Week 5

1. AIMA 13.8. 

   The key to this problem is to notice when the problem is asking for a distribution, and when it is looking for a single probability.  Lowercase variables are single values, uppercase are distributions.

2. AIMA Ex. 13.21.  If the answer is no, explain why not.  If the answer is yes, give the probability (and explain how you arrived at this answer).

The next two problems are taken from http://www-nlp.stanford.edu/~grenager/cs121//handouts/hw2.pdf

3.  In this question we examine the conditional independence assumptions encoded in the
   Bayesian network graph topology. Consider the following Bayesian network:

   

   (a) Write down all the independencies not conditioned on other variables that are enforced by this
   Bayesian network, using the notation A ⊥ B to mean that A is independent of B.
   
   (b) Write down three independencies which do not necessarily hold in this Bayesian network.

   (c) Write down all the conditional independencies that are enforced by this Bayesian network, using the notation A ⊥ B|C to mean that A is conditionally independent of B given C.
   
   (d) Write down three conditional independencies which do not necessarily hold in this Bayesian
   network.

4. In this problem we’re going to prove the conditional independence properties of the following
   Bayesian network:
   
   
   (a) What are the conditional probability distributions (CPDs) that are represented in this Bayesian
   network?
   
   (b) Write down the joint probability distribution over X, Y , and Z as represented by this Bayesian
   network. This expression should be written in terms of the CPDs you enumerated in (a).
   
   (c) Write down an expression in terms of these CPDs for P(X,Z), the marginal probability of X
   and Z (hint: sum the variable Y “out” from the joint distribution you wrote above).
   
   (d) Based on the expression in (c), and the defition of independence, are X and Z independent?
   
   (e) Write down an expression for P(X,Z|Y ).
   
   (f) Based on the expression in (e), and the definition of conditional independence, are X and Z
   conditionally independent given Y ?
5.  Consider the Dog Problem network that we looked at in class.  Calculate the probability that the family is out given that the lights are on and you do not hear the dog barking.