ࡱ> df czo^%` Ibjbj"x"x @@A&fff4^!^!^!h!,"|E2z#~ -".....'/;/ jElElElElElElE$GhIEfv0..v0v0E..E666v0l.f.jE6v0jE66r@T:,fA.n# '^!2bADTE0E A)J4)JA)JfAXG/>/,6/$/G/G/G/EE5^G/G/G/Ev0v0v0v0^!^! Statistics 550 Notes 2 Reading: Section 1.2. I will add one office hour, Wed., 9-10. Prof. Smalls office hours: Tues., 4:45-5:45; Wed., 9-10; Thurs., 4:45-5:45. TA Dan Yangs office hours, 431.2 Huntsman Hall, Tues., 2-3. I. Frequentist vs. subjective probability (Section 1.2) Model: We toss a coin 3 times. The tosses are iid Bernoulli trials with probability p of landing heads. What does  EMBED Equation.DSMT4 mean here? Mathematical definition A probability function for a random experiment is a function on the sample space S of possible outcomes of the experiment that satisfies: Axiom 1: For all (measurable) events E:  EMBED Equation.DSMT4  Axiom 2:  EMBED Equation.DSMT4  Countable additivity Axiom 3: For any sequence of mutually exclusive (measurable) events,  EMBED Equation.DSMT4 . For a single coin toss, the axioms imply that  EMBED Equation.DSMT4 must be between 0 and 1 and the probability of the coin landing tails must be  EMBED Equation.DSMT4 . Meaning of probability as a model for the real world: Frequentist probability In many independent coin tosses, the proportion of heads would be about  EMBED Equation.DSMT4 . The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. Result: 2048 heads, or relative frequency 2048/4040=0.5069 for heads. Around 1900, the English statistician Karl Pearson heroically tossed a coin 24,000 times. Result: 12,012 heads, a relative frequency of 0.5005. While imprisoned by the Germans during World War II, the Australian mathematician John Kerrich tossed a coin 10,000 times. Result: 5067 heads, a relative frequency of 0.5067. Subjective (personal) probability Probability describes a persons degree of belief about a statement. The probability that the coin lands heads is 0.5 Subjective interpretation This represents a persons personal degree of belief that a toss of the coin will land heads. Specifically, if offered the chance to make a bet in which the person will win  EMBED Equation.DSMT4 dollars if the coin lands heads and lose EMBED Equation.DSMT4 dollars if the coin lands tails, the maximum  EMBED Equation.DSMT4  the person would play the game with is  EMBED Equation.DSMT4  dollars. In general, a persons subjective probability of an event  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , is the value of  EMBED Equation.DSMT4 for which the person thinks a bet in which she will win  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur is a fair bet (i.e., has expected profit zero). Subjective probability rejects the view of probability as a physical feature of the world and interprets probability as a statement about an individuals state of knowledge. Coins dont have probabilities, people have probabilities. Persi Diaconis. An advantage of subjective probability is that it can be applied to things about which we are uncertain which cannot be envisioned as being part of a sequence of repeated trials: Will the Philadelphia Eagles win the Super Bowl this year? Was there life on Mars 1 billion years ago? Did Lee Harvey Oswald act alone in assassinating John F. Kennedy? What is the 353rd digit of  EMBED Equation.DSMT4 ? If I say the probability that the 353rd digit of  EMBED Equation.DSMT4 is 5 is 0.2, I mean that I would consider it a fair bet for me to gain 0.8 dollars if the 353rd digit of  EMBED Equation.DSMT4 is 5 and lose 0.2 dollars if the 353rd digit of  EMBED Equation.DSMT4 is not 5. II. Coherence and the axioms of probability A rational person should have a coherent system of subjective probabilities: a system is said to be incoherent if there exists some set of bets such that the bettor will take the set of bets but will lose no matter what happens. If a persons system of probabilities is coherent, then they must satisfy the axioms of probability. Example: Proposition 1: If  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 are mutually exclusive events, then  EMBED Equation.DSMT4 . Proof: Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 be a persons subjective probabilities for these events. Suppose that  EMBED Equation.DSMT4 differs from  EMBED Equation.DSMT4 . Then the person thinks that the following bets are fair: (i) a bet in which the person will win  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur; (ii) a bet in which the person will win  EMBED Equation.DSMT4 if  EMBED Equation.DSMT4 occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur; and (iii) a bet in which the person will win  EMBED Equation.DSMT4 if  EMBED Equation.DSMT4 occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur Say,  EMBED Equation.DSMT4 and let the difference be  EMBED Equation.DSMT4 . A gambler offers this person the following bets: (a) the person will lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 occurs and win EMBED Equation.DSMT4 if  EMBED Equation.DSMT4 does not occur; (b) the person will win  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4  occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur; (c) the person will win  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4  occurs