I think this is mistaken in that eliminating the HT and TTT possibilities isn’t the only update SB can make on seeing heads. Conditioning on a particular sequence of flips, an observation of heads is certain under the HH or THH sequences, but only 50% likely under the THT or TTH sequences, so SB should adjust probabilities accordingly and consequently end up with no new information about the initial flip.
HOWEVER. The above logic relies on the assumption that this is a coherent and useful way to consider probabilities in this kind of anthropic problem, and that’s not an assumption I accept. So take with a grain of salt.
I think this is mistaken in that eliminating the HT and TTT possibilities isn’t the only update SB can make on seeing heads.
It’s the Sherlock Holmes Axiom that the original post was suggesting we use.
Conditioning on a particular sequence of flips, an observation of heads is certain under the HH or THH sequences, but only 50% likely under the THT or TTH sequences, so SB should adjust probabilities accordingly and consequently end up with no new information about the initial flip.
This would be SB deciding that she is randomly selected from the reference class of SBs. In other words, it’s SSA, only with a much smaller reference class than I’d suggest using.
If she uses a larger reference class, she’d realize that she’s about twice as likely to wake up in a room during the experiment if the coin landed on tails, and would conclude that there’s a nearly 2⁄3 probability of the coin landing on tails.
I think this is mistaken in that eliminating the HT and TTT possibilities isn’t the only update SB can make on seeing heads. Conditioning on a particular sequence of flips, an observation of heads is certain under the HH or THH sequences, but only 50% likely under the THT or TTH sequences, so SB should adjust probabilities accordingly and consequently end up with no new information about the initial flip.
HOWEVER. The above logic relies on the assumption that this is a coherent and useful way to consider probabilities in this kind of anthropic problem, and that’s not an assumption I accept. So take with a grain of salt.
It’s the Sherlock Holmes Axiom that the original post was suggesting we use.
This would be SB deciding that she is randomly selected from the reference class of SBs. In other words, it’s SSA, only with a much smaller reference class than I’d suggest using.
If she uses a larger reference class, she’d realize that she’s about twice as likely to wake up in a room during the experiment if the coin landed on tails, and would conclude that there’s a nearly 2⁄3 probability of the coin landing on tails.