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[This is a crosspost for the same piece on LessWrong. Links to other pieces are on LessWrong.]
[This is the first piece in a series on the Parametric Parameter Assumption (PPA), a new model of anthropics. This particular piece is self-contained and for most people will contain all they need to know.
TLDR:
This is the first piece in a series introducing the Parametric Process Assumption (PPA), a new model of anthropics which allows us to resolve very nearly all anthropic paradoxes I am aware of. With the right prior, the PPA rejects both the Doomsday Argument and the Presumptuous Philosopher and can operate seamlessly over both finite and infinite universes.
The motivating idea behind the PPA is that there are really two different ways to interpret a statement like "the probability of X":
The first interpretation makes more sense for some types of questions, while the second approach makes more sense for others. But it turns out that if you take the standard framing for anthropic problems - where a universe is first produced and then you are assigned an observer identity within it - you will get different answers depending on whether you model the universe in the first way or the second. In fact, a fully Frequentist, infinitely repeating universe yields the SIA, while a fully Bayesian one yields the SSA.
The PPA, however, says that the solution is to compose both interpretations. We imagine the entire universe as a random process progressing according to the Frequentist definition, but the laws of physics - the things which actually control the dynamics of the process, like the weak nuclear force or the strength of gravity - are treated in the Bayesian sense. The Bayesian portion we call the "parameters," and we refer to the rest as the "process."
In practice, since anthropics is mostly focused on the number of observers, we actually get to absorb almost all the important aspects of the laws of physics into a single parameter which we'll call the "propensity for life" - for any given laws of physics, propensity for life is the expected number of times we expect intelligent life to emerge. Anything else, such as how long life lasts once it's here, we treat as random, since that is governed less by the highest-level laws of physics and more by the local dynamics of whatever patch of spacetime that life happens to exist in.
The end result a model that is neither "halfer" nor "thirder."[1] Rather, the PPA effectively composes the SIAish process portion inside the SSAish parameter portion, inheriting the positives of both approaches without making any of the sacrifices of other competing models, including the various "double halfer" methods existing in the literature. [2]
That said, the PPA is not without weaknesses. The biggest of these is that, by virtue of defining certain variables as parameters, the PPA gives us an extra degree of freedom. Having this extra decision point opens the PPA up to accusations of overfitting, and this makes it extra important to properly motivate the priors we use. I do not know if all decisions this series makes in that regard will be unimpeachable, but if nothing else the PPA's complete lack of paradoxes should serve as an existence proof of a better way forward for anthropics.
This introductory piece will be self-contained, and should give you a good idea of how the PPA works, albeit with some details left out. The rest of this series will flesh out and build on these ideas, while also serving as an exploration of anthropics as a whole. Much of what I took for granted when I first started reading about this topic turned out to be wrong or exaggerated, and I suspect I am not the only one who made these mistakes[3].
The best way to motivate the PPA is, naturally, with an appeal to the Sleeping Beauty problem (SBP). Recall that in the SBP, Sleeping Beauty is put to sleep by scientists on a Sunday, after which a coin is flipped. On heads, she is woken on Monday, and on tails she is woken on Monday and Tuesday, with an amnesia drug administered in between so that she has no knowledge of which day it is. The SBP asks “upon waking, what should be the Beauty’s credence that the coin went heads?”
One answer to this question - the “thirder” answer associated with the SIA - is that we ought to give 1/3 credence to the coin going heads. There are a few arguments for this stance, but the most relevant one for our purposes is the “argument from repetition.” If we were to imagine the Sleeping Beauty experiment being repeated some arbitrarily large number of times, then we would find that 1/3 of Sleeping Beauty’s wake-ups were on heads. Since the beauty is “assigned” one of these wake-ups at random, upon waking she should give 1/3 credence to it being a “heads” day.
We’ll discuss the validity of this logic in the context of wake-ups some other time. For now, taking it at face value, this argument seems to be a straightforward proof of the SIA[4].
The impetus behind the PPA (at least this particular version of it) is that, as a matter of fact, generalizing the argument from repetition does not actually yield the exact SIA. It gets results consistent with the SIA for the types of problems that anthropics usually likes to consider, but if you apply it universally that won't always be the case.
To see why, consider a modification of the Sleeping Beauty problem where instead of flipping a coin, the scientists compute the googolth (10^100) digit of pi. If the googolth digit is even, then Sleeping Beauty is woken up once. If it is odd, she is woken twice. Sleeping Beauty, we can assume, neither knows the googolth digit of pi nor knows how to compute it.
On its face, this is an extremely similar problem to the vanilla SBP. Because Sleeping Beauty has no particular reason to assume the digit of pi is more likely to be even than odd, she could reasonably assign 50% credence to either possibility before the experiment starts. However, if we were to actually apply the argument from repetition to this experiment, we would find that there is no possible way to derive a credence of 1/3.
