It’s hard to make a sensible notion of probability out of “fraction of waking moments”. Two subsequent states of a given dynamical system make for poor distinct elements of a sample space: when we’ve observed that the first moment of a given dynamical trajectory is not the second, what are we going to do when we encounter the second one? It’s already ruled “impossible”! Thus, Monday and Tuesday under the same circumstances shouldn’t be modeled as two different elements of a sample space.
As Wei Dai and Roko have observed, that depends on why you’re asking in the first place. Probability estimates should pay rent in correct decisions. If you’re making a bet that will pay off once at the end of the experiment, you should count the fraction of branches. If you’re making a bet that will pay off once per wake-up call, you should count the fraction of wake-up calls.
That’s the wrong way to look at it. A certain bet may be the “correct” action to perform, or even a certain ritual of cognition may pay its rent, but it won’t be about the concept of probability. Circumstances may make it preferable to do or say anything, but that won’t influence the meaning of fixed concepts. You can’t argue that 2+2 is in fact 5 on the grounds that saying that saves puppies. You may say that 2+2 is 5, or think that “probability of Tuesday” is 1⁄3 or 1⁄4 in order to win, but that won’t make it so, it will merely make you win.
Subjective probability is not a well-defined concept in the general case. Fractions are well-defined, but only after you’ve decided where you are getting the numerator and denominator from.
Let us not sacrifice effectiveness of our concepts in order to make them mathematically elegant. If reality gives you problems where you win by reasoning anthropically, but ordinary probability theory is not up to the job of facilitating, then invent UDT and use that instead.
It’s hard to make a sensible notion of probability out of “fraction of waking moments”.
If reality gives you problems where you win by reasoning anthropically, but ordinary probability theory is not up to the job of facilitating, then invent UDT and use that instead.
The winning thing might be better than the probability thing, but it won’t be a probability thing just because it’s winning. Also, UDT weakly relies on the same framework of expected utility and probability spaces, defined exactly as I discuss them in the comments to this post.
It’s hard to make a sensible notion of probability out of “fraction of waking moments”. Two subsequent states of a given dynamical system make for poor distinct elements of a sample space: when we’ve observed that the first moment of a given dynamical trajectory is not the second, what are we going to do when we encounter the second one? It’s already ruled “impossible”! Thus, Monday and Tuesday under the same circumstances shouldn’t be modeled as two different elements of a sample space.
As Wei Dai and Roko have observed, that depends on why you’re asking in the first place. Probability estimates should pay rent in correct decisions. If you’re making a bet that will pay off once at the end of the experiment, you should count the fraction of branches. If you’re making a bet that will pay off once per wake-up call, you should count the fraction of wake-up calls.
That’s the wrong way to look at it. A certain bet may be the “correct” action to perform, or even a certain ritual of cognition may pay its rent, but it won’t be about the concept of probability. Circumstances may make it preferable to do or say anything, but that won’t influence the meaning of fixed concepts. You can’t argue that 2+2 is in fact 5 on the grounds that saying that saves puppies. You may say that 2+2 is 5, or think that “probability of Tuesday” is 1⁄3 or 1⁄4 in order to win, but that won’t make it so, it will merely make you win.
Subjective probability is not a well-defined concept in the general case. Fractions are well-defined, but only after you’ve decided where you are getting the numerator and denominator from.
That fractions are well-defined doesn’t make them probabilities.
Let us not sacrifice effectiveness of our concepts in order to make them mathematically elegant. If reality gives you problems where you win by reasoning anthropically, but ordinary probability theory is not up to the job of facilitating, then invent UDT and use that instead.
The winning thing might be better than the probability thing, but it won’t be a probability thing just because it’s winning. Also, UDT weakly relies on the same framework of expected utility and probability spaces, defined exactly as I discuss them in the comments to this post.