and lose  EMBED Equation.DSMT4 dollars if  EMBED Equation.DSMT4 does not occur. According to the persons subjective probabilities, bets (a)-(c) are all expected to yield a profit, so the person takes all of them. However, (1) Suppose  EMBED Equation.DSMT4 occurs. Then the person loses  EMBED Equation.DSMT4 from bet (a), wins  EMBED Equation.DSMT4 from bet (b) and loses  EMBED Equation.DSMT4 from bet (c). The persons profit is  EMBED Equation.DSMT4  (2) Suppose  EMBED Equation.DSMT4 occurs. The persons profit is  EMBED Equation.DSMT4  (3) Suppose neither  EMBED Equation.DSMT4 nor  EMBED Equation.DSMT4 occurs. The persons profit is  EMBED Equation.DSMT4  Thus, the gambler has put the person in a position in which the person is guaranteed to lose  EMBED Equation.DSMT4 no matter what happens (when a person accepts a set of bets that guarantees that they will lose no matter what happens, it is said that a Dutch book has been set against them). So it is irrational for the person to assign  EMBED Equation.DSMT4 . A similar argument can be made that it is irrational to assign  EMBED Equation.DSMT4 . Similar arguments can be made that a rational persons subjective probabilities should satisfy the other axioms of probability: (1) for an event  EMBED Equation.DSMT4 , EMBED Equation.DSMT4 ; (2)  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 is the sample space. Although, from Proposition 1, it is clear that a rational persons personal probabilities should obey finite additivity (i.e., if  EMBED Equation.DSMT4 are mutually exclusive events,  EMBED Equation.DSMT4 ), there is some controversy about additivity for a countable infinite sequence of mutually exclusive events (see J. Williamson, Countable Additivity and Subjective Probability, The British Journal for the Philosophy of Science, 1999). We assume countable additivity holds for subjective probability. The mathematical axioms of probability, and hence all results in probability theory, hold for both subjective probabilities and frequentist probabilities -- it is just a matter of how we interpret the probabilities. In particular, Bayes theorem holds for subjective probability. Let  EMBED Equation.DSMT4 be an event and  EMBED Equation.DSMT4 be mutually exclusive and exhaustive events where  EMBED Equation.DSMT4 for all  EMBED Equation.DSMT4 . Then  EMBED Equation.DSMT4 . III. The Bayesian Approach to Statistics Example 1 from Notes 1: Yao Mings free throws in the 2007-2008 season and future seasons are iid Bernoulli trials with probability  EMBED Equation.DSMT4 of success. In the 2007-2008 season, Yao made 345 out of the 406 free throws he attempted (85.0%). What inferences can we make about  EMBED Equation.DSMT4 ?  The Bayesian approach to statistics uses subjective probability and regards  EMBED Equation.DSMT4 as being a fixed but unknown object over which we have beliefs given by a subjective probability distribution and then uses the data to update our beliefs using Bayes rule. Prior distribution: Subjective probability distribution about parameter vector ( EMBED Equation.DSMT4 ) before seeing any data. Posterior distribution: Updated subjective probability distribution about parameter vector ( EMBED Equation.DSMT4 ) after seeing the data. Bernoulli trials example: Suppose that  EMBED Equation.DSMT4  are iid Bernoulli with probability of success  EMBED Equation.DSMT4 . We want our prior distribution for  EMBED Equation.DSMT4 to be a distribution that concentrates on the interval [0,1]. A class of distributions that concentrates on [0,1] is the two-parameter beta family. The beta family of distributions: The density function  EMBED Equation.DSMT4 of the beta distribution with parameters  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4  The mean and variance of the Beta(r,s) distribution are  EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  respectively. See Appendix B.2 for details.   Suppose our prior distribution for  EMBED Equation.DSMT4 is Beta(r,s) with density  EMBED Equation.DSMT4  and we observe  EMBED Equation.DSMT4 . Using Bayes theorem, our posterior distribution for  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4  The last expression is proportional to the Beta( EMBED Equation.DSMT4 ) density so the posterior density for  EMBED Equation.DSMT4 is Beta( EMBED Equation.DSMT4 ). Families of distributions for which the prior and posterior distributions belong to the same family are called conjugate families. The posterior mean is thus equal to  EMBED Equation.DSMT4  and the posterior variance is  EMBED Equation.DSMT4 . Returning to our example in which Yao Ming made 345 out of the 406 free throws he attempted (85.0%) in 2007-2008, if we had a Beta(1,1) [uniform] prior, then the posterior distribution for Yaos probability of making a free throw in the next season is (345+1,406-345+1)=Beta(346,62). The posterior mean is 0.848 and the posterior standard deviation is 0.018. A valuable feature of the Bayesian approach is its ability to incorporate prior information. Returning to our example in which Yao Ming made 345 out of the 406 free throws he attempted (85.0%) in 2007-2008, we could use information from Yaos previous seasons. Free Throws MadeFree Throws