The reason is that the googolth digit of pi does not change between repetitions. It’s not even random in the conventional sense of the word, since you cannot simply “re-roll” pi to get a different sequence of digits. The digits of pi are fixed, unalterable truths about our world. “50% chance of even,” is a statement about belief, not the dynamics of the world.
With that in mind, there are two things that can happen:
We are not being placed randomly in a single sequence, but rather being placed with 50% probability into one of two sequences. This means that the possibility of waking on Tuesday is nested within the possibility of an odd digit - one can only ever compute a 25% chance. Thus, in this scenario, we are forced by the argument from repetition to accept the SSA.
The PPA is what you get if you take repetition to its logical conclusion. It operates by essentially decomposing the universe into two layers. The lower level, the “process” (also known as the “likelihood” in standard statistical parlance), is just a progression of random events happening one after the other, like the repeated flips of Sleeping Beauty’s coin, and we use the SIA to place ourselves within it. The outer layer, the “parameters,” represents facts about the universe which are too fixed to be rightly called random, but which we nevertheless do not know; we handle these according to the SSA.
One unfortunate side effect of using the PPA is that it becomes much harder to hold discussion through the usual thought experiments like the Sleeping Beauty Problem. It suddenly becomes necessary to specify which parts of the experiments represent parameters and which parts represent processes, and even once you have made these specifications (by, say, replacing a coin flip with a digit of pi), the resulting scenario rarely feels analogous to whatever question you were originally trying to ask.
In this section, we’re going to try to build a better model. Our focus will be on two of the biggest paradoxes in anthropics:
Throughout this series, we’ll spill a lot of virtual ink on the many subtleties of these two paradoxes, but for now the most important thing to realize is that these two statements are not, in fact, opposites. The Doomsday Argument concerns itself with the amount of time your specific species has left until extinction, and triggers only once you’ve learned information about when you were born relative to the “first” human. The Presumptuous Philosopher effect concerns itself with the amount of life in the entirety of existence, and outside of certain toy problems would not typically disappear just because you learned something about your particular species. Indeed, it is possible for a single assumption of anthropics to produce both effects at once.[6]
The terminology we’re going to use is that the Doomsday Argument pertains to the longevity of your species, whereas the Presumptuous Philosopher pertains to the universe’s propensity for life, or more often just “propensity.” In general, the SSA does a better job for “big universe-level questions” like propensity, but fails at smaller scales where longevity becomes relevant, and the SIA does the reverse.
Our goal will be to build a single universe model that incorporates both of these concepts, which we're going to call the Basic Model of anthropics. The idea is that we will imagine the universe as a gigantic random process that, through randomness, sometimes produces intelligent species. Like the digits of pi, the approximate size of the universe and its probability of randomly producing life is fixed but unknown, while most other facts - including how long an instance of a species lasts once produced - are genuinely random.
It's easiest to illustrate this model through a simplified version, which we'll call the "grid universe." Here, we imagine our universe as a big grid of some unknown size. As in the Basic Model, the size of the grid is not random but, again like the digits of pi, somehow fixed but unknown.
What is truly random, however, is the pattern that will fill this grid. Starting at the top, going down row by row, a random process fills the grid with colored pixels:
The purpose of these pixels is to represent the randomness in our universe, from galaxies to planets to the individual reactions on those surfaces. It's just a bunch of stuff happening, and though it is random is not uniformly so - the colors of the pixels (much like the randomness in our universe) depend on each other in a myriad of complex ways. The one constraint is that - just as time in our universe progresses in only one direction - the colors on each row cannot depend on the colors of any rows below them. The grid is filled top to bottom and the future only ever depends on the past.[7]
At some point, through the progression of this randomness, blue pixels might emerge. The exact likelihood of a blue pixel emerging doesn't particularly matter, as long as it is relatively low, and we can assume it is known or unknown for our current purposes.
Let's also say that these blue pixels, unlike all the other pixels, are “self-aware,” in the sense that they get to look at some of the pixels above and to the left and to the right, anywhere but into the future, and ask themselves hard philosophical questions like "why am I here?" and “should I be surprised at what I see?”
Blue pixels also propagate - if a pixel is blue, there is some fixed chance the pixel directly below it will also be blue. The blue pixels only know that whatever the chance of another blue pixel, it is determined primarily by randomness within the grid world (in other words, propagation is more like a coin toss than a digit of pi). For the sake of argument we can suppose that the blue pixels have looked around and decided that there is an approximately 10% chance that a blue pixel won't be followed by another. The relevant part is that whatever prior the pixel has, it is attributing the final outcome to random variation, not some inherent property of the universe.
