This version: June 02 2026 12:39

Philosophical unification of the Bayesian and frequentist positions is not likely, nor desirable, since each illuminates a different aspect of statistical inference. We can hope, however, that we will eventually have a general methodological unification, with both Bayesians and frequentists agreeing on a body of standard statistical procedures for general use.
— M. J. (Susie) Bayarri and James O. Berger (2004) “The Interplay of Bayesian and Frequentist Analysis.” Statistical Science v. 19. (thanks to Hugh McCague)

Announcements

Today, Thursday, May 28, the class will be on Zoom at https://yorku.zoom.us/j/95480297710?pwd=pg1kP2QbGYuJr7hTe3tGE9kD8DTJhP.1

The course is moving to a new lecture room immediately, i.e. as of Thursday, May 7.

We will meet in Dahdaleh Building room DB 0013 (formerly known as the TEL building).

Calendar

Classes meet on Tuesdays and Thursdays from 1 pm to 4 pm in DB 0013
Instructor: Georges Monette
Office hours: After class
Email: . Messages about the course should be posted to Piazza.

Day 1: Tuesday, May 5, 2026

To understand Bayesian Statistics, we need to approach it from many directions. It is more than a new method, or an application of ‘standard’ statistical theory to a different kind of data. It is a fundamentally different approach to the problem of uncertainty. To develop a good basic understanding in a relatively short course, we need to develop many of these paths in parallel. This is a tentative list of approaches to Bayesian ideas and sources we can use to explore them.

  • Bayesian Statistics as a philosophy of inference:
    • Contrasted with Frequentist Inference. Critiques
      of Bayesian and Frequentist Inference.
    • The relationship of Bayesian Inference with other approaches to statistical inference: Fiducial Inference, Structural Inference, Statistical Decision Theory, Likelihood, Predictive Inference. Statistical methods can be understood in terms of their affinity with Bayesian versus Frequentist paradigms.
    • Sources:
      • History: lukeprog (2011)
      • From Efron (2016) Computer Age Statistical Inference, p. 265 (Bookshelf) BFF triangle
  • The mathematical theory of Bayesian Statistics.
  • Modern workflows for the application of Bayesian Statistics
  • Programming packages and languages for Bayesian Statistics

Assignment 1: What do \(p\)-values mean?

  • Due: Monday, May 11

The purpose of the assignment is to explore the meaning of p-values. Before starting, stop and reflect on what it means for an experiment to ‘achieve’ a p-value of 0.049. What meaning can we give to the quantity ‘0.049’? How is it related to the probability that the null hypothesis is correct?

To keep things very simple suppose you want to test \(H_0: \mu = 0\) versus \(H_1: \mu \neq 0\) and you are designing an experiment in which you plan to take a sample of independent random variables, \(X_1, X_2, ... , X_n\) which are iid \(\textrm{N}(\mu,1)\), i.e. the variance is known to be equal to 1. You plan to use the usual test based on \(\bar{X}_n\) rejecting \(H_0\) for values of \(\bar{X}_n\) that are far from 0.

An applied example would be testing for a change in value of a response when all subjects are submitted to the same conditions and the measurement error of the response is known. In that example \(X_i\) would be the ‘gain score’, i.e. post-test response minus the pre-test response exhibited by the \(i\)th subject.

Let the selected probability of Type I error be \(\alpha = 0.05\). Each of you has a budget that allows you to collect a sample of \(n\) observations as follows:

Who \(n\)
A 5
B 10
C 20
D 100
E 1,000

You are interested in answering the questions below using the following values of \(\mu_1\):

