[This is the seventh piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics. You should start here if you are just seeing this for the first time.
SSA is not "temporally consistent" - learning your brith rank causes an indexical update which can, for instance, change the probabilities of a fair coin that has not yet been tossed
In order to perform an indexical update - the kind which causes distortions in the SSA and removes presumptuousness from the SIA - you need to learn your rank relative to some absolute property defined for all observers.
Simply using a metric like "closest to the sun" would not help because all observers using such a procedure (picking the nearest star) would always find themselves the closest
An actual update would require something like "ranked distance from largest star in the universe" or "life-bearing universe in multiverse with Nth largest gravitational constant," information we could not possibly hope to acquire
One well-established failure mode of the SSA is that it is not "temporally consistent." I believe Nick Bostrom first pointed this out and Joe Carlsmith has an excellent write up on it here.
The simplest example of inconsistency in the SSA is the Late-Flip version of the Sleeping Beauty Problem where the coin is flipped only after Sleeping Beauty is woken on Monday. In this scenario, P(Heads | Monday) seems as if it must be 0.5, since the coin hasn’t even been flipped yet. The SSA, however, assigns 2/3 credence to P(Heads | Monday).
In that sense, what we mean when we say “temporal consistency” is that for some event X with fixed and known probability, like the flip of a known-fair coin, learning our identity should not alter our credence in X if our identity was set before X happened. But there are two catches to this:
This only applies if the future event has fixed and known probability. If you don’t know whether or not the coin is fair, but then learn that it is 75% biased towards heads, then this will change your credence in the coin going heads even though the event has not happened yet.
When learning something about your identity, there is nothing inherently paradoxical about indexical information changing the credences of past events.
Point 1, I hope, is non-controversial. Point 2 is a bit thornier, and I wouldn’t blame you for arguing we don’t want indexical updates to change the past either. Indeed, in order for temporal consistency to hold in all cases, the only credences it would actually be permissible to distort (as in, alter beyond a normal update on the universe prior) are those on events from before the very first observer was created.
And it's also true that, upon learning your full identity, the SIA induces no distortions on either the past or the future, so point 2 is unnecessary in that case.
The situation for the PPA is not much worse, though. The PPA still does not distort any events on the same timeline as the universe you are in; the only thing it actually distorts are the credences you give to the parameters of your model, since those are the only things that behave SSAishly. The idea behind parameters however is that they represent fixed laws of math and physics, so - if they can be said to be temporal at all - they are unambiguously “sampled” before the universe or any observers existed. If you buy even the weakest form of point 2, then there is nothing inherently paradoxical about this.
In fact, the way we're using parameters in the PPA (as the variable for expected count of intelligent species) means that it is virtually impossible for us to make an update that causes an anthropic distortion. It would not be enough simply to learn the size of the universe, or that other forms of life existed on other planets. You would have to specifically learn what we're going to call a "nonindexical label" related to your specific place in the universe, which, as we're about to see, can only be done if you already have significant global knowledge of the entire universe's state.
Nonindexical Labeling
Let’s take a step back and talk about what it takes to even trigger the kind of indexical updates that make this sort of discussion necessary. As we saw in the piece on diversity, there is a lot of information about ourselves or the universe for which our credences are unperturbed by the use of indexicals.
Similarly, not every piece of information we learn about ourself will trigger a significant indexical update. To take a simple example, suppose we are in a Sleeping Beauty-like scenario where we are woken up either one day, two, three or four days in a row, starting on Monday. Learning that it was, say, Tuesday, would indeed cause a large indexical update, but merely learning that there had been one other wake-up would only cause an update in the universe prior.
Learning that the day is Tuesday under the SSA. The update is indexical, in that some of the rows are narrowed.
Learning that there are at least 2 wake-ups under the SSA. The update is only in the universe prior, in that some rows are removed but the widths do not change.
We can see that the first update is indexical because some of the rows are narrowed, whereas the second update is only on the prior because rows are removed but not narrowed. We can also have much smaller indexical updates by only learning, for instance, that there was a wake-up the day before, which instead of narrowing each row to a single observer, merely removes one from each row:
The prior is perturbed a little because we know we are not the first person, but the effect is smaller than a normal SSA update and shrinks as the number of observers increases.
In order to get strong indexical effects it’s not enough to place yourself relative to others; you need to learn your absolute position relative to other observers. This is easy enough to do (in an approximate sense) in the temporal case where your species/experiment has a start time and a well-defined notion of order, but in many cases learning absolute information is almost impossible. In particular, when we’re wondering about things like “how many other intelligent species exist in the universe?” it will be almost impossible to gather information which allows you to do meaningful updates.
