Imagine that you live in a medieval kingdom. Its ruler, King Seyab, is known for his love of mathematics, philosophy, and for cruelty. As a very young king, 40 years ago, he ordered a group of wise men to take an urn and fill it with 1000 white and black balls. The color of each ball was chosen by a coin flip, independently of other balls. There is no reason to doubt wise men's honesty or accuracy in fulfilling king's order. The king examined the contents of the urn and filled another urn with 510 black and 490 white balls. The contents of the two urns is top secret and the subjects of King Seyab never discuss it.
The laws of the kingdom are very harsh, many ordinary crimes are punished by death, and the courts are encouraged to met out the capital punishment. On average, one person is sentenced to death each day. The people sentenced to death cannot apeal for mercy but are given a chance to survive by the following strange decree of the monarch. The prisoner on the death row can sample 999 balls from the original urn. He is told that the second urn contains 1000 balls, 490 of which are white. Then he can either take the last ball from the first urn or take a single random ball from the second urn. The prisoner's life will be spared if the ball turns out to be white.
Now imagine that you have been falsely accused of squaring a circle and sentenced to death. You have sampled 999 balls from the first urn. The sample contains 479 white balls. You have been told the contents of the second urn. Will you take the last ball from the first urn or sample a single ball from the second one?
In view of how the balls were originally
chosen for the first urn, the probability of the last ball being white
is 0.50. The probability of sampling a white ball
from the second urn is only 0.49. It seems that
taking the last ball from the first urn is the optimal
decision.
However, you know that over 40 years, the survival
rate for those who took the last ball from the first
urn is either 48% or 47.9%. The survival rate
for those who sampled from the second urn is about 49%.
So what is your decision?
Is Bayesian
philosophy compatible with patriotism?
The society in King Seyab's land is old fashioned. People believe that there are things more important than one's life - they are one's honor and the country. We will suppose that every person in the country wants to sacrifice his or her life on the altar of the common good, if needed. King Seyab is not revered by his subjects but they still believe that they should do whatever they can for the country. Since the strength of the country is positively and strongly correlated with its population size, every cicitzen will do whatever necessary to make or keep the population as large as possible. We will assume that King Seyab's subjects will totally ignore their personal interest and do what they believe is best for the country.
Suppose that the society consists entirely of frequentists. A frequentist whose sample contains 479 white balls realizes that if all people choose the remaining ball from the first urn, the population will eventually shrink to about 48% of its original size. If all people take a ball from the second urn, the survival rate will be 49%. A frequentist would choose a ball from the second urn, out of his patriotic feelings.
Next assume that all people in the kingdom are Bayesians. A Bayesian who has his contry's interest in his mind would try to maximize the expected value of the country's population. Since the probability that the last ball in the first urn is white is equal to 0.5, the Bayesian will take the last ball from the first urn. The opposite decision is suboptimal from the Bayesian viewpoint because the probability that a ball sampled from the second urn is white is only 49%.
The society consisting of patriotic frequentists will
end up with the population equal to 49% of its original size.
A country of patriotic Bayesians will shrink to 48%
of the original size.
Bayesian philosophy, friendship and loyalty
The following ethical dilemma does not seem to be tied to Bayesian philosophy but neverthless naturally arises in this context.
Imagine that you are a Bayesian and you have a good friend who is also a Bayesian. You already chose a ball (the last ball from the first urn) and it turned out to be white. Your life was spared and you know the exact number of white balls in the first urn.
Prison guards approach you and tell you that your friend examined his sample of 999 balls and was about to sample a ball from an urn but he fainted. A decision in his case has to made immediately and you are asked to make the decision. The guards would not let you see his sample. All you can do is to take the last ball from the first urn (the leftover after your friend's sample) or a ball from the second urn. Your friend's life will be spared only if the ball is white.
First we will argue that you, a Byesian decision maker, should take a ball from the second urn. Since your friend cannot make a decision, you have to make the decision. Given the information you have, the probability that the remaining ball in the first urn is white is strictly less than 49%. Hence, it is best to take a ball from the second urn.
Next we will give arguments in favor of the opposite decision. Let us switch the roles of you and your friend. Imagine that you are incapable of drawing a ball yourself, for any reason, although you have already made a decision. Your friend will draw a ball for you but you cannot communicate with him. You know that he knows the exact number of balls in the first urn (and you know that it is less than 490) but since this information cannot be transferred to you, his knowledge is useless to you. For this reason, you hope that he will draw the last ball from the first urn, i.e., you would like him to do what you would do yourself if you could. Hence, a loyal friend should draw the last ball from the first urn.
Which ball would you draw for your friend?
How does your loyalty translate into practice?
What would you like your friend to do for you,
if you could not draw a ball yourself?