Although statistical techniques are employed in virtually every scientific discipline, there are still several unresolved debates concerning the foundations of statistics. For example, there is considerable disagreement about the value of random sampling and whether the rule for terminating an experiment ought to affect the interpretation of its results. This course is an introduction to these issues and others concerning proper experimental design and norms for statistical/inductive inference. We will focus on the differences between the so-called ``frequentist'' and ``Bayesian'' paradigms and the arguments offered in favor of each.
Because philosophical discussions of statistical techniques often presuppose a particular interpretation of probability, we will begin by reviewing two common interpretations of probability: subjective and propensity. For the remainder of the semester, we will discuss a number of issues that concern the debate between so-called Bayesians and Frequentists. Topics will include the likelihood principle, exchangeability, Bayesian convergence theorems, the value of randomization in experimental design, and stopping rules.
The course has two central goals. Namely, by the end of the semester, students should be able to (1) describe the differences between frequentist and Bayesian methods, and (2) evaluate the arguments offered in favor of (and against) each. By acquiring these two skills, students should be better able to evaluate the strength of various everyday inductive arguments (e.g., those appearing in newspapers) and rudimentary (but common) statistical arguments appearing in scientific journals.
Philosophical thinking is a skill, not unlike playing the piano, riding a bike, or dancing. Learning a new skill requires practice, and the best way to practice philosophical thinking is to write and engage in spirited (but polite) debates with other philosophers. Thus, the central requirement of this course is to write three short papers (between five and seven pages in length) in which you summarize an argument from the readings and defend it from potential objections. More details about the topics of the three papers will be available below as the course progresses.
Additionally, before four of the fourteen classes, you must write a one-page summary of the assigned readings. Please bring both the assigned readings and your summaries with you to class. Some students view frequent assignments/assessments as ``busy work'' or as an instructor's attempt to gauge which students are working hardest. That is is not my intention at all. The philosophy of statistics is a complex subject, and consequently, many of the assigned readings are somewhat difficult. When faced with hard-to-understand texts, it is easy to become discouraged and to give up. One of my central duties, as an instructor, is to ensure that you do not give up when concepts and/or arguments are initially difficult to understand. The purpose of the one-page summaries is threefold: (i) to force you to read the assigned texts closely and actively, (ii) to prepare you for class discussions in which we will clarify and build upon the readings, and (iii) to provide me with feedback about which concepts are most difficult for students to understand.
Finally, one cannot study the philosophy of statistics without knowing a bit about its mathematics. Unfortunately, I cannot teach you statistics and philosophy at the same time, and so if you have not studied elementary statistics recently (or at all), I ask that you to review a bit of basic probability theory and statistics during the course of the semester. I will spend only one class reviewing concepts from a first semester statistics class. On the reading schedule in the pdf copy of the syllabus, I have recommended sections of an introductory statistics book (Larry Wasserman's All of Statistics) that discusses topics with which you should be familiar. These readings are not required, but I will presume knowledge of basic probability theory and elementary classical statistics after a few weeks. In particular, by the end of the first unit, you should able to define basic terms of probability theory (e.g., random variable, conditional probability, independence, expectation, etc.) and to employ these concepts in (very) short calculations and proofs. Discussions during the second and third units of the course will assume that you can define the following terms from elementary (frequentist) statistics: p-value, confidence interval, power, and consistency. Because many students are not introduced to Bayesian methods in introductory statistics courses, I assume no knowledge of such methods for this course.
To turn in assignments, you may either email me or bring a copy of your essay (or summary) to class.
Twice during the course of the semester, I will grant you a two-day an extension on an assignment, no questions asked. That is, instead of turning assignment in on Monday, you may email me a copy of your work some time on Wednesday. If you need more than two additional days to complete an assignment, please talk to me.
If you are a registered student, all of the course files (e.g., syllabus and readings) can be downloaded from here.
Below is a table indicating readings and assignments that are due each class. If you are a registered student in the class, then you can download the readings from the link in the "Course Files" section above. The pdf copy of the syllabus also contains (1) suggested additional readings, and (2) a suggested schedule for reviewing material from an introductory statistics class.
| Date | Topic | Readings | Assignment | |
|---|---|---|---|---|
| 8/4 |
Course Introduction
Lecture 1 Slides |
Hajek. "Interpretations of Probability." Sections 1 and 2.
Carnap. Logical Foundations of Probability. Chapter 1. Suppes. Representation and Invariance of Scientific Structures. Pages 51-70. |
None. | |
| 15/4 |
Subjective Probability: Dutch Books, Comparative Probability, and de Finetti's Theorem
Lecture 2 Slides |
Required:
Kadane. Principles of Uncertainty. Section 1.1
De Finetti. "Foresight: Its Logical Laws, Its Subjective Sources." Pages 97-104. Joyce. "A Non-Pragmatic Vindication of Probabilism" Savage. Foundations of Statistics. Pages 1-26. |
None. | |
| 22/4 | No Class | |||
| 29/4 |
Subjective Probability and Utility: Representation Theorems of Savage, Von Neumann and Morgenstern, and Anscombe
and Aumann
Lecture 3 Slides Notes on Savage |
Savage. Foundations of Statistics. Pages 27-40.
Von Neumann and Morgenstern. Theory of Games and Economic Behavior. Chapter 1, Section 3. Anscombe and Aumann. "A Definition of Subjective Probability." |
None. | |
| 6/5 | Arguments for Conditionalization: Dynamic Dutch Books
Discussion Questions |
Skyrms. "Dynamic Coherence and Probability Kinematics." Sections 1, 2, 5.1, and 6.
Levi. "The Demons of Decision." 193-211. Savage. Foundations of Statistics. Pages 43-44. |
First Summary | |
| 13/5 | The Propensity Interpretation
Lecture 5 Slides |
Suppes. Representation and Invariance of Scientific Structures. Pages 202-225.
Strevens. "Inferring Probabilities from Symmetries" Von Plato. "The Method of Arbitrary Functions" |
Second Summary | |
| 20/5 | Bayesian and Frequentist Methodology |
Wasserman. All of Statistics. Chapter 6, 10.1-10.2.
DeGroot. Probability and Statistics. Sections 6-6.4. |
None. | |
| 27/5 |
Criticisms of Frequentist Methdology
Discussion Questions Lecture 8 Slides |
Goodman. "The P-Value Fallacy"
Howson and Urbach. Scientific Reasoning: The Bayesian Approach. Chapter 5. |
First Paper | |
| 3/6 |
More Criticisms of Frequentist Methodology
Lecture 9 Slides |
Berger and Wolpert. The Likelihood Principle. Chapter 2.
Birnbaum. "On the Foundations of Statistical Inference" |
None. | |
| 10/6 | No Class (University Holiday) | |||
| 17/6 | Frequentism Defended
|
Mayo and Spanos. "Error Statistics." Sections 1-2, Pages 153-183. | Third Summary | |
| 24/6 | Bayesian Inference | Jose Bernardo. "Modern Bayesian Inference: Foundations and Objective Methods." Pages 263-306. | None | |
| 1/7 | Randomization
|
Fisher. "The Design of Experiments." Chapter 2.
Kadane and Seidenfeld. "Randomization in a Bayesian Perspective." |
Second Paper | |
| 8/7 | Stopping Rules
|
Mayo and Spanos. "Error Statistics:"
Kadane, Schervish, and Seidenfeld. "When several Bayesians agree there will be no reasoning to a foregone conclusion." |
Fourth Summary | |
| 29/7 | Third Paper Due for Graduating Students | |||
| 19/9 | Third Paper Due for Non-Graduating Students | |||