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What is the probability you are a Bayesian

Bayesian methodology continues to be widely used in statistical applications. As a result, it is increasingly important to introduce students to Bayesian thinking at early stages in their mathematics and statistics education. While many students in upper level probability courses can recite the differences in the Frequentist and Bayesian inferential paradigms, these students often struggle using Bayesian methods when conducting data analysis. Specifically, students tend to struggle translating subjective belief to the specification of a prior distribution and the incorporation of uncertainty in the Bayesian inferential approach. The purpose of this paper is to present a hands-on activity involving the Beta-Binomial model to facilitate an intuitive understanding of the Bayesian approach through subjective problem formulation which lies at the heart of Bayesian statistics.


Introduction
1) state that, "The Bayesian approach to statistical design and analysis is emerging as an increasingly effective and practical alternative to the frequentist one." Given the increasing popularity of the Bayesian approach to address statistical problems, it is important to expose students to this perspective. In particular, Stangl (1998) provides a nice discussion of some of the issues and challenges for incorporating the Bayesian paradigm into Statistics courses. It is especially important to provide students with meaningful applications and tangible experiences with Bayesian statistics to aid in the learning process. For example, Jessop (2010) presents a simple classroom activity demonstrating Bayesian inference in an introductory statistics course. Albert (2000) utilizes a sample survey project in an introductory course to introduce statistical inference from a Bayesian perspective. There are numerous references like these two that illustrate Bayesian inference using the well-known Bayes' Theorem for events (Casella and Berger 2002, p. 23). However, Bayes' Theorem does not address the generality of the Bayesian approach that is based upon prior and posterior probability distributions which incorporate uncertainty on the parameters (Robert 2001, p. 9). Though not as common, classroom activities have been proposed to illustrate this more general Bayesian approach. Albert and Rossman (2009, Ch. 17) provide a number of problems for studying proportions using continuous probability distributions. Their examples are designed for students in an introductory statistics course who are familiar with basic probability concepts. Kern (2006) describes an entertaining application of Bayesian inference using the game Pass the Pigs ® to model multinomial probabilities with the Dirichlet probability distribution. This class activity is designed for undergraduates and graduates majoring in mathematics and statistics. DiPietro (2004) uses a psychological therapy project to engage students with advanced Bayesian methodology such as posterior simulation, sensitivity analysis, and Bayes factors. This project is designed for first-year graduates in an applied Bayesian methods class that is computationally intensive.
In light of the competing paradigms to statistical inference, many statisticians have been asked, "Are you a Bayesian or Frequentist?" Definitive answers to this question have led to acrimonious debates. A Frequentist is thought of as one who would address a particular statistical inference problem from a perspective which treats the observations as random (from hypothetical replications) and the parameter of interest as an unknown, fixed constant. A Bayesian would address a particular statistical inference problem by regarding the experimental outcome as fixed and treating the parameter of interest as random (reflecting uncertainty in knowledge about the parameter). In this case, the parameter has a probability distribution before the data is collected (prior distribution) and a probability distribution after the data is collected (posterior distribution). Many students in upper division undergraduate or beginning graduate courses in statistics have some familiarity with the Frequentist versus Bayesian debate, but they generally have not been trained in Bayesian statistics. Their background in statistical methodology has been almost exclusively from a Frequentist perspective.