Take another look at the 4 wake-up problem from a moment ago, but now imagine that each of these represents a possible intelligent species somewhere in the universe you might have been born into. If you learned you were only species #2, for instance, that would induce the same updates we had upon learning it was Tuesday in the last scenario. Everything would look the same.The question is how we would actually learn we were species #2.
The answer is that we would need some sort of “species rank.” One way to do this would be to do the same thing we did with birth rank, and just order temporally by species emergence in the universe. But how would we ever learn that information? In order to have any confidence that we were actually, say, the 2nd species in the universe, we would first have to map out the entire universe, identify every species in it, and then study their history until we were sure they emerged at a certain time. Only then would we get something like an indexical update.
Except that would only apply for our universe. What if we were in a multiverse? Would the concept of temporal ordering even exist between multiverses? Presumably not.
But let’s ignore that for a moment and keep the focus on our own universe. We’ve established that it’s difficult to use temporal ordering to learn species rank, but one might wonder if there are other markers we could use to rank species.
In fact, virtually any arbitrary metric could be used to label species. For instance, we might decide to use distance from the sun, in which case we can say that humans, being the only intelligence on Earth[1] get the label of “first.”
But this won’t give us an indexical update. If you think it would, you’re making a mistake known as the "Texas sharpshooter fallacy." The canonical metaphor for the fallacy is a man shooting randomly at his barn, hitting a spot completely at random, and then drawing a target right over where the bullet hit. Then everyone gathers round and thinks to themselves "Wow! What were the chances he would hit that exact target 3.231 feet from the left and 8.654 feet off the ground?"
The lesson for us is that when you condition on a certain event happening (for instance, the bullet hitting some extremely specific spot on the barn, or being closest to the sun), you also need to condition on the process that selected that event. It's not "what is the probability that you would hit a bullseye at coordinate (3.231, 8.654)?" it's "what is the probability that you would hit a bullseye drawn right where the bullet hit?" The answer to the second question is obviously "100%."
In the anthropic case, if every species were to assign a ranking by looking around, finding the closest star/large celestial landmark, and choosing to rank based on that, then they would all rank themselves number 1. Thus, the probability of being rank 1 using this procedure is 100% in all universes, and no update is possible.
Instead, what you would need to do is rank yourself according to some kind of universal anchor that would be identified identically by all species in the universe/multiverse, wherever they are. In a single universe, this could be time since the Big Bang, but that would mostly only effect your prior on the typical number of species within a single universe;[2] if you believed there might be a multiverse you wouldn’t be able to do any serious indexical updating because you wouldn’t know how many other life-bearing universes there were.
And, as we mentioned, indexing those would be basically impossible. Perhaps you could rank multiverses by “largest gravitational constant” or something similar, but updating would still require you to know how much life existed in each and every multiverse. The point is, without significant global information about the entirety of existence, you’re not going to be able to perform indexical updates about your species. If you’re using the SSA, you’ll have trouble moving beyond your prior for species-level information[3] and if you’re using the SIA, you’ll be locked into a “presumptuous” view of the universe where you’re unduly biased towards a multitude of species.
Now, the fact that you need global information to perform indexical species updates is a practical objection, not a theoretical one. You can’t, as far as I can tell, use this fact as a first principle from which to design an anthropic assumption. But it’s still something to keep in mind when we explore the PPA, since it is only these sorts of fully-anchored species-level updates which can actually cause indexical SSAish updates to your model’s parameters. For all intents and purposes, you will never be able to learn the kind of information that would cause the PPA to perform an unusual update.
In the next piece, we’ll discuss the concept of “life density.” In that framing, learning you were the first species in the universe you’re in (or the observable part of it) would just cause you to update on life being low-density across the multiverse, but under a reasonable prior it would not be sufficient to tell you anything about the total amount of life.
Another way to think about it: the SSA and SIA are functions that take a universe prior distribution and transforms it into an indexical distribution. In order for this process to be cogent, it needs to produce the same indexical distribution for all observers. That means the universe prior has to be the only input to the function. If you need to use indexical information to produce indexical labels, you can't deterministically produce a single output distribution.
What about temporal species information? To be nonexchangeable, you must learn not when your species exists int he universe, but your species' rank. In other words, you must learn that you are, e.g. the fourth intelligent species to emerge in the entire universe.
The universe, at least classically, is a Markov Chain.
