It is a great idea to test a hypothesis experimentally. I did your experiment too, and the result is:
Several experiments show that I can extract useful information just by treating myself as a random sample, and thus a view that I can't use myself as a random sample is false.
I think there are some problems here. I think be more accurate claim would be:
You can do experiments that extract useful information about whether you can treat yourself as a random sample (i.e., a representative or "typical" sample) by comparing the result of the experiment to the baserate.
Or at the very least, based on my experiments, for me, the claim seems to be false. I'm not representative enough. But I can't know that without comparing my results to a baserate. I can't use the observations to establish a baserate or make estimations such as expected lifetime.
From a statistical perspective, a random sample means:
You may not be representative in any observable or unobservable dimension for your purpose. And to know if you are representative, you have to look at other samples and then you are back so some kind of baserate.
Using oneself as a random sample is a very rough way to get an idea about what order of magnitude some variable is. If you determine that the day duration is 2 hours, it is still useful information as you know almost for sure now that it is not 1 millisecond or 10 years. (And if I perform 10 experiments like this, one on average will be an order of magnitude off). We can also adjust the experiment by taking into account that people are sleeping at night, so they read LW only during the day, evening, or early morning. So times above 12 or below 2 are more likely.
You are right that the point of the experiments here is not to learn the real time of the day, but to prove that I can treat myself as a random sample in general and after that use this idea in domains where I do not have any information.
I think the basis to treat myself as a random sample is the following:
First of all, your experimental method can really benefit from a control group. Pick a setting where a thing is definitely not randomly sampled from a set. Perform your experiment and see what happens.
Consider. I generated a random word using this site https://randomwordgenerator.com/
This word turned out to be "mosaic". It has 6 letters. Let's test whether it's length is randomly sampled from the number of months in a year.
As 6*2=12, this actually works perfectly, even better than estimating the number of months in a year based on your birth month!
It also works decently to estimate several other things. The number of hours in a day, number of days in a months. Number of minutes in an hour. It gets the order of magnitude right! If we use the number of minutes in your 15:14 timestamp to estimate the number of minutes in an hour we are simularly off.
Worse with number of days in a week and number of days in a year. But it's mistaken by only one order of magnitude, maybe it's also okay?
At this point the problems with your methodology should be clear:
Secondly, if we want to talk about DA or SH, there is the whole jump from
"I can approximate my birth date as a random sample from all the days in the month/months in a year"
to
"I can approximate my birth rank as a random sample from all the births of people throughout history/starting from the moment of knowing about DA".
The latter doesn't follow from the former, even though in semantic terms both can potentially be described as random sampling of a person throughout time.
The cyclical nature of days in a months and months in a year never allows you to be off more than by an order of magnitude. Even if your parents specifically timed your conception to give you birth on the first of January, therefore putting you in a very specific "reference class", you won't be extremely mistaken about these numbers following your methodology.
On the other hand, there is no such gurantee for birth orders of a people which work as natural numbers, not as elements of a finite field.
Don't agree. You chose word length generator as you know that typical length of words is 1-10. Thus not random.
I didn't rejected any results – it works in any test I have imagined, and I also didn't include several experiments which have the same results, e.g the total number of the days in a year based on my birthday (got around 500) and total number of letters in english alphabet (got around 40).
Note that alphabet letter count is not cyclical as well as my distance to equator.
Do not understand this:
Even if your parents specifically timed your conception to give you birth on the first of January, therefore putting you in a very specific "reference class", you won't be extremely mistaken about these numbers following your methodology.
If I were born 1 of January, I would get years duration 2 days which is very wrong.
You chose word length generator as you know that typical length of words is 1-10. Thus not random.
This is not relevant to my point. After all you also know that typical month is 1-12
No, the point is that I specifically selected a number via an algorithm that has nothing to do with sampling months. And yet your test outputs positive result anyway. Therefore your test is unreliable.
I didn't rejected any results – it works in any test I have imagined
That's exactly the problem. Essentially you are playing a 2,4,6 game, got no negative result yet and are already confident about the rule.
Note that alphabet letter count is not cyclical as well as my distance to equator.
