3. Using only the axioms of probability and the definition of conditional independence, prove Proposition 8.5.

Consider the belief network of Figure 8.34. This the “Simple diagnostic example” in the AIspace belief network tool at http://www.aispace.org/bayes/. For each of the following, first predict the answer based on your intuition, then run the belief network to check it. Explain the result you found by carrying out the inference.

(a) The posterior probabilities of which variables change when Smokes is observed to be true? That is, give the variables X such that P(X ∣ Smoke = true) ≠ P(X).

(b) Starting from the original network, the posterior probabilities of which variables change when Fever is observed to be true? That is, specify the X where P(X ∣ Fever = true) ≠ P(X).

(c) Does the probability of Fever change when Wheezing is observed to be true?
That is, is P(Fever ∣ Wheezing = true) ≠ P(Fever)? Explain why (in terms of the domain, in language that could be understood by someone who did not know about belief networks).

(d) Suppose Wheezing is observed to be true. Does the observing Fever change the probability of Smokes? That is, is P(Smokes ∣ Wheezing) ≠ P(Smokes ∣ Wheezing, Fever)? Explain why (in terms that could be understood by someone who did not know about belief networks).

(e) What could be observed so that subsequently observing Wheezing does not change the probability of SoreThroat. That is, specify a variable or variables X such that P(SoreThroat ∣ X) = P(SoreThroat ∣ X, Wheezing), or state that there are none.
Explain why.

(f) Suppose Allergies could be another explanation of SoreThroat. Change the network so that Allergies also affects SoreThroat but is independent of the other variables in the network. Give reasonable probabilities.
(g) What could be observed so that observing Wheezing changes the probability of Allergies? Explain why.
(h) What could be observed so that observing Smokes changes the probability of Allergies? Explain why.
Note that parts (a), (b), and

(c) only involve observing a single variable.