1.11 Exercises

We do not know whether the brain works in a Bayesian way, in an approximately Bayesian fashion, or maybe some evolutionary (more or less) optimized heuristics. Nevertheless, we know that we learn by exposing ourselves to data, examples, and exercises… Well you may say that humans never learn, given our record as a species on subjects such as wars or economic systems that prioritize profit and not people’s well-being... Anyway, I recommend you do the proposed exercises at the end of each chapter:

1. Suppose you have a jar with 4 jelly beans: 2 are strawberry-flavored, 1 is blueberry-flavored, and 1 is cinnamon-flavored. You draw one jelly bean at random from the jar.

   1. What is the sample space for this experiment?

   2. We define event A as the jelly bean drawn is strawberry-flavored and event B as The jelly bean drawn is not cinnamon-flavored. What are the probabilities of events A and B?

   3. Are events A and B mutually exclusive? Why or why not?

2. Previously, we defined a Python function P to compute the probability of an event using the naive definition of probability. Generalize that function to compute the probability of events when they are not all equally likely. Use this new function to compute the probability of events A and B from the previous exercise. Hint: you can pass a third argument with the probability of each event.

3. Use PreliZ to explore different parameters for the BetaBinomial and Gaussian distributions. Use the methods plot_pdf, plot_cdf, and plot_interactive.

4. We discussed the probability mass/density functions and the cumulative density function. But there are other ways to represent functions like the percentile point function ppf. Using the plot_ppf method of PreliZ, plot the percentile point function for the BetaBinomial and Gaussian distributions. Can you explain how the ppf is related to the cdf and pmf/pdf?

5. From the following expressions, which one corresponds to: the probability of being sunny given that it is 9^th of July of 1816?

   1. p(sunny)

   2. p(sunny|July)

   3. p(sunny|9 of July of 1816)

   4. p(9^th of July of 1816|sunny)

   5. 

6. We showed that the probability of choosing a human at random and picking the Pope is not the same as the probability of the Pope being human. In the animated series Futurama, the (Space) Pope is a reptile. How does this change your previous calculations?

7. Following the example in Figure 1.9, use PreliZ to compute the moments for the SkewNormal distribution for a different combination of parameters. Generate random samples of different sizes, like 10, 100, and 1,000, and see if you can recover the values of the first two moments (mean and variance) from the samples. What do you observe?

8. Repeat the previous exercise for the Student’s T distribution. Try values of ν like 2, 3, 500. What do you observe?

9. In the following definition of a probabilistic model, identify the prior and the likelihood:

   

10. In the previous model, how many parameters will the posterior have? Compare it with the model for the coin-flipping problem.

11. Write Bayes’ theorem for the model in exercise 9.

12. Let’s suppose that we have two coins; when we toss the first coin, half of the time it lands on tails and half of the time on heads. The other coin is a loaded coin that always lands on heads. If we take one of the coins at random and get a head, what is the probability that this coin is the unfair one?

13. Try re-plotting Figure 1.12 using other priors (beta_params) and other data (trials and data).

14. Read about the Cromwell rule on Wikipedia: https://en.wikipedia.org/wiki/Cromwell%27s_rule.

15. Read about probabilities and the Dutch book on Wikipedia: https://en.wikipedia.org/wiki/Dutch_book.