Distance to equator is in fact cyclical in a very literal sense. Alphabet letters do not have anything to do with random sampling of you through time.
If I were born 1 of January, I would get years duration 2 days which is very wrong.
It's not more wrong for a person whose parents specifically tried to give birth at this date than for a person who just happened to be born at this time without any planning. And even in this extreme situation your mistake is limited by two orders of magnitude. There is no such guarantee in DA.
For example, If I use self-sampling to estimate the number of seconds in the year, I will get a correct answer of around several tens of millions. But using word generator will never output a word longer than 100 letters.
I didn't understand your idea here:
It's not more wrong for a person whose parents specifically tried to give birth at this date than for a person who just happened to be born at this time without any planning. And even in this extreme situation your mistake is limited by two orders of magnitude. There is no such guarantee in DA.
hich For example, If I use self-sampling to estimate the number of seconds in the year, I will get a correct answer of around several tens of millions. But using word generator will never output a word longer than 100 letters.
Using the month of your birth to estimate the number of seconds in the year also won't work well, unless you multiply it by number of seconds in a month.
Likewise here. You can estimate the number of months in a year by number of letters in the world and then multiply it by number of seconds in a months.
I didn't understand your idea here
Consider this.
Parents of person A tried really hard to give birth to A on the first of January and indeed it happened
Person B just so happened to be born on the first of January
Parents of person C tried really hard to give birth to C on the 15th of June and indeed it happened
Person D just so happened to be born on the 15th of June
Here date of births of B and D can be approximated as randomly sampled, while A and C are not. Your test, however will return that C and D can treat themselves as random sample, making both false positive and false negative errors.
This is because your test simply checks the distance from the mean value which, while somewhat correlated to being a result of random sampling, is a completely different thing.
I meant that if I know only the total number of the seconds which passed from the beginning of the year (around 15 million for today of this year) – and I want to predict the total number of seconds in each year. No information about months.
As most people are born randomly and we know it, we can use my date of birth as random. If we have any suspicions about non randomness, we have to take them into account.
Yes, you can use yourself as a random sample but at best only within a reference class of "people who use themselves as a random sample for this question in a sufficiently similar context to you". That might be a population of 1.
For example, suppose someone without symptoms has just found out that they have genes for a disease that always progresses to serious illness. They have a mathematics degree and want to use their statistical knowledge to estimate how long they have before becoming debilitated.
They are not a random sample from the reference class of people who have these genes. They are from people who have the genes and didn't show symptoms before finding that out and did so during adulthood (almost certainly) and live in a time and place and with sufficient capacity to earn a mathematics degree and of suitable mindset to ask themselves this question and so on.
Any of these may be relevant information for estimating the distribution, especially if the usual age of onset is in childhood or the disease also reduces intellectual capacity or affects personality in general.
Relating back to the original doomsday problem: suppose that in the reference class of all civilizations, most discover some principle that conclusively resolves the Doomsday problem not long after formulating it (within a few hundred years or so). It doesn't really matter what that resolution happens to be, there are plenty of possibilities.
If that is the case, then most people who even bother to ask the Doomsday question without already knowing the answer are those in that narrow window of time where their civilization is sophisticated enough to ask the question without being sophisticated enough to answer it, regardless of how long those civilizations might last or how many people exist after resolving the question.
To the extent that the Doomsday reasoning is valid at all (which it may not be), all that it provides is an estimate of time until most people stop asking the Doomsday question in a similar context to yours. Destruction of the species is not required for that. Even it becoming unfashionable is enough.
Agree that in some situations I have to take into account non-randomness of my sampling. While date of birth seems random and irrelevant, the distance to equator is strongly biased by distribution of the cities with universities which on Earth are shifted North.
Also agree that solving DA can be solution to DA: moreover, I looked at Google Scholar and found that the interest to DA is already declining.
As you mention, the three examples here work regardless of whether SSA or SIA is true because none of the estimated outcomes affect the total number of observers. But the Doomsday Argument is different and does depend on SSA. If SIA is true, the early population of a long world is just as likely to exist as the total population of a short world, so there’s no update upon finding yourself in an early-seeming world.