To be clear: our proximity to the sun is still information that allows us to update our knowledge of the universe. But because that fact is exchangeable with respect other our universe prior, learning it alone cannot cause an indexical update about the number of other species in the universe, only an update accoridng on our prior.
[This is the seventh piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics. You should start here if you are just seeing this for the first time.
The previous piece is "Indexical Models Don't Have Diversity Biases."
TLDR:
The next piece is "The Fermi Paradox, Fine-Tuning, and the SSA."]
One well-established failure mode of the SSA is that it is not "temporally consistent." I believe Nick Bostrom first pointed this out and Joe Carlsmith has an excellent write up on it here.
The simplest example of inconsistency in the SSA is the Late-Flip version of the Sleeping Beauty Problem where the coin is flipped only after Sleeping Beauty is woken on Monday. In this scenario, P(Heads | Monday) seems as if it must be 0.5, since the coin hasn’t even been flipped yet. The SSA, however, assigns 2/3 credence to P(Heads | Monday).
In that sense, what we mean when we say “temporal consistency” is that for some event X with fixed and known probability, like the flip of a known-fair coin, learning our identity should not alter our credence in X if our identity was set before X happened. But there are two catches to this:
Point 1, I hope, is non-controversial. Point 2 is a bit thornier, and I wouldn’t blame you for arguing we don’t want indexical updates to change the past either. Indeed, in order for temporal consistency to hold in all cases, the only credences it would actually be permissible to distort (as in, alter beyond a normal update on the universe prior) are those on events from before the very first observer was created.
And it's also true that, upon learning your full identity, the SIA induces no distortions on either the past or the future, so point 2 is unnecessary in that case.
The situation for the PPA is not much worse, though. The PPA still does not distort any events on the same timeline as the universe you are in; the only thing it actually distorts are the credences you give to the parameters of your model, since those are the only things that behave SSAishly. The idea behind parameters however is that they represent fixed laws of math and physics, so - if they can be said to be temporal at all - they are unambiguously “sampled” before the universe or any observers existed. If you buy even the weakest form of point 2, then there is nothing inherently paradoxical about this.
In fact, the way we're using parameters in the PPA (as the variable for expected count of intelligent species) means that it is virtually impossible for us to make an update that causes an anthropic distortion. It would not be enough simply to learn the size of the universe, or that other forms of life existed on other planets. You would have to specifically learn what we're going to call a "nonindexical label" related to your specific place in the universe, which, as we're about to see, can only be done if you already have significant global knowledge of the entire universe's state.
Nonindexical Labeling
Let’s take a step back and talk about what it takes to even trigger the kind of indexical updates that make this sort of discussion necessary. As we saw in the piece on diversity, there is a lot of information about ourselves or the universe for which our credences are unperturbed by the use of indexicals.
Similarly, not every piece of information we learn about ourself will trigger a significant indexical update. To take a simple example, suppose we are in a Sleeping Beauty-like scenario where we are woken up either one day, two, three or four days in a row, starting on Monday. Learning that it was, say, Tuesday, would indeed cause a large indexical update, but merely learning that there had been one other wake-up would only cause an update in the universe prior.
Learning that the day is Tuesday under the SSA. The update is indexical, in that some of the rows are narrowed.
Learning that there are at least 2 wake-ups under the SSA. The update is only in the universe prior, in that some rows are removed but the widths do not change.
We can see that the first update is indexical because some of the rows are narrowed, whereas the second update is only on the prior because rows are removed but not narrowed. We can also have much smaller indexical updates by only learning, for instance, that there was a wake-up the day before, which instead of narrowing each row to a single observer, merely removes one from each row:
The prior is perturbed a little because we know we are not the first person, but the effect is smaller than a normal SSA update and shrinks as the number of observers increases.
In order to get strong indexical effects it’s not enough to place yourself relative to others; you need to learn your absolute position relative to other observers. This is easy enough to do (in an approximate sense) in the temporal case where your species/experiment has a start time and a well-defined notion of order, but in many cases learning absolute information is almost impossible. In particular, when we’re wondering about things like “how many other intelligent species exist in the universe?” it will be almost impossible to gather information which allows you to do meaningful updates.
Take another look at the 4 wake-up problem from a moment ago, but now imagine that each of these represents a possible intelligent species somewhere in the universe you might have been born into. If you learned you were only species #2, for instance, that would induce the same updates we had upon learning it was Tuesday in the last scenario. Everything would look the same.The question is how we would actually learn we were species #2.