We can also assume the grid is large enough - and the pixel’s visual range limited enough - that there is no real chance of the pixel seeing an edge, nor of the chain terminating because it hits the bottom. Other than that, though, just as we can only really observe our local slice of spacetime, the pixels don’t know anything about the size of the grid.
Now, it’s unclear exactly what mechanism “assigned” consciousness to the blue pixel, or if the word “assign” even makes sense in this context. All the average blue pixel knows is that it exists, it has seen some blue pixels behind it, and there are likely going to be more blue pixels after it. What it wants to know are the two questions we already discussed:
From our perspective as a human outside the grid, certain answers seem obvious. The expected longevity for all chains, for instance, is 10, because the independent 10% chance of terminating means that continuing the chain is equivalent to rolling a 10-sided die over and over. Also, since dice rolls are “memoryless” (they don’t depend on previous rolls - see the Gambler’s Fallacy) if we know a particular chain has already progressed for N pixels, we will expect the whole chain to be N+9 pixels long.
Similarly, from our outside perspective, the propensity to produce blue pixel chains would depend exclusively on the size of the grid and the probability of a blue pixel spontaneously emerging. But if all that happens is that we are shown a randomly selected blue pixel chain and told it was at least size N, we would not have enough evidence to make an informed decision about the size of the grid; the best we might be able to say was that the grid wasn’t so small that blue pixels were unlikely to form. After that, we would just make a reasonable guess, call it a “prior,” and use that to produce our credences.
The interesting thing about these simple conclusions though is that no existing anthropic approach can reproduce them, at least not without unacceptable side-effects. The SSA, especially when your prior gives non-zero weight to small grids, will under-estimate the length of the chain (a.k.a. the Doomsday Argument), while the SIA will strongly overestimate the size of the grid relative to the prior (a.k.a. the Presumptuous Philosopher). Double Halfer methods also fare poorly.[8]
As you might have guessed, the PPA alone reproduces these intuitions, predicting 10 more pixels in the chain while also closely matching our prior on grid size.
The PPA does this by modeling the size of the grid the same way it modeled the googolth digit of pi in our previous example, constructing a prior by assigning different probabilities to different possible sizes. Then, for each possible size, we imagine that the random patterns of the grid are re-sampled an infinite number of times, selecting “ourself” (that is, the blue pixel that we are) at random from the set of all blue pixels in the ever-repeating grid.
In other words, we nest our predictions about the randomness of the pixels inside Bayesian predictions about the size of the grid (with an additional adjustment factor we'll discuss that compensates for the possibility the grid is empty).
My goal for most of this series is to keep the use of math isolated to only a few places, but it’s worth it to take a moment to spell out what the PPA’s distribution actually looks like.
All of our anthropic assumptions start with a "universe prior" , which is the probability of a particular universe before we consider any distortions caused by being a a particular observer. In the Sleeping Beauty problem, for instance, we have , because the entire thing is premised on the coin being fair.
The SIA in this case would be the distribution we get when the probability of being observer in universe is proportional to whenever is an observer in universe , and zero otherwise:
This means that if each universe has observers, , which gives us the SIA’s presumptuousness - large universes get linearly larger. In particular, when there’s something you don’t know and can’t learn - like “propensity for life” in the Basic Model - you can never update away from largeness, and you are forced to assume life is arbitrarily abundant.
The only way we can reduce the largeness bias is to reduce the multiplier to something less than . The SSA does this by dividing by (again when is an observer in ):
giving . But this creates a strong bias in the other direction - the things you do know, like your rank in your species, cause you to update towards smallness. We’ll see there’s actually a lot of subtlety to that statement, but for now you can take it for granted that things like the Doomsday argument rely on this bias.
In the PPA, we resolve these paradoxes by breaking our universes into two components, a process component and a parameter component . The PPA says we should treat the distribution in an SIAish fashion, while nesting it inside an SSAish distribution. Although the full PPA will need an extra term,[9] for now we can think of that as looking like this:
Here, is defined as an SIA distribution built out of .
Moving some terms around, we get this:
That is, we divide the SIA’s prediction by the expected number of observers at a given parameter setting. In the Basic Model, this means finding the expected number of observers given that the laws of physics have a propensity for creating so-and-so many civilizations, so-and-so many large scale simulations, so-and-so many Boltzmann Brains, etc.
What this does, in essence, is divide out the propensity, while leaving everything else intact. You don’t know how many times the observer-generating processes start up, but you act SIAish (and thus in a temporally consistent way) about the process you are currently in.