A total utilitarian observing from outside both worlds will care just as much about the early population of a long world as the total population of a short world, so the expected value of both reference classes is the same. This suggests to me that if I care about myself, I should be indifferent between the possibilities that I’m early and that I’m in a world with a short lifespan. Of course, if my decisions in one world will affect more people, then I should adjust my actions accordingly.
There is a way to escape this by using the universal doomsday argument. In it, we try not to predict the exact future of the Earth, but the typical life expectancy of Earth-like civilizations, that is, the proportion of long civilizations to short ones.
If we define a long civilization as one which has 1000 times more observers, the fact that we find ourselves early means that short civilizations are at least 1000 times more numerous.
In short, it is SSA, but applied to a large set of civilizations.
The universal doomsday argument still relies on SSA, because under SIA, I’m equally surprised to exist as an early person whether most civilizations are short or long. If most civilizations are long, I’m surprised to be early. If most civilizations are short, I’m surprised to exist at all. I could have been any of the late people who failed to exist because most civilizations are short. In other words, the surprise of existing as an early person is equivalent in both cases under SIA, so there's no update. Only under SSA am I certain I will exist but unsure where in the universe I will be.
I think that SIA is generally* valid but it uses all its power to prove that I live in the infinite universe where all possible observers exist. After that we have to use SSA to find in which region of the multiverse I am more likely to be located.
*I think that the logically sound version of SIA is "if I am in a unique position, generated by some random process, then there were many attempts to create me" – like many earth-like-but-lifeless planets are in the galaxy.
Another point is that the larger number of short civilizations can compensate for their "shortness." We can live in the region of multiverse where there are many short civilizations and almost all of them die off.
I looked at my clock and it was 15:14. It gives a 50 percent probability that the total number of hours in a day is 30
I am curious how you got that number.
It seems to me that, for any reasonnable prior, it it more probable that there is 16 hours in a day rather than 30.
Maybe I am misunderstanding your “50 percent probability”?
It was discussed above in comments – see buses example. In short, I actually care about periods, and 50 per cent is for "between 15 and 30" hours and other 50 per cent is for "above 30 hours".
Thank you. It is clearer that way. ^^ I feel like it would be less confusing (more true?) to write “below 30” rather than “30” in the sentence I quoted. ;-)
Gott started this type of confusion than he claimed that Berlin wall will stay 14 more years and it actually did exactly that. A better claim would be "first tens of hours with some given credence"
Note the actual doomsday argument properly applied predicts that humanity is mostly likely to end right now, with probability dropping proportional to total number of humans there have ever been.
To give a simple example why: if you go to a city and see a bus with the number 1546, the number of busses that maximises the chance you would have seen that bus is 1546 busses. At 3000 busses the probability you would have seen that exact bus is halved. And at 3,000,000 it's 2000 times less likely. This gives you a Bayesian update across your original probability distribution for how many busses there are.
Just to clarify, guessing that there are 1546 buses maximizes the probability that you are exactly correct, but it does not minimize your expected error, since you are guessing close to many numbers (everything below 1546) that are impossible. This is known in statistics as the "German tank problem"[1] and the posterior distribution is actually not well-defined in many setups.
From WW2 soldiers trying to estimate enemies' manufacturing capacity based on tank serial numbers
The intuition is that if we both saw bus 1546, and you guessed that there were 1546 buses and I guessed that there were 1547, you would be a little more likely to be correct but I would almost certainly be closer to the real number.
The Bayesian update isn't generally well-defined because you get a divergent mean. Your implicit prior is 1/n which is an improper prior. This is fine for deriving a posterior median, which in this case happens to be about 3,100 buses, and a posterior distribution, which in this case is a truncated zeta distribution with s=2 and k=1546. But the posterior mean does not exist.
I'm not using this is a prior, I'm using it to update my existing prior (whatever that was). I believe the posterior will be well defined, so long as the prior was.
As a worked example, if I start off assuming that chance of there being n busses is 1/2^n (nice and simple, adds up to 1), then the posterior is 1/n(ln(2))(2^n) - multiply the two distributions, then divide by the integral (ln(2)) so that it adds up to 1.