\(\mu_1\) Cohen’s terminology for effect size: \(\mu_1/\sigma\)
0.1
0.2 small effect size
0.5 medium effect size
0.8 large effect size
3
  1. What is the probability that \(p \le 0.05\) if \(H_0: \mu = 0\) is true?
  2. What is the probability that \(p \le 0.05\) if \(\mu = \mu_1\) for each of the values of \(\mu_1\) above.
  3. What is the power of this test if \(\mu = \mu_1\) for each of the values of \(\mu_1\) above.?
  4. Suppose that you collect the data and that the observed \(p\)-value is 0.049. What can you say about the probability that \(H_0\) is true?
  5. Suppose that, before running the experiment, you were willing to give \(H_0\) and \(H_1: \mu = 0.5\) equal probability.
    1. What is the probability that \(H_0\) is true given the event that \(p \le 0.05\)?
    2. What is the probability that \(H_0\) is true given the event that \(p = 0.049\)?
  6. Hypothesis testing is often presented as a process that parallels that of determining guilt in a criminal process. We start with a presumption of innocence, i.e. that \(H_0\) is true, We then hear evidence and consider whether it contradicts the presumption of innocence ‘beyond a reasonable doubt.’ Suppose we quantify the presumption of innocence to mean that \(P(H_0) \ge .95\). How small an observed \(p\)-value do you need to obtain in order to ‘flip’ the presumption of innocence to ‘guilt beyond a reasonable doubt’ if that is defined as \(P(H_0 | \mathrm{data}) \le .05\).
  7. What \(p\)-value would we need if the presumption of innocence and guilt beyond a reasonable doubt correspond to \(P(H_0) \ge 0.999\) and \(P(H_0|\mathrm{data}) \le 0.001\), respectively?
  8. Courts have often adopted a criterion of \(p < 0.05\) in imitation of the typical practice in research. Comment on the possible consequences.
  9. Optional challenge (which you might want to do first): Write a function in the language of your choice that computes the posterior probabilities in this assignment as a function of the true \(\mu_1\), \(n\), and the prior probability of \(H_0\). Use the function to produce interesting graphs. Discuss them in class on Tuesday, May 12.

To delve more deeply into these issues you can read Wasserstein & Lazar (2016) and Wasserstein et al. (2019). Concerns about \(p\)-values have been around for a long time, see Schervish (1996). For a short overview see Bergland (2019). There are two important papers by John Ioannidis (2005), (2019). For a recent overview of Bayesian statistics in psychology and related sciences, read Etz et al. (2018).

See this interesting Youtube video by Julia Galef..

For an entertaining take on related issues see Last Week Tonight: Scientific Studies by John Oliver (2016) (warning: contains strong language and political irony that many could consider offensive – watch at your own risk!).

Preparing for the next class: Read chapters 1 and 2 of BDA3 Post questions or comments on Piazza.

Day 2: Thursday, May 7

Links:

Assignment 2 due May 12 noon

  • Install R or Python on your computer.
  • Install Stan
  • Play with the first 5 or 6 of the ‘Stan tutorials’ here or find some other introductory tutorials. Play with them and post about your experience on Piazza. Discuss your experience at the next class.

Day 3: Tuesday, May 12

Links:

The lectures by Michael Jordan provide a good structure for lectures on the mathematical aspects of Bayesian Inference. The material in the Jordan lectures can be supplemented by references to material in BDA3 or elsewhere. Here’s a list of topics. I’ve posted the list to Piazza where you can select a lecture and a date. They should probably be given in the order below. You can supplement the math with examples in R, Python or STAN.

Construct three assignment questions related to your topic.

Order Topic Sources in Lectures
1 Statistical Decision Theory 3
2 Conjugate Priors 4,5
3 Jeffreys and Reference Priors (overview) 6,7,8
4 Confidence Intervals and Hypothesis Testing 12, 13
5 g-priors for regression and (maybe) Hierarchical Models 13,14

Possible project topics:

  • Bayesian networks
  • Bayesian methods and machine learning
  • Bayesian causal models
  • Expand brms by writing modules to fit lme and nlme models
    • Prepare a presentation (tutorial) on using Stan for longitudinal models with correlation structures: e.g. being able to fit lme and nlme models with ARMA errors of CAR errors.
  • Bayesian approaches to missing data:
    • Challenge: Apply to GLMM Hierarchical data: e.g. migraine data in the spida2 package.
  • Tutorial on some topic, e.g.
    • Dealing with hyposkedasticity in hierarchical models
    • Including potential interactions parsimoniously
    • Any other topic you find interesting

Day 4: Thursday, May 14

Links:

Day 5: Tuesday, May 19

  • Saurabh Panchasara: Statistical Decision Theory
  • An important and sometimes challenging application of Bayesian methods is to multilevel model. Some of you have already studied them and others have not. So we we will review them using the following slides:

Day 6: Thursday, May 21

Day 7: Tuesday, May 26

  • Daniel Dema: Conjugate Priors

Day 8: Thursday, May 28

  • Saurabh Panchasara: Variational Inference

Day 9: Tuesday, June 2

Day 10: Thursday, June 4

  • Duy Tan Doan: Confidence Intervals and Hypothesis Testing

Day 11: Tuesday, June 9

  • Danny Mychakov: g-priors for Regression and Hierarchical Models
  • Project talks

Day 12: Thursday, June 11

  • Project talks

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