1. Show how to derive Bayes’Rule from the de  nition P (A ∧ B) = P (A|B).P (B).

2. Suppose you are give the following information

Mumps causes fever 75% of the time

The chance of a patient having mumps is 115000

The chance of a patient having fever is 11000

Determine the conditional probability of a patient suering from mumps given that they have don’t have a fever, i.e. P (Mumps|¬Fever ).

3. Consider the following statements

Headaches and blurred vision may be the result of sitting too close to a monitor. Headaches may also be caused by bad posture. Headaches and blurred vision may cause nausea. Headaches may also lead to blurred vision.

(i) Represent the causal links in a Bayesian network. Let H stand for“headache”, B  for “blurred vision”, S for“sitting too close to a monitor”, P for“bad posture”and N for “nausea”. In terms of conditional probabilities, write a formula for the event that all  ve variables are true, i.e. P (H ∧ B ∧ S ∧ P ∧ N).

(ii) Suppose the following probabilities are given

P (H|S, P) = 0.8  P (H|S, ¬P) = 0.6 P (B|S, H) = 0.4  P (B|S, ¬H) = 0.2 P (S) = 0.1

P (P) = 0.2

P (N |H, B) = 0.9  P (N |H, ¬B) = 0.5

P (H|¬S, P) = 0.4

P (H|¬S, ¬P) = 0.02

P (B|¬S, H) = 0.3

P (B|¬S, ¬H) = 0.01

P (N |¬H, B) = 0.3

P (N |¬H, ¬B) = 0.7

Furthermore, assume that some patient is suering from headaches but not from nau- sea. Calculate joint probabilities for the 8 remaining possibilities (that is, according to whether S, B, P are true or false).

(iii) What is the probability that the patient suers from bad posture given that they are suering from headaches but not from nausea?

4. Consider the“burglar alarm”Bayesian network from the lectures. Derive, using Bayes’Rule, an expression for P (Burglary|Alarm) in terms of the conditional probabilities represented

in the network. Then calculate the value of this probability.

Is this number what you expected? Explain what is going on.

Check your answer using the AIPython program probVE.py. Answers to method calls such as bn4v.query(B,{A:True}) are in a list form [F,T] giving the probability that B is false and the probability that B is true, in conjunction with the list of conditions (here that A is true).  The desired answer is then calculated by normalization. The Bayesian network is encoded as bn4 in probGraphicalModels.py.

5. Prove the conditional version of Bayes’Rule:  P (B|A, C) =  .  Here C is an added condition to all terms in the original version of Bayes’Rule.