I think talking about ‘observers’ might be muddling the issue here.
That’s probably why you don’t understand the result; it is an anthropic selection effect. See my reply to Academician above.
We could talk instead about creatures that don’t understand the experiment, and the result would be the same. Say we have two Petri dishes, one dish containing a single bacterium, and the other containing a trillion. We randomly select one of the bacteria (representing me in the original door experiment) to stain with a dye. We flip a coin: if it’s heads, we kill the lone bacterium, otherwise we put the trillion-bacteria dish into an autoclave and kill all of those bacteria. Given that the stained bacterium survives the process, it is far more likely that it was in the trillion-bacteria dish, so it is far more likely that the coin came up heads.
That is not an analogous experiment. Typical survivors are not pre-selected individuals; they are post-selected, from the pool of survivors only. The analogous experiment would be to choose one of the surviving bacteria after the killing and then stain it. To stain it before the killing risks it not being a survivor, and that can’t happen in the case of anthropic selection among survivors.
I don’t think of the pi digit process as equivalent.
That’s because you erroneously believe that your frequency interpretation works. The math problem has only one answer, which makes it a perfect analogy for the 1-shot case.
That is not an analogous experiment. Typical survivors are not pre-selected individuals; they are post-selected, from the pool of survivors only. The analogous experiment would be to choose one of the surviving bacteria after the killing and then stain it. To stain it before the killing risks it not being a survivor, and that can’t happen in the case of anthropic selection among survivors.
I believe that situations A and B which you quote from Stuart_Armstrong’s post involve pre-selection, not post-selection, so maybe that is why we disagree. I believe that because the descriptions of the two situations refer to ‘you’ - that is, me—which makes me construct a mental model of me being put into one of the 100 rooms at random. In that model my pre-selected consciousness is at issue, not that of a post-selected survivor.
That’s because you erroneously believe that your frequency interpretation works. The math problem has only one answer, which makes it a perfect analogy for the 1-shot case.
By ‘math problem’ do you mean the question of whether pi’s millionth bit is 0? If so, I disagree. The 1-shot case (which I think you are using to refer to situation B in Stuart_Armstrong’s top-level post...?) describes a situation defined to have multiple possible outcomes, but there’s only one outcome to the question ‘what is pi’s millionth bit?’
A few minutes later, it is announced that whoever was to be killed has been killed. What are your odds of being blue-doored now?
Presumably you heard the announcement.
This is post-selection, because pre-selection would have been “Either you are dead, or you hear that whoever was to be killed has been killed. What are your odds of being blue-doored now?”
The 1-shot case (which I think you are using to refer to situation B in Stuart_Armstrong’s top-level post...?) describes a situation defined to have multiple possible outcomes, but there’s only one outcome to the question ‘what is pi’s millionth bit?’
There’s only one outcome in the 1-shot case.
The fact that there are multiple “possible” outcomes is irrelevant—all that means is that, like in the math case, you don’t have knowledge of which outcome it is.
This is post-selection, because pre-selection would have been “Either you are dead, or you hear that whoever was to be killed has been killed. What are your odds of being blue-doored now?”
The ‘selection’ I have in mind is the selection, at the beginning of the scenario, of the person designated by ‘you’ and ‘your’ in the scenario’s description. The announcement, as I understand it, doesn’t alter the selection in the sense that I think of it, nor does it generate a new selection: it just indicates that ‘you’ happened to survive.
The fact that there are multiple “possible” outcomes is irrelevant—all that means is that, like in the math case, you don’t have knowledge of which outcome it is.
I continue to have difficulty accepting that the millionth bit of pi is just as good a random bit source as a coin flip. I am picturing a mathematically inexperienced programmer writing a (pseudo)random bit-generating routine that calculated the millionth digit of pi and returned it. Could they justify their code by pointing out that they don’t know what the millionth digit of pi is, and so they can treat it as a random bit?
I continue to have difficulty accepting that the millionth bit of pi is just as good a random bit source as a coin flip. I am picturing a mathematically inexperienced programmer writing a (pseudo)random bit-generating routine that calculated the millionth digit of pi and returned it. Could they justify their code by pointing out that they don’t know what the millionth digit of pi is, and so they can treat it as a random bit?
Seriously: You have no reason to believe that the millionth bit of pi goes one way or the other, so you should assign equal probability to each.
However, just like the xkcd example would work better if the computer actually rolled the die for you every time rather than just returning ‘4’, the ‘millionth bit of pi’ algorithm doesn’t work well because it only generates a random bit once (amongst other practical problems).