In this manuscript, we propose an assignment to engage the students in the Bayesian paradigm which leads them away from the 'yes' or 'no' dichotomy of being a Bayesian. We consider the question: What is the probability you are a Bayesian? We will henceforth denote this question as 'Q.' Since Q involves the concept of Bayesian statistics, it is necessary to consider what a Bayesian would do. Also, since Q is inherently subjective, it nicely facilitates the incorporation of a prior distribution. This question also recognizes that personal growth as a statistician is an ongoing process. What was posterior belief in an earlier experiment becomes prior belief in the next (Press 2003, p. 6). We pose Q to students in an upper-level statistical inference course (i.e., a mix of upper division undergraduate and first-year graduate students). Texts for such courses include Berry and Lindgren (1996), Bickel and Doksum (2001), Casella and Berger (2002), and Hogg, McKean, and Craig (2005). Students are expected to use a Bayesian approach to provide their personal answer to Q. Students are asked to provide a prior beta distribution to characterize their subjective belief about being a Bayesian. Next, each student is given a randomly selected inferential problem from a collection of problems appropriate to the particular class. For their problem, each student is expected to collect data and use that the data to conduct statistical inference utilizing either a Frequentist or Bayesian paradigm. The choice of either a Frequentist or Bayesian approach for their randomly chosen problem is treated as a single observation from the Binomial (or Bernoulli) distribution with which they update their beta prior distribution on Q. The transition from prior to posterior distribution utilizes the Beta-Binomial model which is frequently the first model used in teaching Bayesian statistics. For convenience, Table 1 shows the complete assignment. Let 'Q' denote the following question: "What is the probability that you are a Bayesian?" You are to answer Q utilizing a Bayesian inferential approach. Please address the following in your write-up. You have two weeks to complete this assignment.
This manuscript consists of the following sections. Section 2 provides a description of student background in this upper-level course along with some of the issues we commonly encounter as students attempt to wrap their minds around the Bayesian inferential paradigm. Section 3 describes the class instruction prior to giving the assignment where we characterize the population of problems and provides specific insights into the Bayesian approach for this assignment. Sections 4, 5, and 6 provide details about the sections of the assignment including the prior distribution, the experiment, and the posterior distribution, respectively. These types of details arise in office hours and class discussions as students are working through the assignment. Guidance will be given in each of these sections for grading the assignment along with common errors. Example student scenarios will be used in each of these sections to illustrate the results of this project. Section 7 contains concluding remarks, points for class discussion, as well as feedback about the success of this assignment.

Background
Since the intended course for this exercise is an upper-level undergraduate course in statistical inference, students are expected to have a background in probability theory and integral calculus. It is common to initiate the discussion of Bayesian statistics in the context of point estimation once Frequentist approaches (method of moments, maximum likelihood) have been discussed. Prior to going through this exercise, students have been exposed to in-class derivations of exact posterior distributions for the Beta-Binomial model and the Normal-Normal model. When writing the expression 'Distribution1-Distribution2,' students understand that 'Distribution1' denotes the prior distribution while 'Distribution2' denotes the distribution of the population from which the data is sampled. The Beta-Binomial and Normal-Normal models are typically used in course texts (e.g., Casella and Berger 2002, pp. 324-326) to introduce the Bayesian approach since they are conjugate families in which the posterior distribution is from the same family of distributions as the prior distribution. Students have also been taught that there are three main distinctions which characterize the Bayesian approach: The parameter of the population model is treated as a random variable. (B2).
A prior distribution for the parameter is utilized.
The inferential procedure is based upon the experimental outcome.
These distinctions are in opposition to the Frequentist approach in which: (F1) The parameter is treated as a fixed unknown constant; (F2) A prior distribution is not utilized; (F3) The inferential procedure is based upon a collection of outcomes which could occur in hypothetical replications of the experiment. While students can generally recite such differences between the two inferential paradigms, they often struggle making the transition to the Bayesian perspective when actually conducting statistical inference. Many of them need more than discussion points (B1), This assignment can be organized in several ways, but for the purpose of this manuscript, we present the exercise as a single homework assignment which students have two weeks to complete. Students are permitted to ask questions or seek clarifications about the assignment from the instructor. Once the assignment is completed and assessed, a class session is spent discussing results, challenges encountered, and understandings gained from the assignment. Through this exercise, students personally experience a variety of important steps in the Bayesian decision making paradigm. These experiences include: (a) translating subjective belief to the specification of a prior distribution; (b) carrying out a Bayesian analysis with their own data; and (c) incorporating uncertainty in the Bayesian inferential approach.

