Probability is a model, frequency is an observation: Why both halfers and thirders are correct in the Sleeping Beauty problem.


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This post was inspired by a yet another post talking about the Sleeping Beauty problemRepeated (and improved) Sleeping Beauty problem. It is also related to Probability is in the Mind.

(Updated: As pointed out in the comments, If a tree falls on Sleeping Beauty... is an old post recasting this as a decision problem, where the optimal action depends on the question asked: " just ask for decisions and leave probabilities out of it to whatever extent possible")

It is very common in physics and other sciences that different observers disagree about the value of a certain quantity they both measure. For example, for a person in a moving car, the car is stationary relative to them (v=0), yet to a person outside the car is moving (v=/=0). Only very few select measurable quantities are truly observer-independent. The speed of light is one of the better known examples. The electron charge. Some examples of non-invariant quantities in physics are quite surprising. For example, does a uniformly accelerating electric charge radiate? Do black holes evaporate? The answer depends on the observer! Probability is one of those.

In fact, the situation is worse than that. Probability is not directly measurable.

Probability is not a feature of the world. It is an observer-dependent model of the world. It predicts the observed frequency of some event, which is also observer-dependent.

When you say that a coin is unbiased, you model what will happen when the coin is thrown multiple times and predict that, given a large number of throws, the ratio of heads to tails approaches 1. This might or might not be a sufficiently accurate model of the world you find yourself in. As any model, it is only an approximation, and can miss something essential about your future experiences. For example, someone might try to intentionally distract you every time the coin lands heads, and sometimes they will be successful, so your personal counts of heads and tails will be a bit off, and the ratio of heads to tails will be statistically significantly below 1 even after many many throws. Someone who had not been distracted, would record a different ratio. So you would conclude that the coin is biased. Who is right, you or them?

If you remember that frequency is not an invariant quantity, it depends on the observer, then the obvious answer “they are right, I was maliciously distracted and my counts are off” is not a useful one. If you don’t know that you had been distracted, your model of the coin as biased toward tails is actually a better model of the world you find yourself in.

The Sleeping Beauty problem is of that kind: because she is woken up twice per throw that comes up tails, but only once per throw that comes up heads, then she will see twice as many tails as heads on average. So for her this is equivalent to the coin being biased and the heads:tails ratio being 1:2 (the Thirder position). If she is told the details of the experiment, she can then conclude that for the person throwing the coin it is expected to come up heads 50% of the time (the Halfer position), because it is a fair coin. But for the Sleeping Beauty herself this fair coin is observed to come up heads half as often as tails, that’s just how this specific fair coin behaves in her universe.

This is because probability is not an invariant objective statement about the world, it is an observer-dependent model of the world, predicting what a given observer is likely to experience. The question “but what is the actual probability of the coin landing heads?” is no more meaningful than asking “but what is the actual speed of the car?” — it all depends on what kind of observations one makes.