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  • Module reading notes

  • Week 1: Introduction and review of likelihood

  • Week 2: Bayes’ Theorem and Bayesian inference

  • Week 3: Bayesian updating/Conjugate distributions

  • Module Description

    • Bayes-ically, it is all about Bayesian inference. The statistics modules you have taken so far all follow a classical Frequentist approach based on the idea that probability represents a long run limiting frequency. By contrast, in the Bayesian approach probability represents a degree of belief in an event which is conditional on the knowledge of the person concerned. Estimated quantities for the population or system of interest depend on prior knowledge combined with available data. Why is the Bayesian approach so widely used, and why does it provide us with such powerful tools to draw conclusions about whole populations after looking at the available samples?
      In this module you will gain hands-on experience working on real-world problems by applying Bayesian methods in the lab while experimenting with various software packages.

  • Week 4: Conjugate distributions

  • Week 5: Point estimates and credible intervals

  • Week 6: Choosing a prior distribution

  • Week 8: Informative priors, Monte Carlo integration methods and MCMC

  • Week 9: MCMC

  • Week 10: MCMC and posterior predictive probability

  • Week 11: Posterior predictive probability and Bayes factor

  • Week 12: Revision

  • Past exam papers

  • Module aims and learning outcomes

    • This module covers an introduction to Bayesian statistics.  At the end of this module, students should be able to: 

      • analyse the differences between the Bayesian paradigm and frequentist statistical methods, 
      • formulate likelihoods and calculate maximum likelihood estimators, 
      • calculate posterior distributions and related summary quantities, such as posterior median and credible intervals,
      • explain the ideas of the Metropolis algorithm, 
      • describe how missing data problems and hierarchical models may be analysed using Bayesian methods.
      • be able to write R code and interpret the output.

      TOPICS TO COVER

      • Introduction to Bayesian methods; review of likelihood-based methods.
      • Prior distributions; conjugate priors; non-informative priors.
      • Point estimates, credible intervals.
      • Markov chain Monte Carlo.
      • Model choice.
      • Predictive distributions.
      • Missing data; hierarchical models.




  • Syllabus


    • Tentative schedule
      Week  Topic Assignment due

      25/09/2023

      Introduction and review of likelihood

      10/2/2023

      Bayes’ Theorem and Bayesian inference

      10/9/2023

      Conjugate distributions

      10/16/2023

      Point estimates and credible intervals

      Exercise 2  due 10/16 at 11:00

      10/23/2023

      Prior distributions

      10/30/2023

      Markov Chain Monte Carlo Methods (MCMC)

      Exercise 4 due 10/30 at 11:00

      11/6/2023

      		 No lectures 

      11/13/2023

      MCMC implementation 

      Exercise 6 due 11/13 at 11:00

      11/20/2023

      Posterior predictions

      11/27/2023

      Hierarchical models

      Exercise 8 due 11/27 at 11:00

      12/4/2023

      Model choice and Bayes factors

      12/11/2023

      		 Revision

      Exercise 10 due 11/27 at 11:00


  • Assessment information

    Assessment Pattern

    The assessment for this module will involve two components:

    • Five in-term coursework assignments, worth 4% each;
    • A final exam worth 80%.
    In-term coursework assignments

    The coursework, worth 20%, is made up of five sets of exercise sheet questions, to be submitted every two weeks. The first questions to be submitted will be posted at  11:00 on Monday of week 2 (2/10). There will be a link in the week's content on QMPlus to submit your solutions. The first deadline to submit solutions is at 11am on Monday of week 4 (16/10). 

    Format and dates for the in-term assessments - 

     Please, see the tentative schedule in the syllabus section in Important module information.

    You can submit a Word document, pdf or a clearly legible image of hand-written work. The answers can be brief - you only need to answer what is explicitly asked for; but please use some words to state what each part of the answer is. So if the question asks for a maximum likelihood estimate, then say e.g. "The MLE for sigma is ...", rather than just writing a number, or a symbol and number. For questions that use R, you need to submit the R code, and also write out the answers and submit those in a separate document.

    Some of the assessed questions use datasets on QMPlus. There is a different dataset for each student: details will be given in the exercise sheet. You need to be logged in to QMPlus to see the dataset. If you cannot see a file, please send me an email.

