Exercises

We do not know whether the brain really works in a Bayesian way, in an approximate 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. From the following expressions, which one corresponds to the sentence, The probability of being sunny given that it is 9^th of July of 1816?
   • p(sunny)
   • p(sunny | July)
   • p(sunny | 9^th of July of 1816)
   • p(9^th of July of 1816 | sunny)
   • p(sunny, 9^th of July of 1816) / p(9^th of July of 1816)
2. Show 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?
3. In the following definition of a probabilistic model, identify the prior and the likelihood:
4. In the previous model, how many parameters will the posterior have? Compare it with the model for the coin-flipping problem.
5. Write the Bayes' theorem for the model in exercise 3.
6. Let's suppose that we have two coins; when we toss the first coin, half of the time it lands 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?
7. Modify the code that generated Figure 1.5 in order to add a dotted vertical line showing the observed rate head/(number of tosses), compare the location of this line to the mode of the posteriors in each subplot.
8. Try re-plotting Figure 1.5 using other priors (beta_params) and other data (trials and data).

9. Explore different parameters for the Gaussian, binomial, and beta plots (Figure 1.1, Figure 1.3, and Figure 1.4, respectively). Alternatively, you may want to plot a single distribution instead of a grid of distributions.
10. Read about the Cromwel rule on Wikipedia: https://en.wikipedia.org/wiki/Cromwell%27s_rule.
11. Read about probabilities and the Dutch book on Wikipedia: https://en.wikipedia.org/wiki/Dutch_book.