The story of the boy who cried wolf too many times is a good way to illustrate our attitude toward the people who try to warn us about dangers ahead. Be it about a wolf or about climate change, the result is always the same: prophets of doom are not believed (and, sometimes, they are hanged). Here is a version of the story of the boy and the wolf told using Bayesian statistics where I assume, unlike in Aesop's version, that the boy was simply trying to do his best (if you are not familiar with the Bayesian approach, try this link where the story is very well explained). This post, anyway, doesn't pretend to use the Bayesian theory in its full version, it is a "montypythonesque" story to illustrate how politicians and the public alike can't understand statistics. (Image from the witch scene in the Monty Python "Holy Grail" movie)
The Boy who Cried Wolf: A Bayesian Drama in one act
Characters
The Villagers
The Village Chief
The Village Master Statistician
The Boy
Village Chief: Fellow villagers, we have collected here today to discuss about the boy who acts as a lookout for wolves in order to protect our sheep. I know that several of you have been complaining because the boy has been crying wolf at night several times this year, and every time we woke up and went to the village fence to protect our sheep armed with clubs and pitchforks and carrying lighted torches. But we seem to have a problem with that.
Villager: Yeah, yeah, we go there and there is no wolf to be seen!
Another villager: The boy calls us for nothing!
Another villager: We must hang him!
Village chief. CALM DOWN, fellow villagers. You know that a few times we did see a creature that seemed to be a wolf in the light of our torches – although we couldn't be sure.
Villager: It was not a wolf. It was a black sheep!
Another villager: It was a wild boar!
Another villager: Nothing like that. It was just a shadow!
Another villager. The boy works for the wolf! He does!
Other villagers: Hang the boy, hang the boy!!
Village chief. Fellow villagers, PLEASE, be quiet. It is true that sometimes we didn’t see anything: no wolf appearing the light of our torches. And, worse than that, a few times the wolf came, snatched away a sheep or two, and the boy didn't alert us in advance.
Villager: The boy is playing tricks with us!
Another Villager: Yeah, the boy just enjoys seeing us running!
Another Villager. There are no wolves when he calls! The boy is cheating us.
More villagers. Hang him high! Hang him! Yeah! Yeah!
Village Chief. Calm down, fellow villagers, CALM DOWN! This is not the way to discuss this serious matter because it may well be that the boy is doing his best, but the night is dark and the wolf is cunning, so it is not easy to be the village lookout . . .
Villagers, Hang him, hang him!
Other villagers. Yeah, he is paid by the wolf. Hang him!
Village Chief. And I say BE QUIET! Because I called the village’s Master Statistician to help us and he will tell us whether the boy is doing us a good service according to his Art of which every one of us knows he is a good and respected practitioner.
– Enters the Village Statistician –
Village Statistician. Fellow villagers, lend me your ear because I heard your plea and I am a master of an Art that can help you in this difficult matter.
Villagers: Yeah, let’s listen to the statistician, let’s listen to him!
Statistician: Fellow villagers, the problem you have here is that you don’t know for sure whether there is a wolf or not when the boy calls. And, of course, you don’t like to rush to the fence at night and find that there is no wolf there – at least no wolf that you can see. But thanks to my Art, I will be able to tell you things that that you wouldn’t otherwise know. And this Art is the work of a great master statistician whose name is Bayes and who is respected for this all over the world.
Villager: Yes, yes, master, tell us!
Another Villager: Yeah, master. We trust you. Tell us!
Statistician. Fellow villagers, first of all, let me summarize the situation. If there is no alert before the wolf attacks, the villagers usually arrive too late to save their sheep: the wolf is quick and cunning and he is able to snatch a sheep or two and run away. Hence, we need to be alerted well in advance. That's why the boy keeps watch of the village fence.
Villagers. Yeah, master, yeah. What you say is right.
Statistician. Now, being the village statistician, I keep a record of the wolf attacks and this record I have kept for the years when there was no lookout and so this number tells us how many times the wolf comes, on the average. And I can tell you, fellow villagers, that during the past years there was a chance of a little less than 3% per day of a wolf attack.
Villager. Yes, Master, yes. That’s great.
Another Villager. But what does that mean, Master?
Statisticians. It means, fellow villagers, that the wolf comes about 10 times per year.
