Required Texts:
Many articles and resources will be made available as they arise in the
syllabus, either online or as class handouts. Students are responsible to
download and read all web-based articles before the relevant class, as
indicated on the course schedule. The course web page is http://homepages.wmich.edu/~mcgrew/gradprob08.htm
Course Description: The aim of this course is to give students a firm understanding of
elementary probability theory, including its basic applications to longstanding
philosophical questions. Along the way we will give considerable attention to
philosophical issues involving the interpretation of the notion of probability,
the interplay between probability and the history and philosophy of science,
and some central epistemological questions regarding testimonial evidence,
confirmation, and reasonable belief. By the end of the course, students should
have a clear sense of the scope of elementary probability theory, facility in
its use, an understanding of the relations between logic and probability
theory, and an understanding of the epistemological implications of and issues
surrounding the use of Bayes’s Theorem.
Although this course does not have any prerequisites,
it does presuppose a nodding acquaintance with highschool mathematics. We will
move rapidly and the material covered will, at times, be more difficult than
that in the average graduate seminar. Exams will include proofs and
calculations as well as conceptual questions. Students who are unable or
unwilling to do this sort of work will not earn a passing grade and are
strongly urged to take some other course.
Course Requirements: This course meets Tuesday and Thursday of each week from 2:00 to 3:15
p.m. except for scheduled holidays. Late work will not generally be accepted
without a medical excuse. Attendance and class participation are taken into
account in the determination of the final grade. In particular, I reserve the
right to subtract five points from the final semester grade for each unexcused
absence beyond the third. Students are expected to come to class having done
the reading indicated on the syllabus and may be asked (with or without advance
warning) to summarize it for the class.
Academic Integrity: You are responsible for making yourself aware of and
understanding WMU's policies and procedures pertaining to academic integrity,
including cheating, fabrication, falsification and forgery, multiple
submission, plagiarism, complicity and computer misuse. If there is reason to
believe you have been involved in academic dishonesty, you will be referred to
the Office of Student Conduct. You will be given the opportunity to review the
charge(s). If you believe you are not responsible, you will have the
opportunity for a hearing. You should consult with me if you are uncertain
about an issue of academic honesty prior to the submission of an assignment or
test.
Grading:
Aside from attendance, the course grade is based on three exams, occasional quizzes
and homework, and the quality (which is not necessarily measurable by the
quantity) of class participation, including demonstrated preparation for class.
The exams will be weighted equally in calculating the final grade,
and the quizzes and homework will count for not more than 25% of the total
grade. The grading scale is:
|
A 93-100 |
B 83-87 |
C 73-77 |
D 60-67 |
|
B/A 88-92 |
C/B 78-82 |
D/C 68-72 |
E below 60 |
Note: The following course schedule is tentative. Because
the material is difficult, some of it may take longer than the indicated time.
You are expected to do the readings in accordance with the sequence of topics
even if we are off schedule. Please note that we are not following a direct
path through either of the texts; the order of readings is determined by the
syllabus alone. Any alterations in dates for exams will be announced in class
ahead of time.
Week 1: From
Logic to Probability
Limitations of deductive logic; Probability as a
generalization of logic; Distinction between probabilities of conditionals and
conditional probabilities; Joint probability distributions and their uses; Math
for understanding probability
Visualizing probability: Venn diagrams and the
probabilistic poster board
Week 2: The
Basic Rules of Probability
Two constraints: normality (0 ≤ P(x) ≤ 1)
and certainty (P(Ω) = 1); mutually exclusive and
jointly exhaustive propositions; the special and general addition rules; the
special multiplication rule
Week 3:
Conditional probability
Definition of conditional probability; constraints on
conditional probabilities; the general multiplication rule; independent
propositions; screening off
Week 4: The Rules of Probability
More about independence and screening; certainty, contradiction,
and probabilities of 0 and 1; Exclusion and entailment; The Equivalence
Theorem; Partitions; Some questions about the rules of inference
Visualizing probability: Schlesinger squares and
counterexamples
Weeks 5 and 6:
Bayes’s Theorem
Derivation of Bayes’s Theorem; Eight Versions of
Bayes’s Theorem; The base rate fallacy; Swamping of the priors; insensitivity
to marginal changes in the base rate;
Sticky priors and special cases; Updating
probabilities; Questions about conditionalization; Problems with priors and
partitions
Visualizing probability: Bayes nets; McGrew’s square
diagrams
Week 7: The
Theorem on Total Probability and its Significance
Theorem on total probability; Joint distributions
revisited; Plantinga’s “Principle of Dwindling Probabilities”
Visualizing probability: Overlapping regions and
non-exclusive events; probability lattices
Week 8:
Probability, Confirmation, and Old Evidence
Qualitative confirmation theory and its problems;
Defining the concepts of evidence, confirmation, and support; The problem(s) of
old evidence; The measurement of evidential support; Resolution of some
problems of scientific inference
Readings: Christiansen paper
Weeks 9 and 10:
Combined Evidence and Testimony
Combined evidence; Credibility and veracity; Questions
of independence; Hume’s critique of miracles: a Bayesian analysis; Modeling
combined testimony
Visualizing probability: geometric combination of
independent evidence
Week 11: The
Problem of Induction
Hume’s problem; Non-monotonicity: Hume’s problem
reformulated; Solutions to the problem of induction; Bernoulli's theorem; The
fundamental question: success or rationality?
Weeks 12 and 13:
Inference to the Best Explanation
Claims for IBE by its adherents; IBE and theoretical
virtues; The paucity objection; The guiding challenge; Peirce’s schema for
abductive inference; Epistemic accessibility; Ceteris paribus theorems; Theoretical consilience: a Bayesian
account
Week 14:
Probability and Decision
Utilities and their ordering; Calculating expected
utility; Probability defined in terms of utilities; Dominance, independence,
and choice; Strategies in decision making (minimax, satisficing, etc.);
Paradoxes of decision theory