I think the issue is that we need some sort of insider model, otherwise we cannot extract any predictions about what we are supposed to be observing.
So you agree that the predictions your are making are not well founded if using the insider model is not justified in these circumstances?
There is also a statistical point justifying insiders arguing they are randomly selected (as if they had been selected by an outsider). If all the insiders in the class were to reason in this way, and predict—for each of their observations—that there will be about as many observations still to come in the class as there have been so far in the class then most of them will be right, most of the time. Statistically, it seems very reasonable to use a method which works for most of the people most of the time.
But there are patterns in the reasons why an insider would be wrong to pretend to be an outsider. Those patterns should be readily apparent to an outsider. Imagine if the population curve we were discussing were of Europe instead of the whole world. A non-European who randomly selects European experience moments might be justified in making the argument you present, but a European doesn’t seem able to justify pretending that their our experience moment is randomly selected (for reasons that are really obvious to the non-European). Or am I misunderstanding what statistics means by random selection?
Nick Bostrom gives some examples to illustrate the point. Suppose physicists come up with a universe model which is infinite or really-really large. The consequence of the model is that every observation which could be made actually is made by someone, somewhere. Now suppose that the background radiation in this model is uniform and has a temperature of 1 Kelvin; we would generally believe that the model is falsified because we observe a temperature of around 3K. However, we know that at least some observations in the model (albeit a tiny minority) wil be exactly like ours, so how is the model falsified? If we don’t make any assumptions about how our observations are sampled from the whole (in the “insider” model we are atypical; so what?) then we have no way of testing or falsifying such models.
I don’t understand the point of this example. Most importantly, the question assumes that the correct answer is already known. I agree it is quite confusing for every measurement to be wrong when the right answer is independently known. But that’s not anywhere close to the state of science, and we don’t expect scientific observation to ever be in the position of trying to confirm what we already known through non-scientific means.
Even granting the assumption that the true theory is known before observation, there’s an ambiguity about what is meant by “observations that the background is 3K.” Do you mean that every human observation in our light-cone shows 3K? If so, doesn’t Ockam’s Razor suggest that there is something unusual about local space, not that all our measurements (and theories about the background temp) are wrong? In a certain sense, we’ll never know that our observations are caused by strangeness in local space rather than being uniform everywhere, but the difficulty of understanding the universe beyond our light-cone is a fact about the physical universe, NOT a fact about statistics. And if it’s only one erroneous measurement, why are we worried that we can’t show it was in error?
No, background radiation is uniformly 1K, but it just happens that every observation we’ve ever made of it has suggested it’s 3K. The point is that in a really big 1K universe, there are some observers who are in that unfortunate position of thinking it’s 3K. Not very many of them, but how do we know we’re not part of that unfortunate minority?
Well it’s crashingly obvious really that if the universe did have 1K background, then we’d expect to be part of the overwhelming majority who observe 1K rather than the teeny-tiny minority who appear to observe 3K. However, that “crashingly obvious” conceals an observer selection hypothesis, which is Bostrom’s point.
Let’s back up for a moment. There are two propositions at issue here. Following is the first:
If an alien observer took a random sample of human experiences and came up with me and my dorm mates, the observer would be justified in certain belief about the shape of the curve of human population over time. Specifically, the observation would be evidence that a cataclysm occurred within our lifetimes.
I agree with this point. Second proposition:
The process that led me and my dorm mates to be placed together and able to share our experiences is sufficiently random that I am justified in reaching the same conclusion as the hypothetical alien observer. That is, by looking only at the shape of the population curve and my place in it, I can make predictions about the future shape of the curve.
This proposition seems mathematically unjustified. (The true of my assertion is what we are debating, right?). I don’t understand what the background radiation hypothetical does to support the mathematical position I’m rejecting. I agree that we would expect to be part of the 1K-measurement population rather than the 3K-measurement population. But the hypothesis is that the universe is so large that someone “wins” this unlikely lottery.
