There is no such thing as a biased coin
I want to start a mini-series on “Probability Theory: The Logic of Science”. I will try to get to “Causal vs logical dependence”.
There is nothing, no physical property that you can measure that will tell you how likely it is that a coin will turn up heads.
Centre of mass is irrelevant, the coin’s width, length and whatever is irrelevant.
Because probability is in your head, in your heeeeead. (Zoooombies, zoooombies.)
Give me a coin and I will construct a throwing device to approach any frequency I want.
Newtonian physics is deterministic. Measure the initial state and watch the coin follow its computable path.
The same can be said about everything else, with an uncomfortable exception for quantum mechanics. Uncomfortable because some physicists are still thinking about it: and many, many weren’t keen on accepting the quantum effects being probabilistic1. Anyway, I am out of my depth here, so let’s go back to easy stuff.
E.T. Jaynes spent an evening throwing various coins to demonstrate exactly this point. He knew physics and managed to “bias” all of them in any way he pleased.
Probability is subjective: it depends on the observer. Some observers could know more than others. May be they know physics and are capable of calculating the trajectory. May be they know that the experiment is staged or some might have gotten a glimpse of a coin just after it landed. Suddenly probability assignments differ amongst them.
This idea, that probability is built into the physical world around us, that it’s out there to be discovered via a design of clever experiments, via orthodox statistical tools which were painstakingly developed took science astray for at least a 100 years.
The problem with orthodox methods and the orthodox idea of probability being something physical is that it kinda works in many situations. And a Bayesian refinement usually only gets you tiny-little gains in accuracy, so is not worth it. And that’s why this world view hasn’t been abandoned. By the cracks are showing if you push it too much. Not only the methods are capable of producing absurd conclusions, they are also limiting. Such limits are self-imposed. Example:
“How can you apply Central Limit Theorem to arrive at this approximation in number theory? Where is randomness in that?” - I heard today.
I guess the question makes sense, if you insist on viewing probability as something physical, something built into this world, something to be discovered by flicking coins and throwing dice. But numbers aren’t like that. You can just compute them. But the same goes for a trajectory of a coin.
This idea of “randomness as a property of an object or a process” is completely unnecessary. If you don’t know the answer - use probability theory. Because that’s what it is for: dealing with insufficient information. Beautifully derived from the Cox’s theorem. See the details in “Probability Theory: The Logic of Science” by E.T. Jaynes
Now I reserve the right to introduce “fair coins”, “loaded dice”, etc. in pure maths only. I treat it as a shorthand. It is much more elegant than saying “let’s start with an observer who assigned the following probabilities to the following events”. That’s just too much.
But, but, but, but, but… Statisticians must outright taboo such statements. Because such philosophy corrupted science. The moment you say “assume measurement errors are normally distributed” - you are doing it wrong. Why? Here’s some data:
33 7 20 18 45 31 93 52 50 29 11
What distribution should you assign to a process producing it? Answer: ParsingError and exit(1). It is known data, presented in front of you, there is no probability distribution to assign to it as there is no uncertainty.
I can’t wait for a revolution in modern statistics: because it’s inevitable that such treatment of probability will be banished outright in the future. In the meantime, such thinking is too ingrained, too established. Because it kinda works. Thanks, Fisher.
On pragmatic grounds I don’t think everything should be derived from Cox’s theorem. It is just an ideal. Even though we haven’t proved \(P \neq NP\) we still deploy encryption relying on this assumption. However, what has been derived should be used in favour of what hasn’t when such choice is available.
Now isn’t this satisfying? There is no need to worry about what random is, how to construct “random variables” and how to deal with the fact that no matter how hard you try the result is always pseudorandom and imperfect instead.
I mean Born’s rule, not Shrodinger equation. ↩