Sunday, June 16, 2019

The Boy who Cried Wolf: A Bayesian Drama in One Act


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 plight 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.

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.


10 comments:

  1. Wingsuiting the Seneca Cliff – A Collapse Conversation with Ugo Bardi

    Now UP on the Doomstead Diner!

    Get a personal, Ugo's Eye View of ongoing Collapse in Italy, along with Global Issues.

    http://www.doomsteaddiner.net/blog/2019/06/16/wingsuiting-the-seneca-cliff-a-collapse-conversation-with-ugo-bardi/

    RE

    ReplyDelete
  2. Public and politicians do not understand that it is much better to be annoyed by false warnings once in a while than to be completely off guard when the going gets tough.

    ReplyDelete
  3. Just for the sake of being a pain in the ass...

    Where does the statistician get his figures on Wolf appearances? He only gets reports of actual Wolf attacks, not all the times the Wolf is out there lurking. The Boy actually probably DETERS Wolf attacks, simply by his presence, the wolves smell the boy, sense danger and don't attack. This is the effect of the observer changing the outcome of what he is observing (see Relativity Theory). I don't think Bayes analysis accounts for this.

    Unless the Boy actually IS being paid by the Wolves...where do they get the money? Is it printed out of thin air by the Wolf Central Bank (WCB)? Who is in charge of the WCB? Is it Mario Draghi? Were any wolves in attendance at Bilderberg this year? What is the exchange rate between Wolf Chips and the Euro? Should I be Bullish or Bearish on Wolf Chips? Can I Short Wolf Chips? Should I BTFD? Can I exchange my Crypto for Wolf Chips on any of the exchanges?

    RE

    ReplyDelete
    Replies
    1. That's right, but we are in the realm of a very simplified situation. The statistician is implicitly assuming that the wolf doesn't detect the boy, the beast only detects the rush of angry villagers coming for him with pitchforks and torches.

      About the likelihood of the wolf paying money to the boy, I think it is not more improbable than Iraq secretly manufacturing WMDs

      Delete
    2. I forgot to ask if my Wolf Chips Wallet was encrypted and secure against hacking? Also are Wolf Chips safe from inflation? Are Wolf Chips Taxable Income? Can I store Wolf Chips in a Swiss Bank Deposit Box anonymously? What is there to stop Soros from cornering the market on Wolf Chips?

      Most important, can I buy OIL with my Wolf Chips?

      Delete
    3. The boy is paid in rubles: the wolf is, actually, Vladimir Putin who returns to his real shape at night, while wearing sheep clothing during the day

      Delete
    4. Werewolves of Moscow...Warren Zevon should record an updated tune.

      I hear the FSoA MIC is stocking up on Silver Bullets.

      Delete
  4. cool novel :-)

    the boy deters 30% of wolf attacks, then tribe doesn't win anything but tribe win the 0% sheeps losses but in any case they have to wake up in the middle of the night.
    the boy not deters 70% of wolf attacks, then tribe always has a loss about 2% of their sheeps but in any case they have to wake up in the middle of the night

    Mean=(0.3*0)-(0.7*0.02)=-0.014

    The game "wolf vs tribe" is a loser-loser game, in any case people have to wake up in the middle of the night, so I suggest to buy a rifle for the boy, then he can defeat the 100% of real wolf attacks, with no dead sheeps and nobody in the village will wake up in the middle of the night. In case of false wolf attack the boy will shoot in the middle of the night but anibody will get up to see what'up.

    But there's a problem: Aliens don't get scared by rifles, and they abduct 100% of the people, 100% of sheeps, and all the wolf they find, so what if aliens comes in the middle of the night?

    :-)

    ReplyDelete
  5. Very good, too bad it is so close to real life with the village chief acting as a typical 'chief' executive. An enlightened chief would have divided the villagers by the rule of thirds. Perhaps take the boy off the job for a while and have the job of watching for the wolf be done by a rotating group drawn from the unreachable part of the third of villagers and then see if they do any better. Then when they don't, hang one of them!

    So many games it all gets weary; thank you Ugo for the jolt of reality being as that is what math is. It helps.

    ReplyDelete
  6. Please, look at this history record of Early Warning Signals about MEDICANE and violent sea trumpets events. I started to sign record since 2015, but please look the impressive acceleration of those sort of violent events. That's the climate change, no doubt in my mind.

    http://lafrecciadellastoria.blogspot.com/2015/09/in-italia-mini-uragani-mediterranei.html

    how long is it still, before a medicane and / or a sea trumpet devastates some charming town in the Adriatic and / or Tyrrhenian / Ligurian seas?

    What stupid and thief politics will say about a mass destruction of an event of this size?!

    ReplyDelete

Who

Ugo Bardi is a member of the Club of Rome and the author of "Extracted: how the quest for mineral resources is plundering the Planet" (Chelsea Green 2014). His most recent book is "The Seneca Effect" (Springer 2017)