This will appear obscure to you. It isn’t. The errors here are fundamental, and cascade all throughout science. They account for, in part, why science has become so bad. The politics are more important, sure. But those politics needed a lever in bad science. Bad science arises from over-certain science. And over-certain science is created, all too often, from misunderstanding and misapplying probability.
EVENTS, DEAR BOY. EVENTS
Our subject is “unique” events, and whether probability can be “assigned” to them. Interest arose from a post at Marginal Revolution “Allegedly Unique Events“. Specifically about whether there can be a probability of nuclear war arising from our latest moral panic over Ukraine.
We answered the question before about the probability of nuclear war. I won’t repeat all that here, and instead discuss “unique” events.
Here’s the answer: all events are unique. If by event we mean a contingent observable. Something that happens, and can be observed or measured, in the real world. Events proper, though, are only a small subset of propositions around which we can form probabilities. There are an infinite number of propositions that having nothing to do with things being observed. Such as (to name only two) numbers and counterfactuals.
I’ll give examples of all that below. First, why fight over “unique” events and probability? Because of magical thinking and the belief that probability is real or a power—like electricity or magnetism, some kind of force that can act remotely and mysteriously, in a way nobody has defined, nor dares define.
Laugh if you like, but this evolved into the mathematical theory of frequentism, the idea that probability is forever unknown, except at “the limit”, which is to say, never. This is why probability is “estimated” from sequences of events: not known: estimated. Events in frequentism are strange curious things, never quite defined, and left ephemeral.
Events in frequentism must be embedded in an infinite sequence of events which are exactly precisely like all the other events in that sequence, except that all are different, too. Different only in their randomness, which is the power of probability.
Randomness is possessed by events. It is an essential, unremovable part of events. But it cannot measured directly. You can only see it by looking away, as it were, and measuring the behavior of events. And, even then, you never come to a complete knowledge of it—except at infinity.
Ever thought about this before? It is as strange as it sounds. This really is the theory. Stated as starkly as this, it appears ridiculous. But that is only because it is ridiculous. Frequentism is believed because it is taught, and it mathematics emphasized; the nature of events and probability is not much pondered. That’s because just as the student becomes curious in the metaphysics of the system, he is bombarded by more math. Which, if he is usual, he scarcely remembers. Except that it was hard and bizarre.
Take coin flips, the paramount exemplar of a frequentist “event”. It is beloved because it really does seem we can embed flips in an infinite sequence of similar events. We cannot, of course, because we’ll never see the end of them.
Now if we knew the starting conditions and causes of each flip—the weight of the coin, the spin given to it, the force impelled, and all that—we would know the outcome. It would be certain. (It has been done many times in real experiments.) The probability of a head would be “extreme”, i.e. 0 or 1, as the case might be. It would not be 1/2. It would never be any number beside 0 or 1—if we knew the causes.
It is impossible for all the causes operating on the coin not to vary between flips, though most of them are negligible, like minute fluctuations in gravity caused by ancient black holes whose gravity waves are only just now reaching us. Or in the quantum effects or particles gaining more actuality than potentiality, only to sink back into states of predominate potentality. And so on. There is always something going on, and different, between flips.
If we did not know those causes, and all—dear reader: take this word in its strictest sense—and all we know is that there are two outcomes, and only two, and one must happen, then the probability is deduced, and therefore known, to be 1/2.
A frequentist ignores this, and instead “experiments” with flips. He takes no care in the various causes, never really thinks about them at all, except in a loose sense, firmed up only strange times; times which belie either his prejudices or strong experience. He believes that probability, the randomness, is a part of the whole experimental set up. Somehow. He never, not ever, explains how. Because, of course, he cannot.
He begins his “experiment”. After a while, he tires, and ends his flipping. He sums the heads and inputs that number into some odd formula, which allows him to “estimate” “the” probability the coin, the next time it is flipped, will show heads.
Well, in the case of coin flips he could have saved himself the trouble. Not always, though. In situations where we don’t know what the causes are—knowledge of symmetry coupled with what is known about physics is some way toward knowing some of the causes in coin flips—we can use observations to calculate probabilities of future events. Where all that it required is to believe the causes of the thing measured or observed are roughly the same, but which differ in small, unmeasureable and unknown ways. Ways that we can express as uncertainty in unknown similar events using probability.
