The most dispiriting thing I’ve read in a while was this post by John Cook involving coin flips. The discussants are certain they have all the puzzle pieces and all that remains is to arrange them with the right prose. There’s not even a hint of awareness that they might be missing something.
Statisticians talk as though Nature generates coin flips from a model but she does no such thing. Real coin flips are about atoms and electrons. They’re electromagnetic forces, gravity, stray backgrounds fields. They’re wind, and sound, and percussion, impulse and impact. Muscle flexing, ATP-ADP reactions, and nervous system signals. Real coin flips involve all kinds of things, but what they don’t involve is “binomial models”. Those are a figment of the Statistician’s mind.
Since there is no such model in the real world, the key question is:
How easy is it for Mother Nature to fool the Statistician into believing a given model?
To make things precise, we can say data from n flips will fool the Statistician whenever they would fail to reject at the level.
Now comes the interesting part. If the model is then it’s very difficult for Mother Nature to bamboozle anyone since for only about .6% of would do so. The model on the hand is a different story. No matter how big n gets, 99% of would, if observed, fool the Statistician.
In other words, there are countless knuckleheads running around thinking they’ve verified of the “fairness” of a coin, when all they did was choose a model which virtually guarantees they’ll be conned by Mother Nature.
It’s at this point that Statistics becomes sublimely absurd. Flush with pride for having “objectively modeled” the “data generation mechanism” they conclude that after flips each n=100 sequence would come up equally often. Stop for a moment, dear reader, and savor the Alice-in-Wonderland like quality of this. They’re using an unphysical model which will appear correct almost no matter what’s actually happening, to confidently predict the outcome of a trial that would take longer to perform than our solar system will exist.
There is an alternative to this pablum. Suppose I’d like to predict the frequency of heads in the 100 flips I’m about to make and I have no idea what’ll happen because I never measured anything (Statisticians never do!). Since 99.99% of all sequences in have a frequency of heads between .31 and .69, I’ll predict is one of those 99.99% and will lie in the same interval.
The fraction .9999 is a reasonable measure of how strongly my state of knowledge is consistent with the prediction .
This is the best, and most robust, guess I can make under those circumstances (*). It could be wrong, but inventing fantastical claims about flips isn’t going to magically make it more right. If happens to be one of those .01% exceptions, the only way I’ll know ahead of time is by doing some real physics.
Simple, clear, true and focused entirely on real things that actually exist. That’s the alternative.
(*) It’s hard to explain this robustness because most Statisticians are lost-in-the-sauce, but here is a nonsense explanation which will nevertheless convince most scholars. Let be the set of all frequency distributions (for a fixed total number of flips ) on and imagine is a single draw from some . Then the statement is extremely likely to hold almost no matter which distribution in is used. This remains true even for the vast (vast!) majority of ‘s which differ radically from the uniform distribution.
(I just know some knucklehead will respond by claiming Statistics can be put on a sure foundation by imagining that is drawn at random from F. This just goes to prove there’s nothing quite as dumb as a smart person.)