participants: 25 pers.

    Contact information

    Coordinator

    • dr. C. Weniger