So if you’re not updating on the apparent conditional rarity of having a highly ordered experience of gravity, then you should just believe the very simple hypothesis of a high-volume random experience generator, which would necessarily create your current experiences—albeit with extreme relative infrequency, but you don’t care about that.
“A high-volume random experience generator” is not a hypothesis. It’s a thing. “The universe is a high-volume random experience generator” is better, but still not okay for Bayesian updating, because we don’t observe “the universe”. “My observations are output by a high-volume random experience generator” is better still, but it doesn’t specify which output our observations are. “My observations are the output at [...] by a high-volume random experience generator” is a specific, updatable hypothesis—and its entropy is so high that it’s not worth considering.
Did I just use anthropic reasoning?
Let’s apply this to the hotel problem. There are two specific hypotheses: “My observations are what they were before except I’m now in green room #314159265″ (or whatever green room) and ”. . . except I’m now in the red room”. It appears that the thing determining probability is not multiplicity but complexity of the “address”—and, counterintuitively, this makes the type of room only one of you is in more likely than the type of room a billion of you are in.
Yes, I’m taking into account that “I’m in a green room” is the disjunction of one billion hypotheses and therefore has one billion times the probability of any of them. In order for one’s priors to be well-defined, then for infinitely many N, all hypotheses of length N+1 together must be less likely than all hypotheses of length N together.
Edit: changed “more likely” to “less likely” (oops) and “large N” to “infinitely many N”, as per pengvado. Thanks!
This post in seventeen words: it’s the high multiplicity of brains in the Boltzmann brain hypothesis, not their low frequency, that matters.
“My observations are the output at [...] by a high-volume random experience generator”
“My observations are [...], which were output by a high-volume random experience generator”. Since the task is to explain my observations, not to predict where I am. This way also makes it more clear that that suffix is strictly superfluous from a Kolmogorov perspective.
In order for one’s priors to be well-defined, then for large N, all hypotheses of length N+1 together must be more likely than all hypotheses of length N together.
You mean less likely. i.e. there is no nonnegative monotonic-increasing infinite series whose sum is finite. Also, it need not happen for all large N, just some of them. So I would clarify it as: ∀L ∃N>L ∀M>N (((sum of probabilities of hypotheses of length M) < (sum of probabilities of hypotheses of length N)) or (both are zero)).
But you shouldn’t take that into account for your example. The theorem applies to infinite sequences of hypotheses, but not to any one finite hypothesis such as the disjunction of a billion green rooms. To get conclusions about a particular hypothesis, you need more than “any prior is Occam’s razor with respect to a sufficiently perverse complexity metric”.
“My observations are [...], which were output by a high-volume random experience generator”. Since the task is to explain my observations, not to predict where I am. This way also makes it more clear that that suffix is strictly superfluous from a Kolmogorov perspective.
You are correct, though I believe your statement is equivalent to mine.
You mean less likely. i.e. there is no nonnegative monotonic-increasing infinite series whose sum is finite. Also, it need not happen for all large N, just some of them. So I would clarify it as: ∀L ∃N>L ∀M>N (((sum of probabilities of hypotheses of length M) < (sum of probabilities of hypotheses of length N)) or (both are zero)).
Here, let me re-respond to this post.
“A high-volume random experience generator” is not a hypothesis. It’s a thing. “The universe is a high-volume random experience generator” is better, but still not okay for Bayesian updating, because we don’t observe “the universe”. “My observations are output by a high-volume random experience generator” is better still, but it doesn’t specify which output our observations are. “My observations are the output at [...] by a high-volume random experience generator” is a specific, updatable hypothesis—and its entropy is so high that it’s not worth considering.
Did I just use anthropic reasoning?
Let’s apply this to the hotel problem. There are two specific hypotheses: “My observations are what they were before except I’m now in green room #314159265″ (or whatever green room) and ”. . . except I’m now in the red room”. It appears that the thing determining probability is not multiplicity but complexity of the “address”—and, counterintuitively, this makes the type of room only one of you is in more likely than the type of room a billion of you are in.
Yes, I’m taking into account that “I’m in a green room” is the disjunction of one billion hypotheses and therefore has one billion times the probability of any of them. In order for one’s priors to be well-defined, then for infinitely many N, all hypotheses of length N+1 together must be less likely than all hypotheses of length N together.
Edit: changed “more likely” to “less likely” (oops) and “large N” to “infinitely many N”, as per pengvado. Thanks!
This post in seventeen words: it’s the high multiplicity of brains in the Boltzmann brain hypothesis, not their low frequency, that matters.
Let the poking of holes into this post begin!
“My observations are [...], which were output by a high-volume random experience generator”. Since the task is to explain my observations, not to predict where I am. This way also makes it more clear that that suffix is strictly superfluous from a Kolmogorov perspective.
You mean less likely. i.e. there is no nonnegative monotonic-increasing infinite series whose sum is finite. Also, it need not happen for all large N, just some of them. So I would clarify it as: ∀L ∃N>L ∀M>N (((sum of probabilities of hypotheses of length M) < (sum of probabilities of hypotheses of length N)) or (both are zero)).
But you shouldn’t take that into account for your example. The theorem applies to infinite sequences of hypotheses, but not to any one finite hypothesis such as the disjunction of a billion green rooms. To get conclusions about a particular hypothesis, you need more than “any prior is Occam’s razor with respect to a sufficiently perverse complexity metric”.
You are correct, though I believe your statement is equivalent to mine.
Right again; I’ll fix my post.