No, that's not the posterior distribution--clearly, the number of buses cannot be lower than 1546, but that distribution has material probability mass on low integers. I'm not quite sure how you got that equation.
But regardless, I think this shows where we disagree. That prior has mean 2... that's a pretty strong assumption about the distribution of n. If you want to avoid that kind of assumption, you can get posterior distributions but not a posterior expectation.
Sorry, I meant to add in an example where for simplicity you saw the bus numbered 1.
Agreed it's a terrible prior, it's just an easy one for a worked example.
I'm not disagreeing with that categorically--for many priors the posterior distribution is well defined. But all of those priors carry information (in the information theoretical sense) about the number of buses. If you have an uninformative reference prior, your posterior distribution does not have a mean.
You can see the sketch of this proof if you consider the likelihoods of seeing the bus for any given n. If there are 1546 buses, there was a 1/1546 chance you saw this one. If there were 1547, there was a 1/1547 chance you saw this one. This is the harmonic series, which diverges. That divergence is the fundamental issue that's going to cause the mean to be undefined.
You can't make claims about the posterior without setting at least some conditions on what your prior is--obviously, for some priors the posterior expectation is well-defined. (Trivially, if I already think n=2000 with probability 1, I will still think that after seeing the bus.) But I claim that all such priors make assumptions about the distribution of the possible number of buses. In the uninformative case, your posterior distribution is well-defined (as I said, it's a truncated zeta distribution) but it does not have a finite mean.
But I claim that all such priors make assumptions about the distribution of the possible number of buses
I mean, yes, that's the definition of a prior. How to calculate a prior is an old question in bayesianism, with different approaches - kolmogorov complexity being one.
In Gotts' approach, the bus distribution statistic between different cities is irrelevant. The number of buses N for this city is already fixed. When you draw the bus number n, you just randomly selected from N. In that case, probability is n/N, and if we look for 0.5 probability, we get 0.5 = 1546/N which gives us N = 2992 with 0.5 probability. Laplace came to similar result using much more complex calculations of summing all possible probability distribution.
In that case, probability is n/N, and if we look for 0.5 probability, we get 0.5 = 1546/N which gives us N = 2992 with 0.5 probability.
Again, I am confused.
From what you write I understand this :
But from your other comment, it looks like that last step and conclusion is not what you mean. Can you confirm that?
Or do you mean :
Or something else entirely?
In last line there should be
Ok. Thanks. So:
implies
?
If that is your reasoning, I do not see how you go from the former to the latter.
Is it a general fact that:
or does it work only for 0.5?
May be we better take equation (2) from the original Gott's work https://gwern.net/doc/existential-risk/1993-gott.pdf:
1 / 3 t < T < 3t with 50 per cent confidence,
in which T is the total number of buses and t is the number of buses above observed bus number T0. In our case, T is between 2061 and 6184 with 50 per cent probability.
It is a correct claim, and saying that the total number of buses is double of the observed bus number is an oversimplification of that claim which we use only to point in the direction of the full Gott's equation.
Oh, it looks exactly like the kind of reference that everyone here seems to be aware of and I am not. ^^ I will be reading that. Thanks a lot.
No, that is not the definition of a prior. There are priors which imply an expected number of buses, and priors that don't. If you select a prior that doesn't, you can still get a meaningful posterior distribution even if that posterior distribution doesn't have a real-valued mean.
In my view, a proper use here is to compare two hypothesis: there are 2000 buses and 20 000 buses. Finding that the actual number is 1546 is an update in the direction of smaller number of buses.
I think you right that 1546 has the biggest probability compared to other probabilities for any other exact number, that is something like 1:1546. But it doesn't means that it is likely, as it is still very small number.
In Doomsday argument we are interested in comparing not exact dates but periods, as in that case we get significant probabilities for each period and comparing them has meaning.
TL;DR: Several experiments show that I can extract useful information just by treating myself as a random sample, and thus a view that I can't use myself as a random sample is false. But it's still not clear whether this can be used to prove the Doomsday argument.
There are two views: one view is that I can use my random location to predict the total size of the set from which I am selected – and, moreover, it is applicable to predicting future Earth population and thus the Doomsday timing and other anthropic things like the Simulation argument.