Insights into the Bayesian Approach for this Problem
When the assignment in Table 1 has been handed out, students may feel overwhelmed asking, "Now what am I going to do?" It can be reassuring and instructional for the students to take class time to put the assignment in perspective and to remind them of some of the basic tenants of Bayesian inference.
We first consider a population of problems   P from which Q should be answered. Thus, the answer to Q will be conditional on P. Let Q  denote the proportion of problems in P in which the student applies the Bayesian paradigm, or equivalently, the probability that the student is a Bayesian. Restrictions on P should include: 1. the student has the capability to carry out their preferred paradigm for that problem and 2. the student is easily able to collect their own data for that problem. For our particular class, we further limit P to consist of a large collection of proportion problems. Not only do we do this to satisfy our two requirements above, but we also like to have students work with the Beta-Binomial model for both their experiment and for obtaining their answer to Q. This approach allows them to see the similarities and differences between the inferential approach they choose and the Bayesian approach required to answer Q. A course instructor using this assignment would not have to use the same set of proportion problems, or to even use proportion problems to formulate P.
This population P may be quite large or even infinite. In order to make the assignment feasible, we consider a set of N problems    which defines a sampling frame containing a representative collection of problems from P. For implementation of this assignment, our  consists of 100 N  proportion problems involving binary outcomes associated with: 1. tossing objects (i.e., coin, checker, tack), 2. binary survey questions (i.e., credit card holder, smoker, cell phone user), and 3. binary object identifications (i.e., car color, gender, parent). Thus, each problem in  has an associated storyline which varies in problem type, student familiarity with the problem, and the required sampling effort. These characteristics could affect whether a student thinks that a particular problem can best be analyzed from a Frequentist or Bayesian viewpoint. Thus, it is important to have a diverse collection of problems in  in order to have variability with respect to these characteristics. For example, a student might think that for a survey type problem, such as assessing the probability a university student owns a credit card, it is better to treat the probability as random (B1). A student who must assess the probability that a local resident has children might be more inclined to incorporate prior information (B2) if they provide child care. A student who must assess the probability of heads for a coin toss might be less inclined to condition on the experimental outcome (B3) when it is easy to obtain numerous outcomes. A large collection of proportion problems can be developed using Albert andRossman (2009), Dunn (2005), and Albert (2000).
Students are reminded that the probability Q  is represented as a random variable with realizations,  , which lie between 0 and 1. As such, there is uncertainty in Q  . Students are also reminded that uncertainty in the knowledge about Q  is formally incorporated through a probability density function (PDF).
Before conducting an experiment, we consider the prior PDF,     . The prior PDF for our model is obtained from the Beta distribution, . This is a convenient PDF for modeling probabilities since it has support over values from 0 to 1 and since it can have a variety of shapes according to the values of a and b (Albert and Rossman 2009, Ch. 17). The prior PDF is (1) Suitable values for a and b need to be chosen to reflect one's belief about Q  . Subjective knowledge about Q  can come from many sources. Four possible sources are depicted in Figure   1 including background from past course instruction in Bayesian statistics, previous experience applying Bayesian methods, preference (popularity) for the Bayesian paradigm, and the ability or perceived ease of carrying out a Bayesian analysis. These four sources provide information about reasonable values for Q  . These sources also provide uncertainty about Q  since it is not known how each one of them will impact the choice of paradigm when faced with a particular problem from the population P.
As shown in Figure 1, knowledge about Q  also comes from the current behavior observed in an experiment. The experiment consists of the number of Bayesian inferential approaches (successes), X , adopted out of n inferential problems (trials). Thus, the experiment is modeled by the Binomial distribution, The knowledge from the experiment is incorporated through the likelihood function which conveys information about likely values of Q  as well as the uncertainty associated with the observed experimental result.
The likelihood function is given by While our development utilizes the Beta-Binomial model, it is important to remind students that their experiment represents only a single randomly chosen inferential problem, i.e. 1 n  .
Plausible values for Q  are reflected through both prior knowledge and the experimental result.
