Statistical evidence evaluation
Exercises 3
1. Suppose a certain merchandise that is to be sold can have one of three states:
“old age”, “younger age”. “reproduction”. If you sell it, your selling price will be SEK
10000, but this price is only valid if the merchandise is “old age”. For the state “younger
age”, a reasonable price is SEK 2000 and a reproduction has almost no market value at all
(approx. SEK 100). The customer can be expected to return the merchandise and get his
money back if the merchandise is of another state than “old age” (we thus assume he will
have the possibility to deduce the true state). Worse, however is that if the true state is
“reproduction” then you will be fined SEK 5000 if you have tried to sell it for a price
above its market value.
Suppose you use a method to investigate the state of the merchandise. The following is
known for the method:
If the true state of the merchandise is “old age”, the method will render “old age” with
probability 0.8, “younger age” with probability 0.1, and “reproduction” with probability
0.1.
If the true state is “younger age”, the method will render “old age” with probability 0.2,
“younger age” with probability 0.5, and “reproduction” with probability 0.3.
If the true state is “reproduction”, the three states will be rendered with equal probabilities.
It is also known that the prevalence of a merchandise with state “old age” is about 50 %,
while merchandises with the other states are equally probable to find on the market.
You are choosing between two actions: to sell or not to sell. If your method renders “old
age” then what action maximises your expected posterior utility?
2. A screening device for measuring the alcohol content in blood is used by the police when
doing sampling inspection of drunk driving. The device works in such a way that the driver
blows into the device and the instrument should then indicate whether there are signs
of too high blood alcohol (above 0.2 per mille) or not. There is a relationship between the
content of alcohol (ethanol) in exhalation air and the content of alcohol in blood and this
relationship is used to calibrate the screening device..
However, these devices are not precise enough to be used as evidentiary tools. When the
blood alcohol is above 0.2 per mille there is a 90 % probability that the device will indicate
that. Moreover, the device may indicate blood alcohol above 0.2 per mille with probability
3 % when the true level is below 0.2 per mille.
Drunk driving is quite common, and it is estimated that on the average every 20th driver
has too high levels of alcohol in their blood.
In earlier days the standard procedure with the police was that when the results obtained
From screening a driver indicated too high blood alcohol, the driver was sent for blood
sampling (which is an evidentiary procedure). This was a costly process. You had to
arrange with a transport from the road side to a medical centre which could take up to an
hour, and the blood sampling had a cost with it. Let’s say that the total cost for a blood
sample was SEK 2000.
This cost was depreciated if the blood sample gave evidence of too high blood
alcohol, but otherwise remained as a societal cost. The cost of not sending someone for
blood sampling while that person actually has a too high level of blood alcohol is of course
difficult to appreciate. There are very high costs (both monetary and societal) when a
drunk driver causes an accident. On the other hand the probability is far from one that a
drunk driver would cause an accident. However, when he so does one may count with a
cost of SEK 1.7 million per accident.
Moreover, since there was a time lag between screening and blood sampling, several
drivers had metabolized the alcohol to that extent that the blood sample would not show
too high contents even if that was the case when screening. We may assume that in 25 %
of the cases where the blood alcohol is really too high it has dropped under the limit when
it comes to blood sampling.
Try to construct an influence diagram for the decision problem of what action to take (send
for blood sampling or do not send) when the screening device has indicated for too high
blood content. Elaborate with different probabilities for a driver to cause an accident if his
blood content is too high.
3. A general practitioner has a patient showing symptoms sore throat and fever.
These are symptoms that occur very frequently when you have a flu and there is no
specific treatment to be ordered against a flu. Assume the symptoms occur with 99 %
probability if you have a flu. The symptoms however also occur when you have a bacterial
infection in the throat but when the GP examines the throat he cannot see any signs of a
bacterial infection. He then estimates that if there is a bacterial infection the symptoms and
the examination results have a total probability of 50 % to occur. The cost of sending the
patient home with no treatment if the patient actually has a bacterial infection can be taken
as double the cost of treating the patient for a bacterial infection if the patient has a flu. Not
treating someone with a flu is comparable with treating someone with a bacterial infection
and the cost may in both cases be set to zero. At this time of year the probability of having
a flu is 80 times higher than the probability of having a bacterial infection.
Which action, treatment or no treatment, gives the lowest expected posterior loss?