So it unexpectedly turns out to be us. We’ll NEVER know that we are the victims of a galactic improbability. And we’ll NEVER know the true theory of background radiation since we don’t have access to the data necessary to justify the theory. From our point of view, the background temperature really IS 3K. And we’re wrong. I’m not trained on the technical issues, so I don’t understand why this helps the proposition above that I labelled as mathematically unjustified.
PS Of course the assumption is “mathematically unjustified”. Every core assumption in every scientific theory is “mathematically unjustified”. Science is not mathematics.
When I say “mathematically unjustified,” I mean contrary to mathematical rules. Physics relies on induction, which is not mathematically justified in some sense (but Hume doesn’t suggest we should abandon the scientific method). But physics never says anything like “assume 1 + 5 = 7 when talking about quarks.”
Your argument looks at two items (1) the shape of the population curve, and (2) our place in the population curve. From that, you infer something about the shape of the curve in the future. I say that making inferences from those two items requires certain technical definitions of random selection be satisfied. And point (2) does not appear to satisfy those requirements. By contrast, the observations you describe in your parallel comment do appear to satisfy those technical requirements.
I’m not a trained statistician, so my understanding of the precise contours of the technical requirements could be wrong. But saying that relaxing the requirements usually doesn’t contaminate the results is insufficient because it seems to me that the reasons for the strict requirements are highly relevant to your argument, but not all statistical arguments. In short, there’s no way to infer the future shape of the population chart from the chart up to this date. Why should I believe that my experience of the chart is additional evidence, independent of the chart, that justifies additional inferences that the chart could not justify on its own?
Edit: Imagine you are trying to make inferences about the skills of professional baseball players, and you start by assuming that talent for playing baseball is normally distributed. This assumption is almost certainly false because professional baseball players are a selected off subset of all people with the capacity to play baseball. That is, we expect the shape of talent among players who are selected for their skill at playing baseball to resemble one tail of a normal curve, not the entire curve. Applying statistical tests that assume normal distribution will almost certainly lead to incorrect conclusions.
So, by “mathematically unjustified” you meant something like “mathematically inconsistent” in the same way that “1 + 5 = 7” is inconsistent. However, now I’m puzzled, since why is it “mathematically inconsistent” for an insider to model his observation as a random sample from some population?
Provided the sampling model follows the Kolmogorov probability axioms, it is mathematically consistent. And this is true even if it is a totally weird and implausible sampling model (like being a random sample from the population {me now, Joan of Arc at the stake, Obama’s left shoe, the number 27} … ).
In the world of the unlucky physicists, they are assuming that their data is randomly selected. If I understand the thought experiment, this assumption is correct. If this were a computer game, we’d say the physicists have been cursed by the random number generator to receive extremely unlikely results given the true state of the universe. But that doesn’t mean the sample isn’t still random—unlikely occurrences can happen randomly.
Likewise, the doomsday argument assumes that the sample of human experiences is randomly selected. Yet there is no reason to think this is so. You are using your experiences as the sample because it is the only one truly available to you. To me, this looks like convenience sampling, with all the limitations on drawing conclusions that this implies. And if your assumption that your sample is random is wrong, then the whole doomsday argument falls apart.
In short, cursed by the random number generator != nonrandom sample.
What I’m trying to understand is the difference between these two arguments:
Model A predicts that the vast majority of observations of the universe will conclude it has a background radiation with a temperature of 1K, whereas a tiny minority of observations will conclude it has a temperature of 3K. Model B predicts that the vast majority of observations of the universe will conclude a background radiation temperature of 3K. Our current observations conclude a temperature of 3K. This is evidence against model A and in favour of model B.
Model 1 predicts that the vast majority of observations of the universe will be in civilisations which have expanded away from their planet of origin and have made many trillion trillion person-years of observations so far; a tiny minority will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Model 2 predicts that the vast majority of observations will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Our current observations are in a civilisation which is still on its planet of origin, and has made less than 10 trillion person-years of observations so far. This is evidence in favour of Model 2.