The information about Q  from the prior PDF in (1) and the likelihood function in (2) We also see from (4) that the posterior PDF has the Beta distribution, Thus, the class of Beta distributions is a conjugate family for the class of Binomial distributions since the resulting posterior distribution also has a Beta distribution with updated parameter values.
For many students, the overview given above is illuminating in that they now see that the formula in (3)  In order to give an idea of "typical student work" in this manuscript, we introduce three hypothetical students. These students represent three common combinations of the descriptions shown in Figure 1 based upon previous implementations of the assignment. Freddy Frequentist only has background and experience with Frequentist based statistics and has little preference for the Bayesian approach. Betty Bayesian has had positive experiences with the Bayesian approach in previous courses and with research projects which has influenced her preference for this paradigm and given her the ability to carry it out. Naive Ned has limited background and experience with both approaches and has no obvious preference for either paradigm. Thus, it is natural to expect these types of students to differ in their answers to Q.

Formulation of Prior Distributions
Section A (questions A.1-A.2) of the assignment in Table 1 requires students to incorporate the prior information and uncertainty concerning Q  utilizing the Beta PDF in (1). We have found that it takes skill to convert a priori scientific information into a probabilistic distribution. Consequently, students are reminded in class of familiar facts concerning the Beta distribution and how to use those facts to choose their prior. Students are encouraged to explore their choice of a and b graphically by altering these values and observing how the Beta distribution changes. We have also found it beneficial to require students to give the mean, standard deviation and quantiles of their prior distribution. Taking the time to reflect on the meanings of these distributional summaries helps the student strengthen their argument for why they chose the particular values of a and b. More specifically, if where a and b are hyperparameters, then the mean and variance of Q  are given by respectively (see A.2.1). Thus, a can be chosen relative to b so that the mean of the distribution reflects the prior estimate of being a Bayesian.
At this point, one could also bring up the notion that the effective sample size for the Beta prior is a b  (Morita et al. 2008). As such, a b  could be understood to represent the number of hypothetical inferential problems out of which a of those problems are addressed via a Bayesian approach. Students can also see from (5) that when this effective sample size   a b  is large, then the variability of the prior distribution is small. Rather than having students guess the values of a and b, it is possible to select these values so as to obtain mean Q e and uncertainty Q v using Specification of hyperparameters can also be discussed with students for commonly used priors to provide additional insights. A vague prior distribution is dispersed over the unit interval with small a b  while an informative prior distribution is concentrated with large a b  . If a b  , then the PDF is symmetric about 0.5. A uniform prior has 1 a b   and Jeffreys prior has 0.5 a b   (Robert 2001, pp. 129-130). Zhu and Lu (2004) consider the least influential prior with 0 a b   . However, this prior is improper as it does not integrate to 1. This prior can be approximated using the common trick of choosing a and b to be close to 0.
Examples of the prior PDFs for our three students (i.e. Freddy Frequentist, Betty Bayesian, Naive Ned) are plotted in Figure 2 and distributional characteristics are provided in Table 2.   Table 2 shows that when Freddy Frequentist is presented with the population of problems P that on average he would use a Bayesian approach in 1 out of 10 of these problems and no more than 4 out of 10 of these problems (prior probability less than 0.025). In light of (5), 1/10 0.1 Q e   and 0.1(0.9) /11 0.09 Q v   . Along with the "black" profile in Figure 2 and the first line of Table 2, Freddy Frequentist might also provide the following explanation for his choice of prior: "My prior is specified to demonstrate a high probability of using a Frequentist approach with a fair amount of certainty. That is, I imagine I would only use a Bayesian approach in 1 out of 10 inferential problems. Thus, I chose my prior mean for being Bayesian as 0.1 with uncertainty less than 0.1. In fact, there is greater than 50% chance that my probability of being Bayesian will be between 0.03 and 0.15." Once a Beta prior has been specified, values such as those in Table 2 are rather simple to obtain to answer A.2. Thus, the grading of A.2 merely involves checking if these summaries match the specified distribution. Generally, students perform well on this part. On the other hand, A.1 is one of the most challenging and important questions on the assignment. The grader must ascertain whether or not the specified Beta prior matches the one the student intended. Prior justification is critical in this assessment; just as it is in any Bayesian analysis. Thus, it is not surprising that students struggle with this question. We have found that plotting various Beta PDFs, thinking about the numerical summaries in (5), briefly exploring effective sample size, and going over some of the commonly used Beta priors helps to prepare students to adequately justify their prior distribution.

