1. Construct a Bayesian network (showing all nodes, links, and conditional probability tables) that is consistent with the below full joint probability distribution below over four Boolean random variables (Party, Sleep, Study, Pass) and consistent with the following three conditions. Round each probability in the conditional probability tables to the nearest tenth.
• Sleep is conditionally independent of Study given Party.
• Study is conditionally independent of Sleep given Party.
• Pass is conditionally independent of Party given Sleep and Study.
Party
Sleep
Study
Pass
Probability
true
true
true
true
0.0216
true
true
true
false
0.0024
true
true
false
true
0.0224
true
true
false
false
0.0336
true
false
true
true
0.0216
true
false
true
false
0.0144
true
false
false
true
0.0084
true
false
false
false
0.0756
false
true
true
true
0.3024
false
true
true
false
0.0336
false
true
false
true
0.0896
false
true
false
false
0.1344
false
false
true
true
0.0864
false
false
true
false
0.0576
false
false
false
true
0.0096
false
false
false
false
0.0864
1
2. Compute the probabilities below based on the following Bayesian network. Show your work.
a. P(EatRight=true, Exercise=true, Healthy=true, LiveLong=true, Prosper=true)?
b. P(Healthy=true | Exercise=false)?
c. P(LiveLong=true | EatRight=true, Exercise=true)?
d. P(EatRight=true | LiveLong=true, Prosper=true)?
e. P(Prosper | EatRight=false, Exercise=false)?
3. What would be the most likely sample from applying direct sampling to the Bayesian network in Problem 2? What is this sample’s probability?
4. Consider the Bayesian network below, where each of the five random variables have a domain of 4 values. What is the minimum number of probabilities needed to completely describe the full joint probability distribution for this scenario? Justify your answer.
