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Question 1: Health Behaviours
Consider the following causal graphical model involving three Bernoulli random variables, which is a
simple model of health status and behaviours: H (health status), C (cautious behaviour), D (disease).
Peopleβs health status influences whether they adopt cautious behaviour, and
their health status together with their behaviour influence their probability of
disease.
We collect an observational dataset of a population of 1000 people:
H C D #instances
0 0 0 10
0 0 1 13
0 1 0 44
0 1 1 28
1 0 0 689
1 0 1 124
1 1 0 84
1 1 1 8
a) Using maximum likelihood estimation, estimate the conditional probabilities of disease given
cautiousness: P(D=1|C=1) and P(D=1|C=0)
b) Suppose that we can intervene and perfectly persuade people to be cautious or not. Estimate
P(D=1|do(C=1)) and P(D=1|do(C=0)) using maximum likelihood estimation. What is the relative
risk reduction of adopting cautious behaviour? (RRR = 1 β (P(D=1|do(C=1)) / P(D=1|do(C=0)))
Question 2: Fire Hazard
Consider the Bayes Net shown here, with all Bernoulli variables,
which involve building types and their risk of fire. Having a fire has
a utility of -1000 if the building was insured, but has a utility of
-50000 if the building was not insured (or the insurance claim is
denied). Having insurance when there is no fire has a utility of -50,
and not having insurance and no fire has a utility of 0.
R O P(F=T|R, O)
F F 0.005
F T 0.02
T F 0.005
T T 0.01
P(R=F) P(R=T)
0.25 0.75
P(O=F) P(O=T)
0.15 0.85
Use the principle of Maximum Expected Utility and Value of Information to answer the following
questions. For parts a)-c) assume the insurance company pays for 100% of the cases.
a) Given no information about whether a building is residential or whether it is occupied, how
much should the insurance company charge to insure the building to break even, in utility
points?
b) How much should the company charge for insurance if you know for certain that the building
is commercial (R=F) and occupied (O=T) to break even?
c) How much should they charge if the building is residential (R=T)?
d) A company is offering cheaper insurance but has a reputation of rejecting 25% of insurance
claims. How much should they charge for this insurance, to make it competitive with the
insurance offered by the more reliable company? (Hint: Set the cost of the new insurance to
have the same MEU as the other insurance.)
Question 3: Bandits
Consider the following 6-armed bandit problem. The initial value estimates of the arms are given by Q =
{1, 2, 2, 1, 0, 3}, and the actions are represented by A = {1, 2, 3, 4, 5, 6}. Suppose we observe that each
lever is played in turn: (from lever 1 to lever 6, and then start from lever 1 again):
π΄
π‘ = ((π‘ β 1)πππ 6) + 1 (1)
We also observe that the rewards π
seem to fit the following function:
π‘
π
π‘ = 2 cos
Ο
6
β‘ (π‘ β 1)
β£
β€
β¦
(2)
So, the first two action-reward pairs are π΄ , and .
1 = 1, π
1 = 2 π΄
2 = 2, π
2 = 3
a) Show the estimated Q values from π‘=1 to π‘=4 of the trajectory using the average of the observed
rewards, where available. Do not consider the initial estimates as samples.
b) It turns out the player was following an Ο΅-greedy strategy, which just happened to coincide with
the scheme described above in (1) for the first 4 time steps. For each time step t from 1 to 4
(inclusively), report whether it can be concluded with certainty that a random action was selected.
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