A Simple Experiment
Section B of the assignment requires each student to conduct an experiment for their randomly selected problem so that they have data to inform their answer to Q. In this section, and for the remainder of the manuscript, we will refer to the four questions in Section B of the assignment in Question B.1 asks students to randomly generate a number of from 1,..., N which can be done using computer programs. For our student examples, we will assume that Freddy Frequentist draws a problem in which he must assess the probability that a common thumbtack lands point up (thumbtack experiment). Betty Bayesian draws a problem in which she must assess the probability that a local resident has children (child experiment). Naive Ned draws a problem in which he must assess the probability that a car on a major city street contains a passenger talking on a cell phone (cell phone experiment).
Question B.2 asks students to explain whether or not they used a Frequentist or Bayesian inferential approach when conducting inference for their problem. Students should address points (B1), (B2), (B3) from Section 2. Formulas that students might use are provided in the Appendix for convenience for the Frequentist approach (A.1) and for the Bayesian approach (A.2). It is helpful for students to clearly address their inferential approach here as this will dictate their answers to questions B.3 and B.4. Question B.3 asks students to justify a selected value for the number of trials, n, for their randomly selected problem. Sample size selection can be tricky due to its connections with the inference procedure. For example, large sample sizes allow for normal approximations for the Frequentist approach (A.1.3) and for the Bayesian approach (A.2.3). Large sample sizes also decrease the influence of prior information (Press 2003, p. 174) (see A.2.2). While sample size calculation is not a main objective of the assignment, students find it to be a hurdle in carrying out their experiment. The most common question on the assignment is, "What sample size should I use?" For introductory classes, B.3 is not required and the instructor can specify the number of trials (such as 10, 15, or 20). However, we encourage students in advanced classes to think about sample size selection since it is fundamental to experimentation. One approach is to select a sample size so that the uncertainty is below a threshold (see A. Question B.4 asks students to provide a brief summary of their experimental results including the number of successes, point estimate, uncertainty, and a 95% interval estimate for  . Answers to this question are provided in Table 3 for our student examples Freddy Frequentist, Betty Bayesian, and Naive Ned. Freddy used a Frequentist approach for the tack experiment and Ned used a Frequentist approach for the cell phone experiment. They calculated their estimate ˆy n   by evaluating the maximum likelihood or method of moments estimator using their values of n and y (see A.1.2). Their measure of uncertainty corresponds to the estimated standard error (see A.1.3). They found 95% confidence intervals which would capture the true fixed value of  in over 95% of the samples. Freddy obtained samples for his experiment by tossing the tack. It was easy for him to perform a large number of tosses, so he was able to use the large sample confidence interval (see A.1.4). Ned picked a moderately busy street in town and systematically sampled 20 cars to determine how many drivers were talking on cell phones. Ned chose his sample size to be moderately large so that the associated uncertainty would be no larger than 0.125. Not wanting to rely on large sample assumptions, he computed a 95% confidence interval appropriate for a small number of trials (see A.1.4). Betty conducted a Bayesian approach for the child experiment. She specified a Beta prior distribution with 1.5 a  and 2.5 b  as she explains in her answer below. She randomly selected 10 houses and found that 4 residents indicated they had at least one child. She used her observed values of n and y to update the Beta prior to obtain the Beta posterior distribution as in (4). Her measure of posterior uncertainty is the posterior standard deviation (see A.2.2). She also computed a 95% credible interval using the highest posterior density (HPD) region (see A.2.4). (B.1) "My randomly selected problem is to assess the probability that a local resident has children. Thus, I plan to sample residences (by address) within a mile radius of my own. I am interested in this problem given that I enjoy babysitting to earn money." (B.2) "I used a Bayesian approach to assess the probability that a local homeowner has children in order to incorporate my knowledge of the nearby residences. Thus, I let  denote this