In addition to the “Bayesian vs. Frequentist” distinction, the PPA can also be derived in two other, equivalent ways:
This second framing in particular is a good way to understand why the PPA can handle infinite universes in a way that other methods cannot: since we can think of it as only meaningfully operating over our local subprocess, the total number of subprocesses is irrelevant. This happens despite the fact that we are technically modeling the entire universe.
The following is a list of some of the biggest contradictions and anthropic paradoxes I’m aware of. As we will discuss, under the right universe model the PPA resolves every single paradox on this list.
The catch to this, which we will discuss, is an extra degree of freedom in our model. Specifically, we must decide which variables in our prior are "SSAish" and which are "SIAish." Much of our discussion will center around motivating this choice.
This piece has covered most of the important details of the PPA.
The rest of the series will dig in a little deeper, hopefully clarifying or defending any points you didn’t follow or disagreed with. If you don't have time for all that but still want a little more detail, then this piece makes a good follow up, summarizing and synthesizing all of the previous pieces.
Otherwise, the first two thirds of this series are "low-math"; there are no real equations and all the proofs are visuals worked out on simplified examples. It contains several pieces, like the ones on indexical reasoning or improper priors, which include introductions to intermediate-level concepts that you might already know.
Only the last third of this series is fully technical. These parts will include full equations and derivations for the arguments made in the first portions, as well as handling of a few additional scenarios, such as infinite universes.
The pieces are labeled as follows:
And here is the full series.
Though the PPA does technically give Sleeping Beauty a 1/3 chance of heads, it would be unwise to index too hard on this fact.
If you are a halfer, you can instead think of the PPA as what you get when you start with the SSA, resample the universe over and over, and then backfill the resulting structure to your original prior. If you are a thirder, then you can simply think of the PPA as the SIA with the "propensity for life" (mean expected number of observers) divided out, keeping all the original dynamics without presumptuousness.
AI DISCLOSURE: Absolutely no text in this series was written by AI, but most of the figures were either generated completely by AI or with heavy AI assistance.
The vast majority of the text and ideas in this series were fully written/developed without AI several months ago before the existence of models strong enough to meaningfully help. If I had known AI would become so powerful so quickly I would have written this up sooner.
Near the end of writing this series I decided to use Claude to proofread, mostly because recent developments in AI convinced me of the need to publish quickly. AI was used only to flag spelling or factual errors; at no point was it used to make any edits.
While proofreading two of the pieces, Claude ignored my instruction not to provide new ideas and wound up providing some insights that, while minor, were worth mentioning. I have included most of these as footnotes. In general, any time Claude did anything remotely non-trivial, I have tried to point it out so as to make it as clear as possible which contributions are 100% my own.
On a few occasions I did have extended conversations with Claude about this work (using Opus 4.8 at the time), but most of these conversations were unfruitful. Claude did make one significant contribution relating to the PPA in infinite universes, which I point out when we reach it.
I actually don't believe that this logic makes sense for the “many wake-ups” scenario of the Sleeping Beauty problem, but I do think it would hold for, say, an “incubator” version of the SBP where a new set of Sleeping Beauties is created each time. This is much closer to the situations we care about in this series. I plan to expound on this in a future piece.
I talk about these different scenarios a bit in this comment.
This is the Carter-Leslie version of the Doomsday Argument. Other versions, such as Gott's Doomsday Argument, are not built on top of the SSA.
Using the notation introduced just a little farther down, define an indexical distribution . This indexical distribution interpolates between the SSA and SIA in such a way that it manages to reproduce a version of basically every paradox from both approaches simultaneously.
Assuming each row depends only on the row above it, the technical term is that the process has the Markov Property.
Non-indexical Double Halfer approaches will look at a bunch of pixels, devolve into Full Non-indexical Conditioning, and fail the same as the SIA. Bostrom’s Hybrid Model makes the right predictions in this particular case but it can be easily money-pumped in other scenarios.
The full PPA has the denominator . This term allows the PPA to behave reasonably in universes that don’t produce any observers, something which both the SSA and SIA already do.
Updatelessness also has the problem that it actually just reduces to the SIA in practice. This is known as “Full Non-indexical Conditioning," which we'll briefly discuss later in this series.
Specifically, we can handle the cases where (1) there are infinitely many species/simulations in the universe and each of these (2) almost surely goes extinct/terminates eventually
Better yet, assume the grid wraps around on the sides, so there is no edge
It does add a Doomsday bias, but as long as your species doesn't permanently grow exponentially, this effect can be mitigated or even asymptotically disappear by all sorts of additional assumptions. Like a good Bayesian prior, the more evidence you get about how long you believe your species will survive, the weaker the anthropic Doomsday effect.
Though the PPA does technically give Sleeping Beauty a 1/3 chance of heads, it would be unwise to index too hard on this fact.
I want to head off two possible misinterpretations:
