This post will address a problem proposed by Radford Neal in his paper Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical conditioning. In particular, he defined this problem—The Beauty and the Prince—to argue against the halver solution to the Sleeping Beauty Problem. I don’t think that this is ultimately a counter-example, but I decided to dedicate a post to it because I felt that it was quite persuasive when I first saw it. I’ll limit the scope of this post to arguing that his analysis of the halver solution is incorrect and providing a correct analysis instead. I won’t try to justify the halver solution as being philosophically correct as I plan to write another post on the Anthropic Principle later, just show how it applies here.
The Beauty and the Prince is just like the Sleeping Beauty Problem, but with a Prince who is also interviewed and memory-wiped. However, he is always interviewed on both Monday and Tuesday regardless of what the coin shows and he is told whether or not Sleeping Beauty is awake. If he is told that she is awake, what is the probability that the coin came up heads. The argument is that 3⁄4 times she will be awake and 1⁄4 times she is asleep so only 1⁄3 times when he is told she is awake will the coin be heads. Further, it seems that Sleeping Beauty should adopt the same odds as him. They both have the same information, so if he tells her the odds are 1⁄3, on what basis can she disagree? Further, she knows what he will say before he even asks her.
I want to propose that the Prince’s probability estimate as above is correct, but it is different from Sleeping Beauty’s. I think the key here is to realised that indexicals aren’t a part of standard probability, so we need to de-indexicalise the situation. However, we’ll de-indexicalise the original problem first. We’ll do this be ensuring that only one interview ever “counts”, by which we mean that we will calculate the probability of events over the interviews that count. We’ll do this by flipping a second coin if the first comes up tails. If it is heads, only the first interview counts, whilst for tails only the second interview counts. We then get the odds being: 1⁄2 heads + Monday counts; 1⁄4 tails + Monday counts; 1⁄4 tails + Tuesday counts.
We similarly de-indexalise the Prince, though we flip the second coin in the heads case too. Similarly, if it is heads we count the interview on Monday and if it is tails we count the interview on Tuesday, so the four possibilities become mutually exclusive and each have a probability of 25%.
If we look when the first coin is heads, we notice that the Prince’s interview on Monday only counts 50% of the time, whilst Sleeping Beauty’s counts 100% of the time. This means that Sleeping Beauty is calculating her probability over a different event space so we should actually expect her answer to differ from that of the Prince. Suppose we expand the Prince’s probability to include the Sleeping Beauty’s Monday interviews (which all count). Then we get the chance of heads moving from 1:2 = 1⁄3 to 2:2 = 1⁄2.
As we’ve seen, The Beauty and the Prince is not a problem for the halver solution. This does not mean that the halver solution is the correct solution to the Sleeping Beauty Problem, just that The Beauty and The Prince doesn’t provide a counter-example.
Update: I’ve been reading more of the literature. It seems that the technique that I’m using here is actually closer to what Bostrom call the Hybrid Model, then David Lewis’ Halver Solution. The difference is that if you are told it is Monday, Bostrom gets heads being 1⁄2, while Lewis gets heads being 2⁄3.
The Beauty and the Prince
This post will address a problem proposed by Radford Neal in his paper Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical conditioning. In particular, he defined this problem—The Beauty and the Prince—to argue against the halver solution to the Sleeping Beauty Problem. I don’t think that this is ultimately a counter-example, but I decided to dedicate a post to it because I felt that it was quite persuasive when I first saw it. I’ll limit the scope of this post to arguing that his analysis of the halver solution is incorrect and providing a correct analysis instead. I won’t try to justify the halver solution as being philosophically correct as I plan to write another post on the Anthropic Principle later, just show how it applies here.
The Beauty and the Prince is just like the Sleeping Beauty Problem, but with a Prince who is also interviewed and memory-wiped. However, he is always interviewed on both Monday and Tuesday regardless of what the coin shows and he is told whether or not Sleeping Beauty is awake. If he is told that she is awake, what is the probability that the coin came up heads. The argument is that 3⁄4 times she will be awake and 1⁄4 times she is asleep so only 1⁄3 times when he is told she is awake will the coin be heads. Further, it seems that Sleeping Beauty should adopt the same odds as him. They both have the same information, so if he tells her the odds are 1⁄3, on what basis can she disagree? Further, she knows what he will say before he even asks her.
I want to propose that the Prince’s probability estimate as above is correct, but it is different from Sleeping Beauty’s. I think the key here is to realised that indexicals aren’t a part of standard probability, so we need to de-indexicalise the situation. However, we’ll de-indexicalise the original problem first. We’ll do this be ensuring that only one interview ever “counts”, by which we mean that we will calculate the probability of events over the interviews that count. We’ll do this by flipping a second coin if the first comes up tails. If it is heads, only the first interview counts, whilst for tails only the second interview counts. We then get the odds being: 1⁄2 heads + Monday counts; 1⁄4 tails + Monday counts; 1⁄4 tails + Tuesday counts.
We similarly de-indexalise the Prince, though we flip the second coin in the heads case too. Similarly, if it is heads we count the interview on Monday and if it is tails we count the interview on Tuesday, so the four possibilities become mutually exclusive and each have a probability of 25%.
If we look when the first coin is heads, we notice that the Prince’s interview on Monday only counts 50% of the time, whilst Sleeping Beauty’s counts 100% of the time. This means that Sleeping Beauty is calculating her probability over a different event space so we should actually expect her answer to differ from that of the Prince. Suppose we expand the Prince’s probability to include the Sleeping Beauty’s Monday interviews (which all count). Then we get the chance of heads moving from 1:2 = 1⁄3 to 2:2 = 1⁄2.
As we’ve seen, The Beauty and the Prince is not a problem for the halver solution. This does not mean that the halver solution is the correct solution to the Sleeping Beauty Problem, just that The Beauty and The Prince doesn’t provide a counter-example.
Update: I’ve been reading more of the literature. It seems that the technique that I’m using here is actually closer to what Bostrom call the Hybrid Model, then David Lewis’ Halver Solution. The difference is that if you are told it is Monday, Bostrom gets heads being 1⁄2, while Lewis gets heads being 2⁄3.