probability which I am treating as random to recognize the uncertainty I have regarding its value (B1). I chose a conjugate beta prior distribution to incorporate my knowledge concerning  . Since I babysit and I have lived in this area for awhile, I wanted to specify a moderately informative prior. Based upon my knowledge, I expect the probability to be about 0.3 with a standard deviation less than 0.2. Thus, I chose the hyperparameters a and b to achieve this expected value and standard deviation (see equation (5)). From the results of my particular experiment (B3), I observed 4 out of 10 residences which had at least one child. These results produced the posterior distribution for  of Beta(5.5, 9.5) (see equation (4)). I plotted my prior and posterior PDFs (see Figure 3)." (B.3) "As mentioned in (B.2), I wanted to incorporate my prior information into this problem. I also wanted to have a small sample size since I would have to meet with people at the randomly sampled residences. Thus, it seemed to me that I could best achieve these two objectives with a sample size of 10 which places weight that the posterior mean is slightly higher than the prior mean. The plot also shows the 95% HPD credible interval for  which is the shortest interval containing 95% of the posterior probability. Thus, there is 95% posterior probability that  is between 0.141 and 0.602. Uncertainty is assessed using the standard deviation of the posterior distribution which has the value    0.5 0.3667 1 0.3667 0.1205 1.5 3.5 10 1 posterior uncertainty is less than the prior uncertainty due to observing 10 residences. The point estimate of 0.3667 seems to closely agree with my prior estimate. I could incorporate this knowledge in a future experiment using a more concentrated prior." Section B of the assignment contains challenging questions as students are expected to carry out an experiment and interpret the results. The grading of B.1 simply involves checking that the students have their scenario picked out and have adequately provided a brief description of it. Question B.2 may be the most difficult on the assignment. For this reason, we ask students to focus on addressing (B1), (B2), (B3) (or (F1), (F2), (F3)) in their written explanation. This guidance has improved student answers and causes them to focus on the critical distinctions between the two paradigms. The structure also helps the grader to better assess student understanding of this distinction. The assessment of question B.3 depends upon guidelines set by the instructor. This flexibility is necessary given differences between course levels. In advanced courses, we have found it helpful to utilize sample size guidelines (see A.1.2, A.1.3, A.2.2, A.2.3). The grader can then assess whether or not the students have adequately utilized those guidelines. Students typically do well reporting the results from their experiment. However, some students struggle to correctly interpret their interval estimate in light of their chosen paradigm. Thus, the grader should be aware of such interpretation problems. Instructors also may wish to spend class time reviewing the interpretation of inferential results in light of the characterizations (B1), (B2), and (B3). These skills are crucial to the practicing statistician.

The Posterior Distribution
The next portion of the assignment (part C and subsequently questions C.1-C.4) involves determining the posterior distribution of Q  . The data from the experiment utilized to update the prior on Q  is based upon a single Bernoulli trial where 1 n  . The value from the trial is 1 x  if a Bayesian approach is used to conduct inference for the randomly selected problem and 0 x  if a Frequentist approach is used to conduct inference for the randomly selected problem. The Bayesian approach was utilized in the experiment, then 1 x  and the posterior distribution is , and if a Frequentist approach was utilized, then 0 x  and the posterior distribution is   Beta , 1 a b  . The usual Bayes estimate is the posterior mean (see A.2.2) The posterior mean is a weighted average of the prior mean and the maximum likelihood given to the prior mean. The uncertainty in the Bayes estimate corresponds to the standard deviation of the posterior distribution (see A.2.2).
Questions C.1, C.2, and C.3 from Table 1 involve summarizing aspects of the posterior distribution. We once again consider answers to these questions for Freddy Frequentist, Betty Bayesian, and Naive Ned. Table 4 gives summaries of the corresponding posterior distributions for C.1. Figure 4 gives the plot of the posterior distributions requested by C.2. The plots of the posterior PDFs in Figure 4 can be compared to the prior PDFs in Figure 2.   Table 4. Highest posterior density intervals could also be calculated (see A.2.4).