AttemptedFT%2002-20033013710.8112003-20043614460.8092004-20053894970.7832005-20063373950.8532006-20073564130.861Total174421220.822 A way to think about the Beta( EMBED Equation.DSMT4 ) prior is the following: Suppose that if we had no informative prior information on  EMBED Equation.DSMT4 , we would use a uniform (Beta (1,1)) prior. Then a Beta( EMBED Equation.DSMT4 ) prior means that our prior information is equivalent to seeing  EMBED Equation.DSMT4 successes and  EMBED Equation.DSMT4  failures prior to seeing the current data. So, for example, a Beta(82,18) prior is equivalent to saying that our prior information is equivalent to seeing 81 made free throws and 17 missed free throws prior to seeing our data. If we consider all the free throws from seasons prior to 2007-2008 to be equivalent to free throws in 2007-2008, we would use a Beta(1744+1,2122-1744+1) =Beta (1745, 379) prior. If we consider each free throw in prior seasons to be equivalent to half a free throw of information, we would use a Beta (1744*0.5+1, (2122-1744)*0.5+1) =Beta (873, 190) prior.  When there is more data and prior beliefs are less strong, the prior distribution does not play as strong a role. The posterior mean is  EMBED Equation.DSMT4  The posterior mean is a weighted average of the sample mean and the prior mean, with the weight on the sample mean increasing to 1 as  EMBED Equation.DSMT4 . Example 2: A cofferdam protecting a construction site was designed to withstand flows of up to 1870 cubic feet per second (cfs). An engineer wishes to compute the probability that the dam will be overtopped during the upcoming year. Over the previous 25-year period, the annual maximum flood levels of the dam had ranged from 629 to 4720 cfs and 1870 cfs had been exceeded 5 times. Modeling the 25 years as 25 independent Bernoulli trials with the same probability  EMBED Equation.DSMT4 that the flood level will exceed 1870 cfs and using a uniform prior distribution for  EMBED Equation.DSMT4 (which corresponds to a Beta (1,1)), the prior and posterior densities are  IV. Bayesian Inference for the Normal Distribution Suppose  EMBED Equation.DSMT4 iid  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 known, and our prior on  EMBED Equation.DSMT4 is  EMBED Equation.DSMT4 . The posterior distribution is proportional to  EMBED Equation.DSMT4 Thus, the posterior distribution is  EMBED Equation.DSMT4  Note that the posterior mean is a weighted average of the sample mean and the prior mean, with the weight on the sample mean increasing to 1 as  EMBED Equation.DSMT4 . V. Bickel and Doksums perspective on Bayesian models. (a) Bayesian models are useful as a way to generate statistical procedures which incorporate prior information when appropriate. However, statistical procedures should be evaluated in a frequentist (repeated sampling) way: For example, for the iid Bernoulli trials example, if we use the posterior mean with a uniform prior distribution to estimate  EMBED Equation.DSMT4 , i.e.,  EMBED Equation.DSMT4 , then we should look at how this estimate would perform in many repetitions of the experiment if the true parameter is  EMBED Equation.DSMT4 for various values of  EMBED Equation.DSMT4 . More to come on this frequentist perspective in Chapter 1.3. (b) We can view the parameter as random and view the Bayesian model as providing a joint probability distribution on the parameter and the data in a frequentist probability sense. Consider the frequentist probability model  EMBED Equation.DSMT4  for the probability distribution of the data  EMBED Equation.DSMT4 . The subjective Bayesian perspective is that there is a true unknown  EMBED Equation.DSMT4 and our goal is to describe our beliefs about  EMBED Equation.DSMT4 after seeing the data  EMBED Equation.DSMT4 . This requires specifying a prior distribution  EMBED Equation.DSMT4 for our beliefs about  EMBED Equation.DSMT4 ; the posterior distribution describes our beliefs about  EMBED Equation.DSMT4 after seeing the data  EMBED Equation.DSMT4 . Bickel and Doksums viewpoint is to see a Bayesian model as a model for the joint probability distribution of  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 is considered random. Specifically, a Bayesian model consists of a specification of the marginal distribution of  EMBED Equation.DSMT4 , which is the prior distribution  EMBED Equation.DSMT4  and a specification of the conditional distribution of the data  EMBED Equation.DSMT4 , i.e.,  EMBED Equation.DSMT4 . For example, if the data  EMBED Equation.DSMT4  are iid Bernoulli trials with probability  EMBED Equation.DSMT4 of success and the prior distribution for  EMBED Equation.DSMT4  is Beta(r,s), then the joint probability distribution for  EMBED Equation.DSMT4  is generated by: 1. We first generate  EMBED Equation.DSMT4 from a Beta(r,s) distribution. 2. Conditional on  EMBED Equation.DSMT4 , we generate  EMBED Equation.DSMT4  iid Bernoulli trials with probability  EMBED Equation.DSMT4 of success. Although the conceptual basis for regarding  EMBED Equation.DSMT4 as random in a frequentist sense is not clear, this point of view is useful for developing properties of Bayesian procedures.     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