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:
    5. Comment on your results.
Two-Level Diagnosis BN
  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:
    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.