This is never done, though. Except in rare cases. Even Bayesians don’t do this. This is because all Bayesians are first trained as frequentists. They are just as obsessed with “estimating” probability as frequentists—instead of deducing it. Strange, no?
The result of all this is that concentrating on those estimates, instead of moving on to probability, and being vague or wrong about causes, produces massive over-certainties. The reason any good comes from these methods is because not one man anywhere is a consistent frequentist or Bayesian. It is impossible to remain faithfully adherent at all times to the theories. Reality always intrudes, and the uncomfortable parts of the theories are not brought to mind. But never mind all that now.
Know instead of the magical beliefs in “randomness” which lead to “estimates”, and know that we can instead go right to probability. And probability can be deduced for events of any kind—as long as you can supply the premises relating to the event, which describe what you know about it.
Again, all events—observable, measurable contingent bits of Reality—are caused: made to happen. Since the causes change, even by some fractional unimportant among, all events are unique. We can assess probability for events, which nobody disputes, yet since all events are unique, we can assess probability for unique events. QED.
Which you are not likely to if your career hinges on the old, magical beliefs.
There are more than events. Events are only special case of propositions. Propositions can be of any kind. The Marginal Revolution post gave two examples of propositions: a potential measurable event (the world blowing up) and a counterfactual (the world might have blown up long ago). Counterfactuals are not, of course, measureable.
Neither are numbers.
Example: given knowledge of arithmetic and your background knowledge of the definitions of the symbols used, what is the probability of the proposition “4 > 2”?
I know you know, but perhaps have forgotten that numbers are not observable. They are not contingent events. Yet we can form probabilities about propositions regarding them. If you rebel at this, being too well trained in the current academic regimen, change the proposition to this “The nth digit of pi > the nth digit of e, where n = some huge number that you can write down but nobody has yet”. Say, n = 100^100^100^100^100^100 or whatever.
This proposition has no probability. No proposition has a probability. Because probability does not exist!
The proposition only obtains a probability in relation to some premises. When you—yes, you—supply conditions (the premises) on which we can calculate the probability.
You might be a genius mathematician and have worked out the formula for the nth digit of any “irrational” (telling word!) number. In which case your probability is either 0 or 1 depending on what your formula says—even if your formula is mistaken! (If you understand this, you understand all.) Or you might be a plodder like me and only figure the digits have to be 0-9, and that’s it. Then it’s fifty-fifty.
Our unbreakable rule is: change the premises, change the probability.
That’s a long, convoluted way of saying all probability is conditional, just like all logic statements are conditional. If you’re inclined to disagree, don’t forget the tacit premises of knowledge of the symbols and grammar used to write down any proposition are always conditions.
Now let’s do counterfactuals. Here’s a quote from MR:
“Do such respondents really believe that the probability of a nuclear war was not higher during the Cuban Missile Crisis than immediately afterwards when a hotline was established and the Partial Nuclear Test Ban Treaty signed?”
Doubtless you think that counterfactual probability (now) was indeed higher during the Crisis than after. It is no longer an event, because the time has passed. It is now an unobservable counterfactual. But we can still think of it as if the time has not passed, and that we are living in it now, the conclusion uncertain.
To come to a probability, which need not be a number, what would you do? Right. You’d think about all the causes of the thing in question. What were people thinking, and why? What were the nuke capabilities of the time? What was happening in the USSR and what did that mean to the thoughts of its rulers? What was happening in the USA and what did that mean to the thoughts of its rulers? And so on. Causes.
You won’t know them all. Nobody can. So you have to paper over your ignorance of the causes using uncertainty. In some propositions, no knowledge of cause is even possible, and so probability is all we have.
THE FINAL SHOCKING CONCLUSION
It’s cause. For events, anyway. Knowledge of cause is the goal. It’s not always attainable. It is only in the crudest, or simplest of events. If during the Crisis you knew all the causes behind the decisions of all involved, the probability for you would have been certain. Of course, only God himself can know all these causes. The best any man can do is approximate.
Which we do always, all the time, at all moments, really. The future is a string of contingent events. Of which the outcomes are not known with certainty. But we have to act, which means coming to some kind of grasp of the uncertainty of the events relevant to us, or rather that we act upon. Those that act on us unbidden, the unknown unknowns, by definition there’s nothing we can do. Shit happens. To coin a phrase.
For everything else, we have probability.
Buy my new book and learn to argue against the regime: Everything You Believe Is Wrong.