In most pseudorandom generators, you can specify a ‘seed’ which will get you a fixed set of outputs; thus, you could every time restart the generator with the seed that will output ‘4’ and get ‘4’ out of it deterministically. This does not undermine its ability to be a random number generator. One common way to seed a random number generator is to simply feed it the current time, since that’s as good as random.
Looking back, I’m not certain if I’ve answered the question.
Looking back, I’m not certain if I’ve answered the question.
I think so: I’m inferring from your comment that the principle of indifference is a rationale for treating a deterministic-but-unknown quantity as a random variable. Which I can’t argue with, but it still clashes with my intuition that any casino using the millionth bit of pi as its PRNG should expect to lose a lot of money.
I agree with your point on arbitrary seeding, for whatever it’s worth. Selecting an arbitrary bit of pi at random to use as a random bit amounts to a coin flip.
I am picturing a mathematically inexperienced programmer writing a (pseudo)random bit-generating routine that calculated the millionth digit of pi and returned it.
I’d be extremely impressed if a mathematically inexperienced programmer could pull of a program that calculated the millionth digit of pi!
Could they justify their code by pointing out that they don’t know what the millionth digit of pi is, and so they can treat it as a random bit?
I say yes (assuming they only plan on treating it as a random bit once!)
The ‘selection’ I have in mind is the selection, at the beginning of the scenario, of the person designated by ‘you’ and ‘your’ in the scenario’s description.
If ‘you’ were selected at the beginning, then you might not have survived.
Note that “If you (being asked before the killing) will survive, what color is your door likely to be?” is very different from “Given that you did already survive, …?”. A member of the population to which the first of these applies might not survive. This changes the result. It’s the difference between pre-selection and post-selection.
I’ll try to clarify what I’m thinking of as the relevant kind of selection in this exercise. It is true that the condition effectively picks out—that is, selects—the probability branches in which ‘you’ don’t die, but I don’t see that kind of selection as relevant here, because (by my calculations, if not your own) it has no impact on the probability of being behind a blue door.
What sets your probability of being behind a blue door is the problem specifying that ‘you’ are the experimental subject concerned: that gives me the mental image of a film camera, representing my mind’s eye, following ‘you’ from start to finish - ‘you’ are the specific person who has been selected. I don’t visualize a camera following a survivor randomly selected post-killing. That is what leads me to think of the relevant selection as happening pre-killing (hence ‘pre-selection’).
If that were the case, the camera might show the person being killed; indeed, that is 50% likely.
Pre-selection is not the same as our case of post-selection. My calculation shows the difference it makes.
Now, if the fraction of observers of each type that are killed is the same, the difference between the two selections cancels out. That is what tends to happen in the many-shot case, and we can then replace probabilities with relative frequencies. One-shot probability is not relative frequency.
If that were the case, the camera might show the person being killed; indeed, that is 50% likely.
Yep. But Stuart_Armstrong’s description is asking us to condition on the camera showing ‘you’ surviving.
Pre-selection is not the same as our case of post-selection. My calculation shows the difference it makes.
It looks to me like we agree that pre-selecting someone who happens to survive gives a different result (99%) to post-selecting someone from the pool of survivors (50%) - we just disagree on which case SA had in mind. Really, I guess it doesn’t matter much if we agree on what the probabilities are for the pre-selection v. the post-selection case.
Now, if the fraction of observers of each type that are killed is the same, the difference between the two selections cancels out. That is what tends to happen in the many-shot case, and we can then replace probabilities with relative frequencies.
I am unsure how to interpret this...
One-shot probability is not relative frequency.
...but I’m fairly sure I disagree with this. If we do Bernoulli trials with success probability p (like coin flips, which are equivalent to Bernoulli trials with p = 0.5), I believe the strong law of large numbers implies that the relative frequency converges almost surely to p as the number of Bernoulli trials becomes arbitrarily large. As p represents the ‘one-shot probability,’ this justifies interpreting the relative frequency in the infinite limit as the ‘one-shot probability.’
But Stuart_Armstrong’s description is asking us to condition on the camera showing ‘you’ surviving.
That condition imposes post-selection.
I guess it doesn’t matter much if we agree on what the probabilities are for the pre-selection v. the post-selection case.
Wrong—it matters a lot because you are using the wrong probabilities for the survivor (in practice this affects things like belief in the Doomsday argument).
I believe the strong law of large numbers implies that the relative frequency converges almost surely to p as the number of Bernoulli trials becomes arbitrarily large. As p represents the ‘one-shot probability,’ this justifies interpreting the relative frequency in the infinite limit as the ‘one-shot probability.’
