Purpose: Start using R to solve or compute basic probability problems

Objective: Compute the probability of several real-world problems. These probabilities may consist of either unconditional or conditional probabilities but I hope a combination of the two would be useful. Apply the probability theory that you know and refer to the material you read to highlight that you read something to base your solution. Here are some examples but please do not restrict yourself to only these…

• Example 1: Calculate the probability of drawing a full house after four hands (i.e., four players) have been dealt 7 cards.

• Example 2: Calculate the probability that two people will have the same birthday in a classroom of 10, 40, 100, 250, and 500 students.

• Example 3: Calculate the probability that one of your living adult family members will die of cancer (any cancer).

• Example 4: Calculate the probability that all Washington area professional sports teams will win their respective titles (e.g., Capitals win the Stanley cup, Redskins win the Super Bowl, Nationals win the World Series, and the Wizards win the NBA Championship).

These are just examples. Think of your own but be creative. Remember that your presentation needs to conform to the template I provided.

Purpose: Compute condition probabilities in R.

Objective: Apply Bayes theorem to subjective probabilities using R. The conditional probabilities must be subjective to an extent - whereby you don’t really know the real answer but you have a suspicion. Let me give you a few examples so you can all follow my logic.

• Example 1: You believe your spouse is cheating on you but you do not have direct evidence. Instead, you have some pieces of evidence that appear to be indicative of infidelity. Quantify that belief.

• Example 2: Scientists have a hunch that a study will fail to supply a sufficient number of subjects to reach 80% statistical power. Quantify that hunch.

• Example 3: People outside the US believe that Americans are gun-toting, racists, misogynists, or fascists. How can you convince them that those stereotypes are not indicative of Americans?

These are just a few examples but use your own. Find conditional probabilities that enable you to compute some simple values using R. Yes, use R. These exercises are to get you familiar enough with R without having to do anything terribly complicated.

Purpose: Use R package(s) to solve one normal distribution problem.

Objective: Create your own normal data (fabricated is fine) and apply some functions in the bayess package to estimate Bayesian parameters. The purpose of this “project” is to get you familiar with using packages and functions in R. There are no real outcomes that you can achieve at this point but the more familiar you are with R, the easier the rest of the projects will become. Be sure to read the Bayesian Essentials book (Chapter 2 in particular) so you understand what the package does for you. The data you fabricate or find must be viewed as “normal” as in Gaussian Normal with a mean and standard deviation that represent that distribution appropriately.

NOTE: If you find another Bayesian package that does something you prefer, by all means use it. I leave it to each group to use either the bayess package or another package of your choosing. Please restrict the distribution to only normal distributions.

Purpose: Apply Bayes Rule

Objective: To apply what you learned in the assigned reading but, more importantly, to get you to move from simple probabilities to the idea that new information can be used to update your beliefs - the core of Bayesian statistics. Start with a binomial prior (you may select any you desire), use data from either the internet or data you create with your group, update the prior using that data via Bayes rule. Pay attention to the code provided for you and use that to your advantage. The aim is to get you to start thinking like Bayesians. Have a hunch (priors), collect data (likelihood), and then compute your new belief (posteriors) all by simple methods of Bayesian applications.

NOTE: the absence of examples? I'm sure you did. Now, go work as a team!

Purpose: Show me the money! Actually, show me (and others) your results.

Objective: To learn and then demonstrate how to present your results to anyone (including your grandmother). By practice, you learn to use the plotting tools in R more effectively to communicate your results; here, I ask you to show me how far you have come (as a group). Remember to think of what you intend to say and then say it in simple, graphical terms. In Bayesian inference, we start with a belief, we collect data, and then we compute a new belief based upon the combination of our prior (starting belief) and the new evidence. You need to decide first what you intend to communicate and then how you intend to communicate it. Think carefully; this “project” is not one to merely test your ggplot skills but also to press your team to learn how to select what is best to communicate your collective change in uncertainty. Also, think of your audience. Read other articles and peruse the internet for methods to display Bayesian results. You may find a wide variety of really awful methods; choose one that suits your team’s idea of effective communication. Your presentation ought to include some detail about what you did, why you did it, and how you did it (along with the results of all this cogitation). Now, go plot!

Purpose: Compare frequentists and subjectivists results from a single source

Objective: To guide you to think through the differences between frequentists and subjectivists views using data. I recommend you use either an existing dataset or create one that enables you to compute some simple statistics. The aim is to get you to really apply what we discuss in class this week where we compared the methods of both to a single problem. As before, I omitted examples because you are now all familiar with the process and I believe your team can navigate the uncertainty of the project. Think about a simple problem. Use that problem as a case study to illuminate yourselves and your classmates about the differences between these two approaches to data analysis, inference, and logic. Remember, you only have 3 minutes to choose what you plan to present wisely. Your team must be focused on a simple issue.

Purpose: Use Stan to solve a simple Bayesian problem

Objective: To encourage you to move from R to Stan for more complicated Bayesian inference. The aim here is not to make your team (or the rest of the class) experts in Stan. Quite the contrary. I want everyone exposed to Stan and the applications of Stan in Bayesian modeling and inference. You can scan the case studies listed on the Stan website to find a setup suitable for your needs, find some code listed on the GitHub examples, or search the web for other resources. What I expect is that your team comes up with an example that already has code written for Stan and run with results. Your task is to take that code, adapt it to an example you understand - preferably something pertaining to psychological science - and walk us through what you did in 3 minutes. The project may take more time than usual because you need to get Stan running with R (or with any other platform you desire) before you can run your code. Furthermore, you need to read a bit outside the assigned material. I plan to set aside some time on Thursday to address questions that may come up with each team.

Purpose: Calculate and interpret correlations using Bayesian methods

Objective: To push students to think more like Bayesians in the application of a standard tool in social/behavioral sciences - the correlation coefficient. I recommend you either use data you understand (presumably from your lab) or find data from previously published studies. Compute correlations (two or three suffice) via standard methods and then the “Bayesian way” complete with parameter estimation and interpretation. Plot your results and make sure you discuss how you might view correlations from these new methods differently than you would have viewed very similar results before you learned about Bayesian methods.

Purpose: Use Stan and R for general linear model estimation and inference

Objective: To use R and Stan together to solve typical social/behavioral science data analytic problems. Apply Bayesian methods to an ANOVA or MRC (standard, general linear model or GLM - not to be confused with generalized linear model) data analysis problem. Use data that you are familiar with or can easily communicate in 3 minutes (along with all the other bits required for the project presentations). You now have most if not all the tools to run the GLM from a Bayesian perspective. Go do it!

Purpose: A complete application of Bayesian inference using R, Stan, and Python

Objective: To provide you several weeks as a team to pull together all the material we discussed during the semester. The 10th and final project will be one that nobody can throw out because it is the synthesis of the entire class - consider a final exam of sorts. Use Stan, R, and/or Python to create a full analysis of existing, published data. Many articles now include the full data. For those unfamiliar with the practice or for those who are not aware that data exist, I suggest you start with a journal that I know publishes the data along with the article. Check Judgment and Decision Making, PloS One, or other journals where data are readily available. You have enough time to find data, think about the re-analysis, and create the presentation. Remember, your objective is to apply Bayesian methods to data that were not analyzed using these tools. The aim again is to pull together your knowledge and apply it to a replication effort - without collecting new data.