Graphical Models:  Assignment 4
Due February 20, 2017

1. A common kind of Bayesian network used in diagnosis is a two-level, multiple-fault model.  In this kind of Bayesian network, the top layer consists of a set of nodes representing possible failures and the bottom layer consists of a set of observable symptoms.  In the simplest model, all the faults are modeled as independent and all the symptom nodes depend on the faults that can cause the symptoms.  Also, typically, the relationship between the faults and the symptoms is modeled using noisy-OR, or when the failures and symptoms have several degrees of severity, noisy-Max. Consider a Bayesian network with n failure nodes, m symptom nodes, k states (degrees of severity) per failure/symptom and j possible causes for each symptom.
1. Find an expression for the number of the number of probability values needed to specify a fully general joint distribution on m+n nodes with k states per node.
2. Find an expression for the number of probability values needed to specify a general Bayesian network in which there are n independent failure nodes and m symptom nodes, each of which has has j failure nodes as parents.
3. Find an expression for the number of probability values needed to specify a Bayesian network in which the n failure nodes are all independent,  each of the m symptom node has j parents, and the local distributions for the symptoms are modeled as noisy-Max.
4. Make a table comparing the results of parts a, b, and c when:
• n = 5, m = 5, k =2, and j = 2;
• n = 10, m = 10, k = 4, and j = 4;
• n = 100, m = 100, k = 10, and j = 20.

1. Prior to becoming Chief of the Rechtian Intelligence Agency, Hydziz I. Dennity was assigned to work as a secret agent in Depravia.  He built a Bayesian model to classify contacts as Depravian government agents, supporters of the Depravian government, apolitical people (people who don’t care about politics) and dissidents.  Agent Dennity did an extensive study on the characteristics of these different groups of people.  He learned:
• Government supporters and apolitical people rarely criticize the government.  Dissidents often do, as do government agents (because they are trying to lure Rechtian agents into thinking they are dissidents).
• Dissident men sometimes have long hair.  Other men rarely do.
• Government agents and dissidents are usually men.  Apolitical people are usually women.  Government supporters are about equally divided between men and women.
• Government supporters and government agents usually drive expensive cars (because they have plenty of money).
• Dissidents and apolitical people are more likely to be scientists and artists than are other people.
• Dissidents and government agents are likely to own fax machines.
• Most people are either government supporters or apolitical, with apolitical people outnumbering government supporters by a small margin.  About one in every ten people is a dissident, and about one in every hundred people is a government agent.
Build a Bayesian network for Agent Dennity’s inference problem.  Identify the partitions you would use for assessing probability distributions.  Construct your network in a software package, using partition nodes to simplify model specification.
2. Enter evidence for three qualitatively different scenarios for the Bayesian network from Problem 2. Comment on your results.