Homework 2. Bayes AI

Exercise 1 (Two Gambles) In an experiment, subjects were given the choice between two gambles:

Experiment 1
Gamble \({\cal G}_A\) Gamble \({\cal G}_B\)
Win Chance Win Chance
$2500 0.33 $2400 1
$2400 0.66
$0 0.01

Suppose that a person is an expected utility maximizer. Set the utility scale so that u($0) = 0 and u($2500) = 1. person is an expected utility maximizer. Set the utility scale so that u($0) = 0 and u($2500) = 1. Whether a utility maximizing person would choose Option A or Option B depends on the person’s utility for $2400. For what values of u($2400) would a rational person choose Option A? For what values would a rational person choose Option B?

Experiment 2
Gamble \({\cal G}_C\) Gamble \({\cal G}_D\)
Win Chance Win Chance
$2500 0.33 $2400 0.34
$0 0.67 $0 0.66

For what values of u($2400) would a person choose Option C? For what values would a person choose Option D? Explain why no expected utility maximizer would prefer B and C.

This problem is a version of the famous Allais paradox, named after the prominent critic of subjective expected utility theory who first presented it. Kahneman and Tversky found that 82% of subjects preferred B over A, and 83% preferred C over D. Explain why no expected utility maximizer would prefer both B in Gamble 1 and C in Gamble 2. (A utility maximizer might prefer B in Gamble 1. A different utility maximizer might prefer C in Gamble 2. But the same utility maximizer would not prefer both B in Gamble 1 and C in Gamble

Discuss these results. Why do you think many people prefer B in Gamble 1 and C in Gamble 2? Do you think this is reasonable even if it does not conform to expected utility theory?

Exercise 2 (Tarone Study) Tarone (1982) reports data from 71 studies on tumor incidence in rats

  1. In one of the studies, 2 out of 13 rats had tumors. Assume there are 20 possible tumor probabilities: \(0.025, 0.075,\ldots, 0.975\). Assume that the tumor probability is uniformly distributed. Find the posterior distribution for the tumor probability given the data for this study.
  2. Repeat Part a for a second study in which 1 in 18 rats had a tumor.
  3. Parts a and b assumed that each study had a different tumor probability, and that these tumor probabilities were uniformly distributed a priori. Now, assume the tumor probabilities are the same for the two studies, and that this probability has a uniform prior distribution. Find the posterior distribution for the common tumor probability given the combined results from the two studies.
  4. Compare the three distributions for Parts a, b, and c. Comment on your results.

Exercise 3 (Poisson MLE vs Baye) You are developing tools for monitoring number of advertisemnt clicks on a website. You have observed the following data:

y = c(4,1,3,4,3,2,7,3,4,6,5,5,3,2,4,5,4,7,5,2)

which represents the number of clicks every minute over the last 10 minutes. You assume that the number of clicks per minute follows a Poisson distribution with parameter \(\lambda\).

  1. Plot likelihood function for \(\lambda\).
  2. Estimate \(\lambda\) using Maximum Likelihood Estimation (MLE). MLE is the value of \(\lambda\) that maximizes the likelihood function or log-likelihood function. Maximizing likelihood is equivalent to maximizing the log-likelihood function (log is a monotonically increasing function).
  3. Using barplot, plot the predicted vs observed probabilities of for number of clicks from 1 to 7. Is the model a good fit?
  4. Assume that you know, that historically, the average number of clicks per minute is 4 and variance is also 4. Those numbers were valculated over a long period of time. You can use this information as a prior. Assume that the prior distribution is \(Gamma(\alpha,\beta)\). What would be appropriate values for \(\alpha\) and \(\beta\) that would represent this prior information?
  5. Find the posterior distribution for \(\lambda\) and calculate the Bayesian estimate for \(\lambda\) as the expectation over the posterior.
  6. After collecting data for a few days, you realized that about 20% of the observations are zero. How this information would change your prior distribution? This is an open-ended question.

Hint: For part c, you can use the following code to calculate the predicted probabilities for the number of clicks from 0 to 5.

tb = table(y)
observed = tb/length(y)
predicted = dpois(1:7,lambda_mle)

Exercise 4 (Exponential Distribution) Let \(x_1, x_2,\ldots, x_N\) be an independent sample from the exponential distribution with density \(p (x | \lambda) = \lambda\exp (-\lambda x)\), \(x \ge 0\), \(\lambda> 0\). Find the maximum likelihood estimate \(\lambda_{\text{ML}}\). Choose the conjugate prior distribution \(p (\lambda)\), and find the posterior distribution \(p (\lambda | x_1,\ldots, x_N)\) and calculate the Bayesian estimate for \(\lambda\) as the expectation over the posterior.

References

Tarone, Robert E. 1982. “The Use of Historical Control Information in Testing for a Trend in Proportions.” Biometrics, 215–20.