Graphical Models:  Assignment 3
Due February 12, 2019

1. The international community is concerned that Depravia may be developing chemical weapons. Inside sources have reported that chemical weapons research is being conducted at the Crooked Creek Chemical plant, a facility owned and operated by ChemCo. ChemCo is a private company whose board of directors includes many Depravian government officials, including several from the defense and intelligence agencies. Chemical detectors were covertly placed in the area surrounding the Crooked Creek plant.The detectors have picked up some compounds that are used in chemical weapons. These compounds are occasionally used in civilian chemical products, but not in the products ChemCo is known to sell. At a surprise inspection last month, the inspection was delayed for several hours when inspectors asked to visit a part of the plant where intelligence reports suggested that suspicious activity may be occurring. During that time, several military vehicles were seen driving away from the plant.  Calculate the probability that Depravia is developing chemical weapons before seeing any evidence, after learning about the chemical detector reports, then after learning about the inspection delay, and finally after learning about the departing vehicles. Write an explanation of the results of your model.  Do you think the results are reasonable?
1. Develop a Bayesian network structure (nodes, arcs and states) to for the problem of using the evidence given in the problem to infer whether Depravia is developing chemical weapons. Use Bayesian network software to draw your network.
2. Explain your network.  Describe your random variables, states, and arcs. Explain why you chose the structure you did.
3. Develop local probability distributions for your Bayesian network.  Explain your distributions.
   
4. Calculate the probability that Depravia is developing chemical weapons before seeing any evidence, after learning about the chemical detector reports, then after learning about the inspection delay, and finally after learning about the departing vehicles. Write an explanation of the results of your model.  Do you think the results are reasonable?
2. Aisha, Bob and Carlos are designing a robot for an interdisciplinary robotics challenge. They will be graded on whether their robot is able to navigate a maze. Aisha, an electrical engineering major, is designing the radar sensor. Bob, a computer science major, is designing the software. Carlos, a mechanical engineering major, is designing the drive system.
   
   • If all three systems function correctly, there is a 90% chance the robot will successfully navigate the maze.
     
   • If Aisha's radar sensor does not work properly but all other components work correctly, there is a 13.5% chance the robot will successfully navigate the maze.
   • If Bob's software does not work properly but all other components work correctly, there is a 10.8% chance the robot will successfully navigate the maze.
   • If Carlos' drive system does not work properly but all other parts work correctly, there is a 22.5% chance the robot will successfully navigate the maze.
1. Use a noisy-or with leak to model their probability of failing the navigation challenge. Find the conditional distribution for passing/failing the challenge given all possible combinations of working properly / not working properly for the three components. You may verify your answer with a software package that implements noisy-OR, but you must do the calculations and show your reasoning. 
2. Suppose each student's component has an 85% chance of working properly. Given that the robot fails to navigagte the maze, what is the probability that each of the components has failed to work properly? If we learn that the drive system has failed, what is the probability that each of the other two components has failed?  Discuss the results. You may use a Bayesian network package to answer this question.