    Late submission: I will be posting solutions to the in-term coursework as soon as the deadline passes, so I won't be able to accept late submissions.  If you are prevented from submitting by circumstances beyond your control, then you can submit an extenuating circumstances claim (talk to the Maths office about this).  If this is accepted, then this coursework can be excluded from your final module mark.


    Format of final assessment -

    The remaining 80% of the assessment is the final exam, which is a traditional written exam, not computer-based. This will take place in January 2024.

    You are allowed to bring  3 pages of A4 notes as well as a  non-programmable calculator.

    Past exams

    The past exam papers  uploaded in Module Content will give you some idea of the style of the final exam.

    Description of Feedback - 

    Your feedback comes in many forms

    - Exercise sheets every two weeks with feedback
    - Weekly lab tutorials
    - Extra exercise sheet to work through each week for practice
    - Participation in the IT classes
    - Personalised feedback during office hours of the organiser  

  • Teaching team

    • Instructor: Eftychia Solea
      Office:  MB-324
      E-mail: e.solea@qmul.ac.uk                 

      TA:



  • Hints and tips

    Other points to note are:

    • there is an appendix with common probability distributions;
    • in that appendix are listed two percentiles of the standard normal distribution, below the entry for the normal distribution - the exam will not use statistical tables, as nothing else from the tables apart from these percentiles will be needed;
    • calculators are permitted.

  • Useful R resources

    Swirl

    Swirl is a package within in R that contains interactive lessons

    There is a website r-bloggers.com, contributed to by various R users. It has a page of online R resources:

    https://www.r-bloggers.com/2015/12/how-to-learn-r-2/

    This includes some online video material, but most of this is paid-for. A couple of free options (at least for now) that it lists are:

    https://www.udemy.com/course/introduction-to-r

    https://www.coursera.org/learn/probability-intro?specialization=statistics

    The Cross Validated question and answer website is a good place to look for help, by searching previous questions:

    https://stats.stackexchange.com/

    If you ask a question yourself, you need to be precise about what the problem is, and what you have tried already.

    The online manual on the official R (Cran) website is organized like a book, although it goes into more details of technicalities than you might want:

    https://cran.r-project.org/doc/manuals/R-intro.html

    There is a lot of other free online written material about R:

    https://cran.r-project.org/


  • Early feedback questionnaire

  • Assessment information

    • Assessment Pattern

      The assessment for this module will involve two components:

      • Five in-term coursework assignments, worth 4% each;
      • A final exam worth 80%.
      In-term coursework assignments

      The coursework, worth 20%, is made up of five sets of exercise sheet questions, to be submitted every two weeks. The first questions to be submitted will be posted at  11:00 on Monday of week 2 (2/10). There will be a link in the week's content on QMPlus to submit your solutions. The first deadline to submit solutions is at 11am on Monday of week 4 (16/10). 

      Format and dates for the in-term assessments - 

       Please, see the tentative schedule in the syllabus section in Important module information.

      You can submit a Word document, pdf or a clearly legible image of hand-written work. The answers can be brief - you only need to answer what is explicitly asked for; but please use some words to state what each part of the answer is. So if the question asks for a maximum likelihood estimate, then say e.g. "The MLE for sigma is ...", rather than just writing a number, or a symbol and number. For questions that use R, you need to submit the R code, and also write out the answers and submit those in a separate document.

      Some of the assessed questions use datasets on QMPlus. There is a different dataset for each student: details will be given in the exercise sheet. You need to be logged in to QMPlus to see the dataset. If you cannot see a file, please send me an email.

      Late submission: I will be posting solutions to the in-term coursework as soon as the deadline passes, so I won't be able to accept late submissions.  If you are prevented from submitting by circumstances beyond your control, then you can submit an extenuating circumstances claim (talk to the Maths office about this).  If this is accepted, then this coursework can be excluded from your final module mark.


      Format of final assessment -

      The remaining 80% of the assessment is the final exam, which is a traditional written exam, not computer-based. This will take place in January 2024.

      You are allowed to bring  3 pages of A4 notes as well as a  non-programmable calculator.

      Past exams

      The past exam papers  uploaded in Module Content will give you some idea of the style of the final exam.

      Description of Feedback - 

      Your feedback comes in many forms

      - Exercise sheets every two weeks with feedback
      - Weekly lab tutorials
      - Extra exercise sheet to work through each week for practice
      - Participation in the IT classes
      - Personalised feedback during office hours of the organiser  

  • Assessment Information

  • Q-Review

  • Reading List Online