Question C.3 asks students to describe the change in the prior and posterior distributions using the absolute relative mean difference, given by prior mean posterior mean posterior std.dev.
Equation (8) measures the change from prior mean to posterior mean relative to the posterior uncertainty. For our hypothetical students, the difference is most pronounced for Ned and least for Fred. This relative change is 11% for Freddy Frequentist, 31% for Betty Bayesian, and 71% for Naive Ned.
An example of an answer to C.4 is given below from the perspective of Naive Ned.
"Based upon my vague prior with 1 a  and 1 b  , and the result of my experiment in which I used a Frequentist analysis so that 0 x  , my posterior distribution for the probability of being a Bayesian is   Beta 1, 2 . This means my posterior estimate of this probability is 0.333 and a 95% credible interval based upon the percentiles is (0.013, 0.842). The 95% credible interval means that there is a 95% chance that my probability lays between these endpoints. Thus, there is a large amount of uncertainty in my answer to Q. However, this is not surprising given my vague prior and what I learned from conducting a single experiment. Nevertheless, the result from this experiment has refined my answer to Q given that my prior estimate was larger with a value 0.500 and my 95% prior percentile interval was wider with values (0.025, 0.975) (see Table 2)." Questions C.1, C.2, and C.3 utilize summaries based upon the posterior distribution. Question C.3 requires a summary involving both the prior and posterior distribution. Evaluation of these questions involves checking that the summaries are correct for the given posterior distributions. Students generally do well on these questions. The key part is C.4 where the student is asked to answer the ultimate question Q. A complete answer to this question involves the interpretation of the posterior distribution and how it pulls together prior information and the experimental result. This is a daunting task for the student, especially one being introduced to a Bayesian perspective. Yet, this is the purpose behind the assignment. Thus, the grader needs to assess if the student has adequately extracted the information and characterized the uncertainty from the posterior distribution to provide a practical answer to Q. An example of an adequate answer is provided above by Ned. A poor answer would be one in which summaries of the posterior distribution are merely reported without an accurate interpretation of these summaries or a description of how these summaries were obtained in light of the prior specification and the experimental result. Students are aided in this process through the individual questions in parts A, B, and C which break down their answer into smaller parts. As a result, question C.4 requires a short summary of the Bayesian approach that led to their answer to Q.

Conclusions
The purpose of the proposed exercise is for students to gain personal experience with the Bayesian paradigm. We have found that students struggle making the transition from the mechanical presentation of Bayes Theorem to practical implementation of the Bayesian paradigm. These challenges include defining prior distributions, understanding differences between Frequentist and Bayesian approaches to statistics, the rationale behind extending Bayes theorem to incorporate uncertainty, and the commonly held view that one has to be either a Frequentist or a Bayesian. This assignment attempts to address each of these difficulties. Thus, it can be quite helpful to spend a large portion of a class period to reflect upon these challenges.
As mentioned by Robert (2001, p. 106), specification of the prior distribution is the most important, and perhaps the most difficult aspect of the Bayesian analysis. Thus, it is not surprising to find that many students say that while they have information for their prior distribution, they have a hard time quantifying and characterizing that information. Plots of the student prior distributions are shown in class. Relating the uncertainty in the prior to student background, experience, preference, and simplicity as in Figure 1 aids in student understanding when it comes to prior specification. Discussions regarding informative and non-informative priors naturally come up. Students could also be asked to think of ways to specify a prior for a client. For example, it might be convenient to have the client think about the expectation, or balance point, of their prior distribution, along with their assessment of prior uncertainty. These values can then be put in equation (6) to obtain the hyperparameters. Plots of prior distributions could then be shown to the client. In addition, it may be helpful for the client to specify the mode and percentile of the prior distribution as in Branscum, Gardner, and Johnson (2005). Uncertainty concerning the hyperparameters a and b can also be accommodated using a hierarchical model (Robert 2001, p. 143) (Press 2003, p. 212).