You have things backwards. The “relative frequency in the infinite limit” can be defined that way (sort of, as the infinite limit is not actually doable) and is then equal to the pre-defined probability p for each shot if they are independent trials. You can’t go the other way; we don’t have any infinite sequences to examine, so we can’t get p from them, we have to start out with it. It’s true that if we have a large but finite sequence, we can guess that p is “probably” close to our ratio of finite outcomes, but that’s just Bayesian updating given our prior distribution on likely values of p. Also, in the 1-shot case at hand, it is crucial that there is only the 1 shot.
But not post-selection of the kind that influences the probability (at least, according to my own calculations).
Wrong—it matters a lot because you are using the wrong probabilities for the survivor (in practice this affects things like belief in the Doomsday argument).
Which of my estimates is incorrect—the 50% estimate for what I call ‘pre-selecting someone who happens to survive,’ the 99% estimate for what I call ‘post-selecting someone from the pool of survivors,’ or both?
You can’t go the other way; we don’t have any infinite sequences to examine, so we can’t get p from them, we have to start out with it.
Correct. p, strictly, isn’t defined by the relative frequency—the strong law of large numbers simply justifies interpreting it as a relative frequency. That’s a philosophical solution, though. It doesn’t help for practical cases like the one you mention next...
It’s true that if we have a large but finite sequence, we can guess that p is “probably” close to our ratio of finite outcomes, but that’s just Bayesian updating given our prior distribution on likely values of p.
...for practical scenarios like this we can instead use the central limit theorem to say that p’s likely to be close to the relative frequency. I’d expect it to give the same results as Bayesian updating—it’s just that the rationale differs.
Also, in the 1-shot case at hand, it is crucial that there is only the 1 shot.
It certainly is in the sense that if ‘you’ die after 1 shot, ‘you’ might not live to take another!
That’s probably why you don’t understand the result; it is an anthropic selection effect. See my reply to Academician above.
That is not an analogous experiment. Typical survivors are not pre-selected individuals; they are post-selected, from the pool of survivors only. The analogous experiment would be to choose one of the surviving bacteria after the killing and then stain it. To stain it before the killing risks it not being a survivor, and that can’t happen in the case of anthropic selection among survivors.
That’s because you erroneously believe that your frequency interpretation works. The math problem has only one answer, which makes it a perfect analogy for the 1-shot case.
Okay.
I believe that situations A and B which you quote from Stuart_Armstrong’s post involve pre-selection, not post-selection, so maybe that is why we disagree. I believe that because the descriptions of the two situations refer to ‘you’ - that is, me—which makes me construct a mental model of me being put into one of the 100 rooms at random. In that model my pre-selected consciousness is at issue, not that of a post-selected survivor.
By ‘math problem’ do you mean the question of whether pi’s millionth bit is 0? If so, I disagree. The 1-shot case (which I think you are using to refer to situation B in Stuart_Armstrong’s top-level post...?) describes a situation defined to have multiple possible outcomes, but there’s only one outcome to the question ‘what is pi’s millionth bit?’
Presumably you heard the announcement.
This is post-selection, because pre-selection would have been “Either you are dead, or you hear that whoever was to be killed has been killed. What are your odds of being blue-doored now?”
There’s only one outcome in the 1-shot case.
The fact that there are multiple “possible” outcomes is irrelevant—all that means is that, like in the math case, you don’t have knowledge of which outcome it is.
The ‘selection’ I have in mind is the selection, at the beginning of the scenario, of the person designated by ‘you’ and ‘your’ in the scenario’s description. The announcement, as I understand it, doesn’t alter the selection in the sense that I think of it, nor does it generate a new selection: it just indicates that ‘you’ happened to survive.
I continue to have difficulty accepting that the millionth bit of pi is just as good a random bit source as a coin flip. I am picturing a mathematically inexperienced programmer writing a (pseudo)random bit-generating routine that calculated the millionth digit of pi and returned it. Could they justify their code by pointing out that they don’t know what the millionth digit of pi is, and so they can treat it as a random bit?
Not seriously: http://www.xkcd.com/221/
Seriously: You have no reason to believe that the millionth bit of pi goes one way or the other, so you should assign equal probability to each.
However, just like the xkcd example would work better if the computer actually rolled the die for you every time rather than just returning ‘4’, the ‘millionth bit of pi’ algorithm doesn’t work well because it only generates a random bit once (amongst other practical problems).
In most pseudorandom generators, you can specify a ‘seed’ which will get you a fixed set of outputs; thus, you could every time restart the generator with the seed that will output ‘4’ and get ‘4’ out of it deterministically. This does not undermine its ability to be a random number generator. One common way to seed a random number generator is to simply feed it the current time, since that’s as good as random.