Villager. Yeah, yeah, master. We understand that.
Statistician. Very good, fellow villagers. And we shall call that number, 3%, the PRIOR, according to my Art as taught by master Bayes. Remember that carefully!
Villagers: yeah, yeah, master. We remember that!
Statistician. Now, I need the boy who acts as a lookout to help me. Come in, boy!
- Enters the boy -
Boy: Master, I am here at your bidding.
Villagers. Hang him, hang him!
Other villagers. Yeah, yeah, hang him!
Village chief. BE QUIET, I say.
Statistician. Boy, let me ask you, how many times did you see the wolf coming this year?
Boy. Master, Every time I thought I saw a wolf I marked a sign with my knife on the bark of the tree on which I stand at night. And I counted these signs, and there were 20 of them.
Statistician. Very good, my boy. So, dividing this number by the number of days in a year, we see that every day there is a chance of 6% that the boy calls. Therefore, according to my Art, we call this number the EVIDENCE.
Villagers. Master, does that mean we should hang the boy?
Village Chief. QUIET, I say.
Statistician. Fellow villagers, the art of master Bayes is going to help you, but I need some more work. Now I need to know how many times the wolf came unannounced this year. That is, the boy didn’t call, but the wolf came. And you told me that it appeared 4 times. With that, I can calculate the LIKELIHOOD according to my Art. And this likelihood is the number of times the wolf is announced when it comes, divided by the number of times when the beast comes, no matter whether unannounced or announced. So, my data tell me that the wolf comes 10 times per year, whereas it came unannounced 4 times this year. It means its venue was correctly announced six times. In this case, the likelihood will be 6/10, which is 0.6.
Villager. Yeah, yeah, that’s right. That’s right. It means we should hang the boy, right?
Another villager. Hang the boy! Hang him! The Wolf will be very unhappy!!
Village chief. QUIET, fellow villagers. Statistician, what can you tell to us, now?
Statistician. (takes out a charcoal stick and rolls open a tanned sheepskin, starting to write on it). I can now use the formula that the Master of the Art, the much esteemed Thomas Bayes developed. So, the formula tells me that I have to multiply the PRIOR by the LIKELIHOOD and divide by the EVIDENCE. And the final result is .03/.06*.60= 0.3 or 30%
--- silence --
Villager. Shouldn’t we just hang the boy?
Village Chief. KEEP QUIET. Master Statistician, please explain to us what you just said.
Statistician. Fellow villagers, it means that when the boy calls, the wolf will be there once every three times, approximately.
Village Chief: But that means, Master, that many times we rush to the fence for nothing, right?
Statistician: That's true. Two times out of three.
Villagers. It is what we said! The Boy is tricking us
Other Villagers. Hang the boy, hang him!
Other Villagers. Yeah, yeah. The boy works for the wolf!
Other villagers: Yeah, yeah, let's hang him!!
- The villagers take hold of the boy and take him away. The boy screams.
Statistician. Chief, this is not good. You should explain to the people of the village that they shouldn't behave like the members of the evil sect we call the Frequentists. Without the boy, every day the probability for the wolf to be there would be only 3%. With the boy, you have 30% when he calls. And it is much better when everyone has to rise up during the night and rush to the fence.
Village Chief. Dear Statistician, I think the villagers are right. The boy should be hanged: he might be working for the wolf, after all!
– Exeunt –
NOTE
The Bayesian analysis is a powerful tool and it can be used to study climate change. It is especially powerful when it is used to correlate the rise of carbon dioxide with temperature increases, as it is done, for instance, in this paper. Just as an example, think of the concept of abrupt climate change and the correlated mass extinctions. We know that there have been five major mass extinctions during the past 500 million years or so. Then, from a "frequentist" viewpoint, you could say that the probability that a new mass extinction during the next century has a probability of about 100/100,000,000, that is one in a million and you would feel safe. But if you take into account the correlation with the CO2 rise during the mass extinctions, then the Bayesian analysis tells you a completely different story when you compare with the current CO2 spike. I think the data available are not good enough so far for a complete quantitative analysis, but that gives you some idea of the power of the method. The problem is that neither the public nor politicians understand it. And they'd rather hang someone as the culprit instead of doing something useful about the problem.