The second challenge is for students to understand differences between the Frequentist and Bayesian approaches. Thus, the class is informed of the number of students who used a Bayesian approach in their experiment which is typically less than one-third for these proportion problems. According to (A.2.2), the posterior mean is The sample size n can be pre-selected to produce specific weights on the prior mean as a and b are known a priori. We see that for large n , relative to a and b, the weight on the sample mean is large. Equation (9) also shows that the Frequentist answer can be obtained with an improper prior with 0 a b   for any n while an approximate Frequentist answer is obtained when a and b are small relative to n . While these two approaches to statistical inference differ in interpretation and philosophy, they do not necessarily differ in terms of answers. Thus, it is important when expressing the differences between the Bayesian and Frequentist paradigms to focus on the Bayesian characterizations (B1), (B2), and (B3). It can also help students to hear some of the better answers to question B.2 during the class discussion.
The third challenge refers to the incorporation of uncertainty in Bayes theorem. Plots can be shown of student posterior distributions and compared to the prior distributions. These probability density functions illustrate a students' answer to Q both in terms of the mean and in terms of the uncertainty. Many students express surprise by how little their posterior distribution changes from their prior distribution. This provides a good opportunity to point out the amount of information these students have given to their prior distributions (effective sample size) relative to the single Bernoulli trial from the tack experiment. These students can also be reminded of their ongoing growth as statisticians. While it might be expected that the uncertainty in their prior distribution will decrease with added experience, background, and ability, the opposite may be true for many statisticians whose preference can vary widely between inferential problems. Students are reminded that for conjugate families, it is not too difficult to mathematically incorporate uncertainty through the posterior PDF as shown in Section 3. This is because the required integration in (3) is rather straightforward and the posterior distribution can be obtained simply by updating the hyperparameters of the prior distribution by the experimental result. In cases outside of conjugate families, it may not be so trivial to obtain the posterior PDF. In these cases, numerical techniques are needed to solve integration(s) like those in (3). These points provide a nice way to introduce students to popular computational techniques such as numerical integration, Monte Carlo methods, and Markov Chain Monte Carlo methods (Metropolis-Hastings, Gibbs Sampling) (Robert 2001, Ch. 6). These advanced computational methods are merely a tool to implement the Bayesian approach.
This assignment arose out of the perceived need for students to gain practical experience using the extended form of Bayes Theorem in (3) by the instructor of this statistical inference course.
In this particular course, other inference topics are later discussed from the Bayesian framework including hypothesis testing, interval estimation, and prediction. Admittedly, students do not need this assignment to carry out the mathematical details required for these topics. However, students who have completed the assignment are more likely to ask about specifying the priors for these inference procedures and can more easily interpret Bayesian results. This is especially true for those students who have a low probability of being Bayesian. While these students may not have a preference for Bayesian methodology, the assignment provides them with some background, experience, and ability to carry out this approach. Thus, the students who have been trained mostly in Frequentist methodology are better prepared to comprehend and to utilize Bayesian concepts. The instructor has also noticed that it is easier for students who have completed the assignment to utilize Bayesian methods, including applications in consulting, research, and in other advanced statistical methods courses. Many students tend to see the assignment as challenging, so it is important to remind them of the practical experience they have gained from this assignment. It is also helpful to refer students back to this assignment when additional Bayesian topics are discussed later in the course.
interval for  in small samples can be obtained using the Clopper-

Appendix 2 Formulas for the Bayesian Inferential Approach
Consider the scenario described in Appendix 1. In Appendix 2, let  denote the random variable for the true probability of a success. 1 sd | 1 ny a n y b y n a b n a b The uncertainty can be shown to be maximized at   * / 2 y n b a n    . The sample size can be chosen so that the uncertainty is below threshold  when   computed by integrating the posterior PDF or evaluating the corresponding Beta CDF (Casella and Berger 2002, p. 379). This probability can also be evaluated using the large sample posterior distribution in (A.2.3).