What I’m saying is that there are two probabilities there, and they are both the correct probabilities, but they are the correct probabilities of different things. These different things seem like answers to the same question because the English language isn’t meant to deal with Sleeping Beauty type problems. But there is a difference, which I’ve done my best to explain.
Given that, is there anything your nitpicking actually addresses?
Sleeping Beauty may want to maximize the probability that she guesses the coin correctly at least once, in which cases she should use the probability 1⁄2. Or she may want to maximize the number of correct guesses, in which case she should use the probability 2⁄3.
That looks like two “probabilities” to me. Could you explain what the probabilities would be of, using the usual Bayesian understanding of “probability”?
I can try to rephrase what I said, but I honestly have no clue what you mean by putting probabilities in quotes.
2⁄3 is the probability that this Sleeping Beauty is waking up in a world where the coin came up tails. 1⁄2 is the probability that some Sleeping Beauty will wake up in such a world. To the naive reader, both of these things sound like “The probability that the coin comes up tails”.
I put the word “probability” in quotes is because I wanted to talk about the word itself, not the type of logic it refers to. The reason I thought you were talking about different types of logic using the same word was because probability already specifies what you’re supposed to be maximizing. For individual probabilities it could be one of many scoring rules, but if you want to add scores together you need to use the log scoring rule.
To the naive reader, both of these things sound like “The probability that the coin comes up tails”.
Right. One of them is the probability that the coin comes up tails given some starting information (as in a conditional probability, like P(T | S)), and the other is the probability that the coin comes up tails, given the starting information and some anthropic information: P(T | S A). So they’re both “P(T),” in a way.
Hah, so I think in your original comment you meant “asking “but what’s P(T), really?” isn’t helpful,” but I heard “asking “but what’s P(T | S A), really?” isn’t helpful” (in my defense, some people have actually said this).
If this is right I’ll edit it into my original reply so that people can be less confused. Lastly, in light of this there is only one thing I can link to.
What I’m saying is that there are two probabilities there, and they are both the correct probabilities, but they are the correct probabilities of different things. These different things seem like answers to the same question because the English language isn’t meant to deal with Sleeping Beauty type problems. But there is a difference, which I’ve done my best to explain.
Given that, is there anything your nitpicking actually addresses?
By “two probabilities” you mean this? :
That looks like two “probabilities” to me. Could you explain what the probabilities would be of, using the usual Bayesian understanding of “probability”?
I can try to rephrase what I said, but I honestly have no clue what you mean by putting probabilities in quotes.
2⁄3 is the probability that this Sleeping Beauty is waking up in a world where the coin came up tails. 1⁄2 is the probability that some Sleeping Beauty will wake up in such a world. To the naive reader, both of these things sound like “The probability that the coin comes up tails”.
Ah, okay, that makes sense to me now. Thanks.
I put the word “probability” in quotes is because I wanted to talk about the word itself, not the type of logic it refers to. The reason I thought you were talking about different types of logic using the same word was because probability already specifies what you’re supposed to be maximizing. For individual probabilities it could be one of many scoring rules, but if you want to add scores together you need to use the log scoring rule.
Right. One of them is the probability that the coin comes up tails given some starting information (as in a conditional probability, like P(T | S)), and the other is the probability that the coin comes up tails, given the starting information and some anthropic information: P(T | S A). So they’re both “P(T),” in a way.
Hah, so I think in your original comment you meant “asking “but what’s P(T), really?” isn’t helpful,” but I heard “asking “but what’s P(T | S A), really?” isn’t helpful” (in my defense, some people have actually said this).
If this is right I’ll edit it into my original reply so that people can be less confused. Lastly, in light of this there is only one thing I can link to.