The answer is that we would need some sort of “species rank.” One way to do this would be to do the same thing we did with birth rank, and just order temporally by species emergence in the universe. But how would we ever learn that information? In order to have any confidence that we were actually, say, the 2nd species in the universe, we would first have to map out the entire universe, identify every species in it, and then study their history until we were sure they emerged at a certain time. Only then would we get something like an indexical update.
Except that would only apply for our universe. What if we were in a multiverse? Would the concept of temporal ordering even exist between multiverses? Presumably not.
But let’s ignore that for a moment and keep the focus on our own universe. We’ve established that it’s difficult to use temporal ordering to learn species rank, but one might wonder if there are other markers we could use to rank species.
In fact, virtually any arbitrary metric could be used to label species. For instance, we might decide to use distance from the sun, in which case we can say that humans, being the only intelligence on Earth[1] get the label of “first.”
But this won’t give us an indexical update. If you think it would, you’re making a mistake known as the "Texas sharpshooter fallacy." The canonical metaphor for the fallacy is a man shooting randomly at his barn, hitting a spot completely at random, and then drawing a target right over where the bullet hit. Then everyone gathers round and thinks to themselves "Wow! What were the chances he would hit that exact target 3.231 feet from the left and 8.654 feet off the ground?"
The lesson for us is that when you condition on a certain event happening (for instance, the bullet hitting some extremely specific spot on the barn, or being closest to the sun), you also need to condition on the process that selected that event. It's not "what is the probability that you would hit a bullseye at coordinate (3.231, 8.654)?" it's "what is the probability that you would hit a bullseye drawn right where the bullet hit?" The answer to the second question is obviously "100%."
In the anthropic case, if every species were to assign a ranking by looking around, finding the closest star/large celestial landmark, and choosing to rank based on that, then they would all rank themselves number 1. Thus, the probability of being rank 1 using this procedure is 100% in all universes, and no update is possible.
Instead, what you would need to do is rank yourself according to some kind of universal anchor that would be identified identically by all species in the universe/multiverse, wherever they are. In a single universe, this could be time since the Big Bang, but that would mostly only effect your prior on the typical number of species within a single universe;[2] if you believed there might be a multiverse you wouldn’t be able to do any serious indexical updating because you wouldn’t know how many other life-bearing universes there were.
And, as we mentioned, indexing those would be basically impossible. Perhaps you could rank multiverses by “largest gravitational constant” or something similar, but updating would still require you to know how much life existed in each and every multiverse. The point is, without significant global information about the entirety of existence, you’re not going to be able to perform indexical updates about your species. If you’re using the SSA, you’ll have trouble moving beyond your prior for species-level information[3] and if you’re using the SIA, you’ll be locked into a “presumptuous” view of the universe where you’re unduly biased towards a multitude of species.
Now, the fact that you need global information to perform indexical species updates is a practical objection, not a theoretical one. You can’t, as far as I can tell, use this fact as a first principle from which to design an anthropic assumption. But it’s still something to keep in mind when we explore the PPA, since it is only these sorts of fully-anchored species-level updates which can actually cause indexical SSAish updates to your model’s parameters. For all intents and purposes, you will never be able to learn the kind of information that would cause the PPA to perform an unusual update.
The next piece is "The Fermi Paradox, Fine-Tuning, and the SSA"
Well, until recently anyway
In the next piece, we’ll discuss the concept of “life density.” In that framing, learning you were the first species in the universe you’re in (or the observable part of it) would just cause you to update on life being low-density across the multiverse, but under a reasonable prior it would not be sufficient to tell you anything about the total amount of life.
This is only asymptotically true for large universes, see the next piece.
Another way to think about it: the SSA and SIA are functions that take a universe prior distribution and transforms it into an indexical distribution. In order for this process to be cogent, it needs to produce the same indexical distribution for all observers. That means the universe prior has to be the only input to the function. If you need to use indexical information to produce indexical labels, you can't deterministically produce a single output distribution.
Our universe's location in the entire multiverse, if it existed, would still be exchangeable.
What about temporal species information? To be nonexchangeable, you must learn not when your species exists int he universe, but your species' rank. In other words, you must learn that you are, e.g. the fourth intelligent species to emerge in the entire universe.
The universe, at least classically, is a Markov Chain.
To be clear: our proximity to the sun is still information that allows us to update our knowledge of the universe. But because that fact is exchangeable with respect other our universe prior, learning it alone cannot cause an indexical update about the number of other species in the universe, only an update accoridng on our prior.