Looking back, I’m not certain if I’ve answered the question.
I think so: I’m inferring from your comment that the principle of indifference is a rationale for treating a deterministic-but-unknown quantity as a random variable. Which I can’t argue with, but it still clashes with my intuition that any casino using the millionth bit of pi as its PRNG should expect to lose a lot of money.
I agree with your point on arbitrary seeding, for whatever it’s worth. Selecting an arbitrary bit of pi at random to use as a random bit amounts to a coin flip.
I’d be extremely impressed if a mathematically inexperienced programmer could pull of a program that calculated the millionth digit of pi!
I say yes (assuming they only plan on treating it as a random bit once!)
If ‘you’ were selected at the beginning, then you might not have survived.
Yeah, but the description of the situation asserts that ‘you’ happened to survive.
Adding that condition is post-selection.
Note that “If you (being asked before the killing) will survive, what color is your door likely to be?” is very different from “Given that you did already survive, …?”. A member of the population to which the first of these applies might not survive. This changes the result. It’s the difference between pre-selection and post-selection.
I’ll try to clarify what I’m thinking of as the relevant kind of selection in this exercise. It is true that the condition effectively picks out—that is, selects—the probability branches in which ‘you’ don’t die, but I don’t see that kind of selection as relevant here, because (by my calculations, if not your own) it has no impact on the probability of being behind a blue door.
What sets your probability of being behind a blue door is the problem specifying that ‘you’ are the experimental subject concerned: that gives me the mental image of a film camera, representing my mind’s eye, following ‘you’ from start to finish - ‘you’ are the specific person who has been selected. I don’t visualize a camera following a survivor randomly selected post-killing. That is what leads me to think of the relevant selection as happening pre-killing (hence ‘pre-selection’).
If that were the case, the camera might show the person being killed; indeed, that is 50% likely.
Pre-selection is not the same as our case of post-selection. My calculation shows the difference it makes.
Now, if the fraction of observers of each type that are killed is the same, the difference between the two selections cancels out. That is what tends to happen in the many-shot case, and we can then replace probabilities with relative frequencies. One-shot probability is not relative frequency.
Yep. But Stuart_Armstrong’s description is asking us to condition on the camera showing ‘you’ surviving.
It looks to me like we agree that pre-selecting someone who happens to survive gives a different result (99%) to post-selecting someone from the pool of survivors (50%) - we just disagree on which case SA had in mind. Really, I guess it doesn’t matter much if we agree on what the probabilities are for the pre-selection v. the post-selection case.
I am unsure how to interpret this...
...but I’m fairly sure I disagree with this. If we do Bernoulli trials with success probability p (like coin flips, which are equivalent to Bernoulli trials with p = 0.5), I believe the strong law of large numbers implies that the relative frequency converges almost surely to p as the number of Bernoulli trials becomes arbitrarily large. As p represents the ‘one-shot probability,’ this justifies interpreting the relative frequency in the infinite limit as the ‘one-shot probability.’
That condition imposes post-selection.
Wrong—it matters a lot because you are using the wrong probabilities for the survivor (in practice this affects things like belief in the Doomsday argument).
You have things backwards. The “relative frequency in the infinite limit” can be defined that way (sort of, as the infinite limit is not actually doable) and is then equal to the pre-defined probability p for each shot if they are independent trials. You can’t go the other way; we don’t have any infinite sequences to examine, so we can’t get p from them, we have to start out with it. It’s true that if we have a large but finite sequence, we can guess that p is “probably” close to our ratio of finite outcomes, but that’s just Bayesian updating given our prior distribution on likely values of p. Also, in the 1-shot case at hand, it is crucial that there is only the 1 shot.
But not post-selection of the kind that influences the probability (at least, according to my own calculations).
Which of my estimates is incorrect—the 50% estimate for what I call ‘pre-selecting someone who happens to survive,’ the 99% estimate for what I call ‘post-selecting someone from the pool of survivors,’ or both?
Correct. p, strictly, isn’t defined by the relative frequency—the strong law of large numbers simply justifies interpreting it as a relative frequency. That’s a philosophical solution, though. It doesn’t help for practical cases like the one you mention next...
...for practical scenarios like this we can instead use the central limit theorem to say that p’s likely to be close to the relative frequency. I’d expect it to give the same results as Bayesian updating—it’s just that the rationale differs.
It certainly is in the sense that if ‘you’ die after 1 shot, ‘you’ might not live to take another!