Formally, these look identical, but it seems you accept the first argument yet reject the second. And the difference is… ?
In both cases, the inferences being drawn rely on the fact that the observation was randomly selected.
In the physics example, the physicist started with no observation, made a random observation, and made inferences from the random observation.
In the population example, we start with an observation (our own lives). You treat this observation as a random sample, but you have no reason to think that “random sample” is a real property of your observation. Certainly, you didn’t random select the observation. Instead, you are using your own experience essentially because it is the only one available.
But then why do you assume that the physicist made a “random observation”? The model A description just says that there are lots of observations, and only a tiny minority are such as to conclude 3K. If both model A and model B were of deterministic universes, so that there are strictly no “random” observations in either of them (because there are no random processes at all) then would you reverse your conclusion?
Is your basic objection to the application of probability theory when it concerns processes other than physically random processes?
If the physicists are not receiving random samples of the population of possible observations, then their inferences are also unjustified. And if random processes are impossible because the universe is deterministic . . . my head hurts, but I think raising that problem is changing the subject. I don’t really want to talk about whether counter-factuals (like scientists proposing a different theory than the one actually proposed) are a coherent concept in a deterministic universe.
Is your basic objection to the application of probability theory when it concerns processes other than physically random processes?
That could be, but I’m not familiar with the technical vocabulary you are using. What’s an example of a non-physical random process?
This discusses lots of different interpretations of “random”. The general sense seems to be that a random process is unpredictable in detail, but has some predictable properties such that the process can be modelled mathematically by a random variable (or sequence of random variables).
Here, the notion of “modelling by a random variable” means that if we take the actual outcome and apply statistical tests to check whether the outcome is drawn from the distribution defined by the random variable, then the actual outcome passes those tests. This doesn’t mean of course that it is in an objective sense a random process with that distribution, but it does mean that the model “fits”.
P.S. For the avoidance of doubt, you can assume that models A and B involve pseudo-random processes, and these obey the usual frequency statistics of true random processes.
OK, let me ask you a question. Suppose that physicists have produced these two models of the universe.
Model A has a uniform background radiation temperature of 1K.
Model B has a uniform background radiation temperature of 3K.
Both models are extremely large (infinite, if you prefer), so both models will contain some observers whose observations suggest a temperature of 3K, as well as observers whose observations suggest a
temperature of 1K.
Our own observations suggest a temperature of 3K.
In your view, does that observation give us any reason at all to favour Model B over Model A as a description of the universe? If so, why? If not, how can we do science when some scientific models
imply a very large (or infinite) universe?
So you agree that the predictions your are making are not well founded if using the insider model is not justified in these circumstances?
But there are patterns in the reasons why an insider would be wrong to pretend to be an outsider. Those patterns should be readily apparent to an outsider. Imagine if the population curve we were discussing were of Europe instead of the whole world. A non-European who randomly selects European experience moments might be justified in making the argument you present, but a European doesn’t seem able to justify pretending that their our experience moment is randomly selected (for reasons that are really obvious to the non-European). Or am I misunderstanding what statistics means by random selection?
I don’t understand the point of this example. Most importantly, the question assumes that the correct answer is already known. I agree it is quite confusing for every measurement to be wrong when the right answer is independently known. But that’s not anywhere close to the state of science, and we don’t expect scientific observation to ever be in the position of trying to confirm what we already known through non-scientific means.
Even granting the assumption that the true theory is known before observation, there’s an ambiguity about what is meant by “observations that the background is 3K.” Do you mean that every human observation in our light-cone shows 3K? If so, doesn’t Ockam’s Razor suggest that there is something unusual about local space, not that all our measurements (and theories about the background temp) are wrong? In a certain sense, we’ll never know that our observations are caused by strangeness in local space rather than being uniform everywhere, but the difficulty of understanding the universe beyond our light-cone is a fact about the physical universe, NOT a fact about statistics. And if it’s only one erroneous measurement, why are we worried that we can’t show it was in error?
