The other day I heard someone say something like the following:
I can’t believe how people don’t understand probability. They don’t realize that if a coin comes up heads 20 times, on the next flip there’s still a 50-50 chance of it coming up tails.
But if I saw a coin come up heads 20 times, I’d suspect it would come up heads the next time.
There are two levels of uncertainty here. If the probability of a coin coming up heads is θ = 1/2 and the tosses are independent, then yes, the probability of a head is 1/2 each time, regardless of how many heads have shown before. The parameter θ models our uncertainty regarding which side will show after a toss of the coin. That’s the first level of uncertainty.
But what about our uncertainty in the value of θ? Twenty flips showing the same side up should cause us to question whether θ really is 1/2. Maybe it’s a biased coin and θ is greater than 1/2. Or maybe it really is a fair coin and we’ve just seen a one-in-a-million event. (Such events do happen, but only one in a million times.) Our uncertainty regarding the value of θ is a second level of uncertainty.
Frequentist statistics approaches these two kinds of uncertainty differently. That approach says that θ is a constant but unknown quantity. Probability describes the uncertainty regarding the coin toss given some θ but not the uncertainty regarding θ. The Bayesian models all uncertainty using probability. So the outcome of the coin toss given θ is random, but θ itself is also random. It’s turtles all the way down.
It’s possible to have different degrees of uncertainty at each level. You could, for example, calculate the probability of some quantum event very accurately. If that probability is near 1/2, there’s a lot of uncertainty regarding the event itself, but little uncertainty about the parameter. High uncertainty at the first level, low uncertainty at the next level. If you warp a coin, it may not be apparent what effect that will have on the probability of the outcome. Now there’s significant uncertainty at the first and second level.
We’ve implicitly assumed that a single parameter θ describes the uncertainty in a coin toss outcome. Maybe that’s not true. Maybe the person tossing the coin has the ability to influence the outcome. (Some very skilled people can. I’ve heard rumors that Persi Diaconis is good at this.) Now we have a third level of uncertainty, uncertainty regarding our model and not just its parameter.
If you’re sure that a parameter θ describes the coin toss, but you don’t know θ, then the coin toss outcome is an known unknown and θ is an unknown unknown, a second-order uncertainty. More often though people use the term “unknown unknown” to describe a third-order uncertainty, unforeseen factors that are not included in a model, not even as uncertain parameters.
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