CS 3600 Project 4
Bayesian Networks
Due: 11:59:59pm, Nov. 16, 2005
(Or in class that day)

Part 1: Book problems (in the green Russell & Norvig book)


Part 2: Bayesian Networks

"The Apprentice" is a TV reality show where contestants compete to become part of one of Donald Trump's companies. The contestants are divided into two teams. Each week, the teams are given a task, which ranges from selling lemonade to designing a new line of toys. A team leader is chosen by the team (not by Donald Trump), who then manages the other members of his/her team that week (the leader also helps with the actual task). Then, each team is evaluated on their performance; the winning team gets a reward (such as eating dinner with Donald Trump!), while the losing team is sent to the board room. There, Donald Trump decides, based on individual performance and the performance of the team leader, who to eliminate from the game - "You're fired."

You are a contestant on "The Apprentice", and you want to know what the probability is that you will get fired this week. For concreteness, suppose that each team currently has 5 members (including you), and that the task for this week is to sell official Donald Trump wigs - "The most luxurious wig on the planet." Each person gets 10 wigs which they want to sell for as much as possible. For simplicity, and in order to keep the number of parameters small, monetary amounts have been restricted to a small number of possible values.

You have decided on the following nodes (and their meanings):
Keep in mind that you should only add edges which reperesent direct influence - a fully connected network is not an acceptable answer (and would make the calculations horrible!). Make sure to include all nodes, and most nodes should only have 2-3 parents (and probably none should have more than 4).

We assume that a priori the teams are approximately equal, as are the contestants (including you). Thus, P(TeamWins = Yes) = 0.5, and P(DeserveFired? = Yes) = 0.2. The following inequalities should also hold in your completed network (here, >> and << denote "significantly larger" and "significantly smaller", respectively):
  1. P(TeamWins? = Yes|Expertise = High) > P(TeamWins? = Yes)
  2. P(TeamWins? = Yes|Leadership = High) > P(TeamWins? = Yes)
  3. P(You'reFired! = Yes|Leadership = High) < P(You'reFired! = Yes)
  4. P(Leadership = High|You'reFired! = No) > P(Leadership = High)
  5. P(You'reFired! = Yes|OtherTeamSales = Little) < P(You'reFired! = Yes)
  6. P(RandomDT? = Yes|You'reFired! = Yes, YourSales = Lots, LeaderPerformance = Good) > P(RandomDT? = Yes)
  7. P(RandomDT? = Yes|You'reFired! = No, YourSales = Little, LeaderPerformance = Bad, Leader? = Yes) > P(RandomDT? = Yes)
  8. P(OtherTeamSales = Little|Leader? = Yes, LeaderPerformance = Bad, You'reFired! = No, YourSales = Little, RandomDT? = No) >> P(OtherTeamSales = Little)
  9. P(Leader? = Yes|YourSales = Average,OtherTeamSales = Little, RandomDT? = No, You'reFired! = Yes) >> P(Leader? = Yes)
  10. P(DeserveFired? = Yes|You'reFired! = Yes) >> P(DeserveFired? = Yes|You'reFired! = Yes, RandomDT? = Yes)
You should be able to satisfy each of these inequalities using "reasonable" parameters.

Your Tasks:
  1. Build the network.
  2. Assign "reasonable" probabilities (All probabilities should be non-zero - even if you are convinced you possess great expertise or leadership skills)
  3. Answer the following questions
    1. What is the probability that ...your team will win? ...that you'll be fired? ...that you'll be fired GIVEN that your team does not win?
    2. What about after you know if you're the leader (Given (c))?
    3. This is the second last week of the season (so there is 1 more competition before the big winner). If you are fired, you will win nothing, and face the humiliation of your peers (worth -10). If you are on the winning team, the prize this week is worth about 8 to each team member (10 if you are the leader). In any case, if you're not fired, you will go on to compete next week. The possible outcomes (and rewards) next week are fired (-25), winning team (100), winning leader (200) (Yes, I realize that the actual show doesn't work like this). You figure that, if you are not fired this week, the probabilities next week are 10% that you're fired, 60% that you're on the winning team, and 25% that you're the leader of whatever team you're on. What is your expected value as of now (before you are given anything about this week's competition)? What if you know that selling the wigs will be an easy task (Given (e) = "Easy")?
    4. Things are not going well. Your sales are low ((f) = "Little") and your teammates are not doing much better ((g) = "Average"). Now what is the probability that your team will win? ...that you'll be fired? ... that you'll be fired GIVEN that your team does not win? Remember, by this point, you'll know if you're the leader (Given (c)). These questions only refer to the wig sales competition (not the following week, as in part 3).
    5. Since things are not going well (part 4 applies), you are considering acting on a piece of advice you were given: The Donald likes model trains. You figure that if you were to buy him a model train (i.e. bribe him), he may behave more favorably to you. Specifically, P(RandomDT = Yes) will increase to 99% (which may be good or bad, depending on how he takes the bribe). Unfortunately, all the model trains that you can find are very expensive. Given the reward structure mentioned in part c, how much would you be willing to pay for a train?
Since you are free to fill in the probabilities as you see fit, there is no right or wrong answer (numerically, at least; the calculations can be right or wrong). As such, it is important to show all your work if you want to receive credit. I would prefer you type everything, but anything you don't type *MUST* be legible to receive credit. Also, any hardcopies must be handed in during class on Nov 14th, but anything submitted electronically (via webwork) is not due until that night.