No, background radiation is uniformly 1K, but it just happens that every observation we’ve ever made of it has suggested it’s 3K. The point is that in a really big 1K universe, there are some observers who are in that unfortunate position of thinking it’s 3K. Not very many of them, but how do we know we’re not part of that unfortunate minority?
Well it’s crashingly obvious really that if the universe did have 1K background, then we’d expect to be part of the overwhelming majority who observe 1K rather than the teeny-tiny minority who appear to observe 3K. However, that “crashingly obvious” conceals an observer selection hypothesis, which is Bostrom’s point.
Let’s back up for a moment. There are two propositions at issue here. Following is the first:
I agree with this point. Second proposition:
This proposition seems mathematically unjustified. (The true of my assertion is what we are debating, right?). I don’t understand what the background radiation hypothetical does to support the mathematical position I’m rejecting. I agree that we would expect to be part of the 1K-measurement population rather than the 3K-measurement population. But the hypothesis is that the universe is so large that someone “wins” this unlikely lottery.
So it unexpectedly turns out to be us. We’ll NEVER know that we are the victims of a galactic improbability. And we’ll NEVER know the true theory of background radiation since we don’t have access to the data necessary to justify the theory. From our point of view, the background temperature really IS 3K. And we’re wrong. I’m not trained on the technical issues, so I don’t understand why this helps the proposition above that I labelled as mathematically unjustified.
PS Of course the assumption is “mathematically unjustified”. Every core assumption in every scientific theory is “mathematically unjustified”. Science is not mathematics.
When I say “mathematically unjustified,” I mean contrary to mathematical rules. Physics relies on induction, which is not mathematically justified in some sense (but Hume doesn’t suggest we should abandon the scientific method). But physics never says anything like “assume 1 + 5 = 7 when talking about quarks.”
Your argument looks at two items (1) the shape of the population curve, and (2) our place in the population curve. From that, you infer something about the shape of the curve in the future. I say that making inferences from those two items requires certain technical definitions of random selection be satisfied. And point (2) does not appear to satisfy those requirements. By contrast, the observations you describe in your parallel comment do appear to satisfy those technical requirements.
I’m not a trained statistician, so my understanding of the precise contours of the technical requirements could be wrong. But saying that relaxing the requirements usually doesn’t contaminate the results is insufficient because it seems to me that the reasons for the strict requirements are highly relevant to your argument, but not all statistical arguments. In short, there’s no way to infer the future shape of the population chart from the chart up to this date. Why should I believe that my experience of the chart is additional evidence, independent of the chart, that justifies additional inferences that the chart could not justify on its own?
Edit: Imagine you are trying to make inferences about the skills of professional baseball players, and you start by assuming that talent for playing baseball is normally distributed. This assumption is almost certainly false because professional baseball players are a selected off subset of all people with the capacity to play baseball. That is, we expect the shape of talent among players who are selected for their skill at playing baseball to resemble one tail of a normal curve, not the entire curve. Applying statistical tests that assume normal distribution will almost certainly lead to incorrect conclusions.
So, by “mathematically unjustified” you meant something like “mathematically inconsistent” in the same way that “1 + 5 = 7” is inconsistent. However, now I’m puzzled, since why is it “mathematically inconsistent” for an insider to model his observation as a random sample from some population?
Provided the sampling model follows the Kolmogorov probability axioms, it is mathematically consistent. And this is true even if it is a totally weird and implausible sampling model (like being a random sample from the population {me now, Joan of Arc at the stake, Obama’s left shoe, the number 27} … ).
In the world of the unlucky physicists, they are assuming that their data is randomly selected. If I understand the thought experiment, this assumption is correct. If this were a computer game, we’d say the physicists have been cursed by the random number generator to receive extremely unlikely results given the true state of the universe. But that doesn’t mean the sample isn’t still random—unlikely occurrences can happen randomly.
