Problem Set #1, Due Wednesday, January 26, 2011

Problem Set #2, Due Monday, January 31, 2011

Problem Set #3, Due Monday, February 7, 2011

Problem Set #4, Due Monday, February 14, 2011

Here is a copy of the study sheet for the first quiz.

Problem Set #5, Due Wednesday, March 9, 2011

Problem Set #6, Due Wednesday, March 23, 2011

Group Project: Outline due April 1, Presentations begin April 25

Here is a copy of the study sheet for the second quiz.

Here is the handout, "How to Get Rich Slowly, But Almost $urely," discussed on Monday, April 4.

From time to time, I will put up links to useful web pages here.

Here is the link to the class blog. I will post comments there from time to time, and you can use it to give me feedback, discuss what we do in class, and ask questions.

An Introduction to Bayesian Inference

Life is full of situations where decisions must be made even though the information one has available to make those decisions is incomplete or uncertain, and the consequences of making the wrong decision may be significant. Questions such as: Should I invest in the stock market, and if so, what should I buy, and how much should I invest? If I make the investment, how much can I expect to gain or lose? If I am seriously sick, which of several treatment should I select if they have different side-effects and probability of cure? When sitting on a jury, should I vote that the defendant is guilty or not guilty? As a scientist, should I publish a paper that reports an important new result, even though I cannot be absolutely certain that it is correct? Probability and decision theory can be an important tool in helping us to analyze questions of this sort and make informed decisions. It provides a systematic tool for deciding how our opinions on various issues ought to change as we learn new data. Although the basic principles are very simple, they can be applied in many diverse circumstances, so the tools can be applied to a wide variety of situations. In this course we will investigate how probability and decision theory can help us make important decisions in problems that arise in science, business, the law, medicine, and even daily life.

The texts for this course are *Why Flip a Coin?* by H.W.
Lewis, *Calculated Risks,* by Gerd Gigerenzer, and Predictably Irrational, by Dan Ariely.
I will
also assign readings relevant to the course taken from newspapers,
magazines, journals or other sources.

- How do we make decisions? We look into the major factors that go into making decisions, both small and large. We use our knowledge to make estimates of numbers, both important and unimportant.
- What is probability? We will discuss the concepts of probability first by considering some puzzles and games in which probability plays a role, and then from a more mathematical point of view, discussing the rules of probability. We will also discuss the notion of probability as a way of measuring our degree of belief in a proposition.
- Axioms of probability. We will learn the basic rules of probability
and how to manipulate the mathematical formulas of probability
in simple cases (mostly discrete). We will study the joint and
marginal distributions and learn what they mean. We will learn
what we mean by
*coherence*and why it is important. - Conditional probability. We will see how probability is conditional
on other knowledge that we may have, and is therefore
*subjective*. We will learn the relationship between joint, marginal, and conditional probabilities. We will learn how to use these concepts in practical situations. - Bayes' theorem. We will learn what it is and how to apply it to problems of inference. We will learn about prior probabilities and how to estimate them. We will learn the relationship between prior and posterior probabilities, using Bayes' theorem, and discuss probability theory as a theory of ideal learning. We will discuss the degree to which human beings achieve this ideal. We will discuss the difference between induction and deduction.
- Utility and Loss. These tell us how much we stand to gain or lose depending upon which outcome in an uncertain situation is actually realized. We will see how Utility and Loss are related to each other. We will learn how to calculate the expected utility or loss in specific decision problems, and how this informs our decisions.
- Decision trees. These are a graphical way of representing the components of a decision problem. We will see how to construct them and how to use them in specific decision problems.
- Case studies. We will launch student projects by discussing

* Organizing the project and deciding what question is to be answered

* Where to locate data that would be relevant in making the decision

* How to organize the investigation

* How to present the results - Applications to real-life situations, e.g., investments, jury duty, public policy decision making, gambling.
- We will also touch on some other interesting aspects of probability and decision making, e.g., game theory and the Prisoner's Dilemma, paradoxes of probability, the Bayesian Ockham's razor, etc.

Grading

Two one-hour exams, 15% each, no make-ups possible

One two-hour final exam, 15% (Optional, can count as make-up exam)

Problem sets and papers, 20%

Journal 20%

Group Project, 15%

Classroom participation, 15%

When computing the final grade, I will drop the *lowest*
of your three exam grades. This allows you to (a) recover from
a bad test or (b) make up a test that you unavoidably missed.
It also gives you the option of skipping the final if you don't
think that taking it will improve your grade. Because of this
policy I will *not* give make-up exams.

We will have frequent activities and discussions in class illustrating the topics we are studying. There will also be ungraded quizzes and we will discuss the answers to assist you in understanding the material of the course. It is important that you do the problem sets in a timely fashion, since our discussions will often refer to them. I require you to work in small groups of 3 or 4 students on the problems, and each group should turn in one write-up (making sure that if there are disagreements within the group, they are noted clearly). In addition, your small group will be your study group outside of class.

I will take attendance. Classroom participation is based on attendance as well as participation in the discussions.

This is a Significant Writing Component course, and a significant
part of your grade comes from your writing, including your Journal.
You are required to keep a *journal*. Every week I want you
to write in your journal a 3-4 page essay (this means at least
three full pages but not more than four pages), to be described
below. I want you to use a *12-point* *proportionally-spaced
serif font* such as Times Roman, *double spaced*. This
is easier to read than other fonts such as Helvetica or Courier.
You should keep your journal in looseleaf form, such as a paper
binder. Please do not use a three-ring type binder...these are
bulky and I don't want to have to carry them around. Instead,
use a simple heavy paper binder like the one I showed you in class.
Insert your weekly essays so that the most recent essay is at
the *front* of the binder. Each week (generally on Friday
unless special circumstances obtain) your complete journal to
date should be turned in for me to read over the weekend. I will
comment on what you have written and return the journal the following
week.

Unless otherwise specified, the topic of your weekly journal essay will be to describe in your own words the most interesting or surprising discovery that you made during our discussions in the previous week. If some material was puzzling or even unbelievable, you should address that in your essay as well. Please explain why you chose the particular topics you did, and discuss how the subject of your essay might be important in making decisions. You should write the essay using standard English sentences and paragraphs, with a beginning, a middle, and an ending. I may ask you to revise the essay based on the comments that I make.

I prefer you to turn in your journal in its binder. Under unusual
circumstances (e.g., you must unavoidably miss the class when
the journal is due) I will accept the essay by E-mail; If you
send it by E-mail, please send it as a *text* file for compatibility
reasons. Do not send Microsoft Word or other formatted file formats.

Religious Holidays: Students have the right to practice the religion of their choice. Each semester students should submit in writing to their instructors by the end of the second full week of classes their documented religious holiday schedule for the semester. Faculty must permit students who miss work for the purpose of religious observance to make up this work.

This page is under construction. Stay tuned for new material.

. My home page is located here.
My office is 107 Lord (in the basement). Office hours MWF 11:00-12:00, or by appointment. For convenience, I can meet students
immediately after class in the Honors College, instead of hiking
across campus to the Math Department (it's a 10-15 minute walk).

All materials at this website Copyright (C) 1994-2011 by William H. Jefferys. All rights reserved.