Evidence for an event cannot be the horrific consequences of that event.
Suppose we’re interested in the probability of a Y = “Horrific plague”. Since nothing has a probability, we need first supply evidence probative of Y with which to calculate a probability. I only mean “calculate” loosely; not all probability is strictly quantifiable.
Suppose this evidence: X = “Horrific plagues are horrific”. Then what is
Pr(Y | X) = ?
I am praying it is obvious that X is not probative of Y. X indeed is merely a restatement of the quality of Y: horrific plagues are horrific. This is a tautology. It is not new evidence. X is no way gives us any evidence of how Y might come about, about how Y is caused, about the circumstances of Y in any way. X merely restates that horrific is horrific.
Thus Pr(Y | X) has no solution; the probability is, if you like, the entire unit interval (this is deduced via obscure arguments not of interest here). We cannot get from X to Y. Agreed?
Suppose instead our evidence is: X = “Horrific plagues have fat tails.” Now what is
Pr(Y | X) = ?
Some seem to think that, suddenly, this probability is now greater than 0; perhaps small, perhaps not a tangible number, but a number greater than 0 just the same. And since a number greater than 0 is real enough, Y suddenly becomes possible. We have moved from complete ignorance in whether Y could happen to definite information that Y might happen.
Then comes step two: since Y might happen, given our assumed X, and Y is by definition horrific, then we must take immediate steps to stop or mitigate Y!
Now this all came up in a conversation I had with a fellow, a rabid fan of Taleb, and I can only hope my brigade of boosters is as tenacious, who kept insisting, via his master, that because horrific plagues “have” “fat tails” we should move to protect against them. He used the analogy of buying fire insurance for his home. He acknowledged he only had a dim idea of the probability of his home burning, but he knew well the consequences of it. So he bought insurance.
The two scenarios are not equivalent. (I informed him his insurance company had such evidence and knew the price, which is how they set the price.)
First, by “fat tails” Taleb is mixing up probability of events and the consequences of those events, sort of. Horrific plagues, for instance, are rare: we observe they do not often happen, though they do happen on occasion, and (again) by definition they are horrific. Taleb thinks that, somehow never mind how, these rare consequential events actually possess the property of “fat tailness”.
Since it is a property, it can be measured. And since it can be measured, Y suddenly becomes possible, and since Y is possible and horrific, it should be protected against. Like Black Swans Attacking From Outer Space.
Do we even need to dissect this? The answer is: yes; yes, we do. Because somehow this argument “feels” convincing to some.
What can “fat tails” mean? Well, it can mean “rare”. Thus X = “The event Y is rare” then
Pr(Y | X) = small.
Nothing wrong with this mathematically. But it’s circular since X assumes the probability we wish to deduce. We wanted the probability of Y with respect to some evidence, and the evidence dictated the rarity.
So “fat tails” meaning “rare” is out. How about “fat tails” means “horrific”, as in consequential? What’s Pr(Y|X) now?
We already did that. We can’t deduce the probability of the event by discussing the consequences of the event.
We have rejected both “rare” and “horrific” (or other similar large consequential words). How about “fat tails” means “rare and horrific”? Then what is Pr(Y|X)? Easy: Pr(Y | X) = small. But that’s again circular and thus of no help.
Saying “fat tails” is absolutely useless, except as a shorthand to say “bad things happen, and really bad unpredicted things happen rarely.” Which everybody always knew. If we knew how to predict the really bad things, we’d predict them. If follows that because we can’t predict them (always), we can’t predict them (always).
So what are people really doing when they’re fretting over, say, this new coronavirus? Some are doing this: X = (something like) “Y is really terrible, and I worry I’ll die if I get the bug, or that many will die if they get it, as some have already died and where the media is implying huge numbers might die, so we better do something to prevent its further spread.” Then
Pr(Y | X) = modest to large.
This X is a confusion of many different pieces of evidence, some of which are truly probative, others of which are restatements of “fat tails”.
An epidemiologist has a better X. He uses historical evidence of outbreaks, ties this together with mathematical this and that, which also uses the plain evidence that the outbreak is now occurring. With this he pegs a Pr(Y|X). This may be high or low depending on what model the epidemiologist uses. It has nothing to do with “fat tails” in any ontological sense, though everything to do with probability (when “fat tails” means rare).
You care about the coronavirus now because you’re hearing about it now. But two weeks ago, or whenever it was, before you heard about it you didn’t care about it. There was no X most were willing to consider, except historical X, about possible new plagues. It’s very reasonable for somebody to argue X = “Horrific plagues have happened before, and here are a list of reasons, such as easy air travel, to why they may happen again.” We can get a Pr(Y|X). We can calculate, roughly anyway, costs of protection and so on. Fun stuff.
You didn’t two weeks ago, or whenever, say “Oh my! Horrific plagues have fat tails and anything might happen, which we know because we don’t know what can happen, so let’s ban air travel now just in case!” That’s a pure Talebism, relying on the precautionary principle. I quote myself:
That we don’t know what we don’t know is known, or should be, and is thus a given. But because we don’t know what we don’t know does not make what we don’t know bad. It could also be good, or benign. To say it could only be bad is the PP [precautionary principle] Fallacy.
It is incoherent to run in a circle screaming “We know nothing so we must do something!” Can it really be that “fat tails” means “unpredicted”? That Pr(Y| Y unpredicted) = not small? That’s what the precautionary principle does for you: truly something out of nothing.
You care now because you have heard definite evidence in favor of the proposition Y. That evidence may be good or bad, who knows at this point. It is not irrational, conditional on these X, to say the probability this plague will be horrific is not small. But you’re saying this because of the definite evidence of infection and deaths you have heard about. You’re not saying it because horrific plagues have “fat tails” and “fat tails” are scary.
We come to the main two points: all probability is conditional, and the conditions are what are important. No event “has” a probability. All probability depends on the evidence we assume. And it is that evidence which we should always be debating. Probability (except for the math bits) is always trivial once the evidence is laid down.
So I ask you, what is the X = actual evidence for this actual outbreak to be an actual Y, i.e. a horrific plague? If you only go with the negatives, your Pr(Y|X) will be high. If you factor in nervousness and instant news cycles hungry for sensation, your Pr(Y|X) will be much lower.
Which X are the correct X? That is the right question.
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