Download Statistical evidence evaluation Exercises 2

Statistical evidence evaluation
Exercises 2
1. For each of the Bayesian networks in a) and b) calculate
(i) Pr (E = E1 | H = H1)
(ii) Pr (H = H1 | E = E1)
a)
H
H Pr
H1 0.6
H2 0.4
E
Pr
H: H1 H2
E: E1 1 0.05
E2 0 0.95
b)
H
H Pr
H1 0.6
H2 0.4
F
E
Pr
H: H1 H2
F: F1 0.5 0.1
F2 0.5 0.9
Pr
F: F1 F2
E: E1 1 0.05
E2 0 0.95
2. Divergent connection
At a working place, approximately 1% of the personnel get injured each year. Some
statistics:
All Injured Non-injured
Gender Female 40 % 30 %
?
Male
60 % 70 %
?
Age
18-30
35%
45 %
?
31-50
30 % 20 %
?
5135 % 35 %
?
a) Complete the table, i.e. replace the question marks
b) Construct a Bayesian network for this setup with Injured/Non-injured as parent node
and Gender and Age as two non-linked children of that node.
c) Use the Bayesian network to show that with this construction Gender and Age are
conditionally independent given information about whether an employed is injured or
not.
d) Use the Bayesian network to compute with this construction the age distribution among
females.
e) Use the Bayesian network to compute the probability that a randomly selected
employed male between 31 and 50 years old will be injured.
f) Is the construction realistic?
3. Serial connection
Assume the prevalence of a disease is 5%. Among individuals having the disease, approx.
30% will have a specific cough. Persons not having the disease can be assumed not to have
this cough. Among individuals with the cough, approx. 10% asks for a prescript cough
tincture (for that cough).
a) Construct a Bayesian network with a serial connection for this setup.
b) Use the network to show that given information about whether cough is present the
event “the individual has the disease” and the event “the individual ask for prescript
cough tincture” are conditionally independent
c) Suppose an individual asks for cough tincture. What is the conditional probability
he/she has the disease?
4. Converging connection
Suppose you watch a person at distance and you notice that the person wears a hoodie.
Hoodies are more common to be worn by men, say that 20% of all men and 10% of all
women wear hoodies. Further, you seldom see hoodies when the temperature is above
20°C, you see them more often when it is below 20°C and very often when it is below
10°C. Let us assume that you have 5% chance to see a hoodie when the temperature is
above 20°C, 10% chance to see a hoodie when it is between 10°C and 20°C and 30 %
chance to see a hoodie when it is below 10°C.
a) Construct a Bayesian network for this setup, where gender and temperature are unlinked
parent nodes and your observation is the child node.
b) Choose arbitrary probabilities for the 3 states of temperature (for instance 0.30, 0.50 and
0.20) and let woman and man be equally likely states for the gender node.
c) Use the network to investigate whether the nodes gender and temperature become
conditionally dependent when the child node is instantiated.
5. “Reconstruct” the Bayesian network that gives rise to the formulas
(
)
Pr (E H ) = Pr (E F , H ) ⋅ Pr (F H ) + Pr E F , H ⋅ Pr (F H )
a) 
Pr (E ) + Pr (E ) = 1; Pr (F ) + Pr (F ) = 1; Pr (H ) + Pr (H ) = 1
(
)
Pr (E ) = Pr (E H , G ) ⋅ Pr (H ) ⋅ Pr (G ) + Pr E H , G ⋅ Pr (H ) ⋅ Pr (G ) +

b) + Pr E H , G ⋅ Pr (H ) ⋅ Pr (G ) + Pr E H , G ⋅ Pr (H ) ⋅ Pr (G )

Pr (E ) + Pr (E ) = 1; Pr (H ) + Pr (H ) = 1; Pr (G ) + Pr (G ) = 1
(
)
(
)
6. In the Bayesian network below, each node has two states
(i.e. H , H ; F , F ; G, G ; E1 , E1 ; E 2 , E 2 ).
H
F
G
E1
E2
(
a) Deduce the complete expressions for Pr (E1 , E 2 H ) and Pr E1 , E 2 H
)
b) Suppose the probability table for node G is
Pr
H:
H
F: F F
G: G 1 0
G 0 1
H
F
0
1
F
1
0
How do your deduced formulas in a) change when you take these probabilities into
account?