And the second view is that there will always be a person at the beginning of any large ordered set of observers who will be surprised by their early location (in the case of DA). Thus, the fact of the surprise is non-informative. Or, as a variant, it should be ignored based on Updateless Decision Theory considerations.
Here I will not argue about the theoretical validity of these two views. Instead, I will perform a series of practical experiments to test the central claim.
Let's start from a simple experiment – please check the time of the day now and use it as a random sample to try to predict the typical duration of a day in hours. When I did it the first time, I looked at my clock and it was 15:14. It gives a 50 percent probability that the total number of hours in a day is 30, which is reasonably close to 24.
The history of anthropic thought – at least in one of its lines – started from a practical experiment: R. Gott claimed jokingly in 1975 that he could predict that the Berlin Wall would exist for around 100 percent more of its current age at the time of the prediction (it was 14 years old at the time of the joke). When it fell in 1989, Gott was surprised and wrote a theoretical underpinning of such prediction method. In some sense, Laplace's Sunrise Problem also is based on experimental observation that the Sun rises every day.
Gott continued to insist that his prediction method could be experimentally tested and used it to predict the durations of Broadway shows, and it worked.
However, Gott's method is not explicitly based on some observer sampling assumption, but on the observation of the duration of the existence of external things where the observer continues to exist after they disappear. Therefore, the application of it to the future duration of humanity's existence – the Doomsday Argument – is questionable.
Here I will perform several practical experiments. I will test the central claim: that some useful information can be extracted just from my random location, that is, that I can treat myself as a random sample. This idea is often attacked from different angles (perspective-based reasoning, linear progression argument, full conditioning requirement, and the idea that future observers are not determined yet and can't be sampled).
1. How many months are in the year?
Given my date of birth, can I estimate the total number of months in a year, as if I do not know it? My birth month is September, the 9th month. This means that with 50 percent probability, according to me-as-a-random-sample logic, the total number of months in a year is 18. This is close to the real number, 12.
2. Earth size
I will try another, more difficult one. I will measure the size of the Earth, taking as input my birth location and I will take from it only the surface distance from my location to the nearest point on the equator (that is, similar to latitude, but in km). For me, it is 6190km. I then assume that it is a random sample, and will ignore anything from spherical geometry and population distribution. In that case, I assume that I was born in a random place between the equator and pole. Therefore, the surface distance to the pole, according to random sampling of me, should also be 6190 km, and the total surface distance from pole to equator will be 12380 km. The real distance is 10000 km. So here again I get a good approximation of real data by treating myself as a random sample from all observers.
However, if we apply this to time, there is a problem: future observers do not exist yet. Leslie thought that this was the real problem of DA and spent a lot of time proving determinism. If determinism is correct, there is no difference between real and future observers, and random sampling works perfectly; DA will work. In other words, DA will work in a block-time universe. But what about MWI?
One trick to escape this problem is predicting the time of the existence of a typical observer and then applying it to myself as a typical observer.
3. Predicting typical human life expectancy based on my age
For example, if I take a random alien and learn that his age is 1500 sols, I can't directly predict that this alien will live 3000 sols, but I can say that the average life expectancy of such aliens is 3000 sols, and thus this alien also has this average life expectancy as he likely is an average alien. This works for me too. My current age is 50 (at the time of writing), and this predicts that average human life expectancy is around 100. Not a bad guess, given that real human life expectancy is 70-80 years.
Here I use a trick with median life expectancy which does not change the prediction. The same trick is used in the so-called universal doomsday argument.
But can it be applied to the real Doomsday Argument?
Ape in the coat suggested that we can't think about ourselves as random samples from human history, as observers appear consequently. Any civilization discovers the Doomsday argument in its 20th-century-analogue and is surprised. But it is wrong: I am not randomly located in the history of human civilization; I am located near the date it first discovered the Doomsday Argument. In other words, thinking about DA pinpoints the specific moment in history and kills random selection. This looks like a refutation of the DA at first glance.
However, I can also say that I am randomly selected from all observers in Earth history who will ever think about DA. These observers appeared in the 1970s and the number of them has been growing at least until 2010. This suggests that such observers will disappear in a few decades from now.