Likewise, the doomsday argument assumes that the sample of human experiences is randomly selected. Yet there is no reason to think this is so. You are using your experiences as the sample because it is the only one truly available to you. To me, this looks like convenience sampling, with all the limitations on drawing conclusions that this implies. And if your assumption that your sample is random is wrong, then the whole doomsday argument falls apart.
In short, cursed by the random number generator != nonrandom sample.
What I’m trying to understand is the difference between these two arguments:
Model A predicts that the vast majority of observations of the universe will conclude it has a background radiation with a temperature of 1K, whereas a tiny minority of observations will conclude it has a temperature of 3K. Model B predicts that the vast majority of observations of the universe will conclude a background radiation temperature of 3K. Our current observations conclude a temperature of 3K. This is evidence against model A and in favour of model B.
Model 1 predicts that the vast majority of observations of the universe will be in civilisations which have expanded away from their planet of origin and have made many trillion trillion person-years of observations so far; a tiny minority will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Model 2 predicts that the vast majority of observations will be in civilisations which are still on their planet of origin and have made less than 10 trillion person-years of observations so far. Our current observations are in a civilisation which is still on its planet of origin, and has made less than 10 trillion person-years of observations so far. This is evidence in favour of Model 2.
Formally, these look identical, but it seems you accept the first argument yet reject the second. And the difference is… ?
In both cases, the inferences being drawn rely on the fact that the observation was randomly selected.
In the physics example, the physicist started with no observation, made a random observation, and made inferences from the random observation.
In the population example, we start with an observation (our own lives). You treat this observation as a random sample, but you have no reason to think that “random sample” is a real property of your observation. Certainly, you didn’t random select the observation. Instead, you are using your own experience essentially because it is the only one available.
But then why do you assume that the physicist made a “random observation”? The model A description just says that there are lots of observations, and only a tiny minority are such as to conclude 3K. If both model A and model B were of deterministic universes, so that there are strictly no “random” observations in either of them (because there are no random processes at all) then would you reverse your conclusion?
Is your basic objection to the application of probability theory when it concerns processes other than physically random processes?
If the physicists are not receiving random samples of the population of possible observations, then their inferences are also unjustified. And if random processes are impossible because the universe is deterministic . . . my head hurts, but I think raising that problem is changing the subject. I don’t really want to talk about whether counter-factuals (like scientists proposing a different theory than the one actually proposed) are a coherent concept in a deterministic universe.
That could be, but I’m not familiar with the technical vocabulary you are using. What’s an example of a non-physical random process?
Maybe take a look at the Wikipedia entry http://en.wikipedia.org/wiki/Randomness
This discusses lots of different interpretations of “random”. The general sense seems to be that a random process is unpredictable in detail, but has some predictable properties such that the process can be modelled mathematically by a random variable (or sequence of random variables).
Here, the notion of “modelling by a random variable” means that if we take the actual outcome and apply statistical tests to check whether the outcome is drawn from the distribution defined by the random variable, then the actual outcome passes those tests. This doesn’t mean of course that it is in an objective sense a random process with that distribution, but it does mean that the model “fits”.
Hope that helps...
P.S. For the avoidance of doubt, you can assume that models A and B involve pseudo-random processes, and these obey the usual frequency statistics of true random processes.
OK, let me ask you a question. Suppose that physicists have produced these two models of the universe.
Model A has a uniform background radiation temperature of 1K. Model B has a uniform background radiation temperature of 3K.
Both models are extremely large (infinite, if you prefer), so both models will contain some observers whose observations suggest a temperature of 3K, as well as observers whose observations suggest a temperature of 1K.
Our own observations suggest a temperature of 3K.
In your view, does that observation give us any reason at all to favour Model B over Model A as a description of the universe? If so, why? If not, how can we do science when some scientific models imply a very large (or infinite) universe?