[boreas]

For those who might be interested, and in a slightly different vein than the examples in the article, there's the "sleeping beauty" paradox: https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

Basically, an agent is put to sleep and told they will be woken up once or twice, depending on the results of a fair coin flip, without the ability to remember other awakenings.

What probability does the agent assign to the event that the coin landed heads?

The intuitive response is 1/3, but this poses obvious epistemological problems. The agent has, ostensibly, no new information at all, and their prior is surely 1/2. Hope someone else finds this as interesting as I do!

[colonelxc]

I mostly find it interesting in that people could think that the chance is 1/3 (and that it may even be obvious!). After reading the description I can understand what they are getting at, but I think the conditional probability is messed up.

Instead of P(Monday | Heads) = P(Monday | Tails) = P(Tuesday | Tails) it is really P(Monday | Heads&Awake) = P(Monday | Tails&Awake) = P(Tuesday| Tails&Awake) or something like that. But the interviewer isn't asking about that, they are asking for the probability of the coin. The 3 positions are only exhaustive given that you are awake to be interviewed about them, not exhaustive of possible states (it's missing P(Tuesday | Heads&Asleep)). Since you're always awakened at least once, I find the argument that being awake has 'given you information that it is not tuesday AND heads' is pretty weak. While true, both heads and tails expect to be awoken while it is not both tuesday AND heads.

[roenxi]

The outcomes can be easily enumerated. If Sleeping Beauty always answers "Heads" she will be right 33% of the times she is asked. This is pretty close to the definition of a 33% chance.

She hasn't been given new information by waking up, she also knows as she goes into the experiment - "most of the outcomes where I am being interviewed involve the coin toss coming up tails".

[tzs]

Here is how one might decide that 1/3 is obvious. Imagine that N people simultaneous undergo the experiment, for a very large N.

Half of them end up in the heads group. They wake up on Monday and are questioned. Then they sleep until Wednesday and are released.

The other half end up in the tails group, and so are questioned twice (Monday and Tuesday) then released on Wednesday.

Because we gain no information during the experiment, we can make our decision before the experiment.

Let's count. There will be 3N/2 interviews conducted. N/2 of the will be 'heads' interviews and N will be 'tails' interviews. So going in, we can see that when someone experiences the event 'being asked about the coin', 1/3 of the time the coin will be heads and 2/3 o the time it will be tails. Hence, our credence in the coin being heads should be 1/3.

Here is a counterargument. Imagine a slightly different experiment. The people are not asked what their credence in the coin being heads is. They are asked to guess if it is heads or tails. If they are right, the experiment continues and they are eventually released. If they are wrong, this is noted, and the experiment continues until Wednesday, and then they are killed and their home planet is destroyed.

As before, we gain no information during the experiment, and so can decide our answer beforehand. No matter what strategy one picks for making that decision, there is a 50/50 chance that one ends up with a destroyed planet. That indicates that our credence in heads should be 1/2.

[ouid]

You will be awoken twice as many times because the coin comes up tails as you will because the coin comes up heads. If you can manipulate the formulae to tell you something different, that only means you have failed to manipulate the formulae correctly.

[bo1024]

It's very interesting and I don't think there's an obvious correct answer. It's hard to formally model mathematically.

Here's a game-theoretic perspective. In general, when an event has a 1/3 chance of happening, an idealized gambler would be indifferent between the following two bets or lottery tickets: (A) win $2 if the event happens; (B) win $1 if the event doesn't happen. (Notice her average payoff is 2/3 no matter which bet she takes.)

Now in the sleeping beauty problem where tails is two awakenings and heads is one, a gambler would be indifferent between (A) winning $2 every time she wakes up and the coin is heads, and (B) winning $1 every time she wakes up and the coin is tails. This suggests that her "belief" is 1/3.

Another way to put it might be that for a risk-neutral agent, doubling the payoff in one state of the world is equivalent to doubling its "perceived probability". In the sleeping beauty problem, doubling the payoff is like experiencing everything twice.