[This is the eleventh 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 "Deriving the PPA."

This piece is part of the technical portion of this series.

TLDR:

• Most of this piece contains proofs for the statements in this piece
• We prove the PPA is negligibly presumptuous, in the sense that, when modeling life emergence as a Poisson point process, we have a presumptuousness of exactly one additional species
• We define anthropic temporally consistency in terms of random processes and show that the PPA, liked the SIA, is temporally consistent

The next piece will be on the using the PPA with infinite universes, and is coming soon.]

At the start of this series, we introduced the “Basic Model” of anthropics. In the Basic Model, the universe is a large random process, and as this process progresses it sometimes spawns subprocesses that generate observers. The parameter of our model, which we call the “propensity for life,” is the average number of times one of these sub-processes emerges.  “Propensity” is a stand-in for the size of everything that exists across any potential multiverse, and it is epistemic - fixed but otherwise unknown, like the digits of pi.

The actual progression of one of these observer-generating processes, once they are kicked off, is random/aleatoric. The typical example of one of these “sub-processes” would be a species that emerges and keeps breeding until it either goes extinct or perhaps colonizes the stars. Other examples might be one-off Boltzmann Brains or large-scale simulations. As is often the case with priors, our criteria for defining an observer-generating process is somewhat subjective, but - with one subtle exception - most choices will give us reasonable results as long as we don’t go out of our way to do anything too pathological.

For instance, it makes little difference if we measure our processes according to “species” or “life-bearing planets,” since both of those things represent well-defined sets of observers with beginnings and ends. But we don’t want to go too small; defining each observer to be its own process (so that propensity = numbers of observers in the universe, i.e. ) would just devolve the PPA into the SSA. The goal really is to select the largest granularity where we feel that the size of the process is primarily determined not by unknown laws of physics, but by randomness within the laws we already have. The speed of light, for example, is probably not relevant to whether our species continues to exist for another 10 years or another 10,000, and neither is the existence of a multiverse for that matter. But the size of the multiverse obviously would matter if we wanted to count the number of species throughout existence.

To be more explicit, our model will comprise three different sets of variables, which you may have already seen in a previous piece:

• (Epistemic) Propensity for Life: How many times does an observer-creating process emerge, on average?
• (Aleatoric) Longevity: For each emergence, how many observers does the process produce before it terminates? 
• Diversity: Everything else. When a process starts, what do the observers experience? What do they look like? etc.

Notation

We will eventually expand the model to include large-scale simulations and Boltzmann Brains as additional types of subprocesses, but for most of this discussion we’re just going to focus on “species” subprocesses, and this notation will focus only on that for now. It should be understood though that “species” is a shorthand for “observer-generating process that works approximately like what we see on Earth.” If you want to expand it to, say, all life on the planet, then that doesn’t change the model even if the word “species” becomes less applicable.

We’ll represent the epistemic propensity parameter as , which for the time being will be a single non-negative number telling us the average number of species. Propensity is only an average, but the actual number of species produced by a given aleatoric universe will be represented by the random variable , where . 

A universe produces exactly species , each having a “longevity” of observers. This means that (ignoring simulations and Boltzmann brains) the total number of observers in any given universe is . We will often use as a shorthand to indicate whichever species we currently belong to. The expected longevity over all species in all possible universes is a known constant . For now we will assume is finite but will discuss the infinite case in the next piece.

Diversity we will also consider aleatoric, and we will represent it with the random variable . We're going to play this variable a little fast and loose because it's not actually that important. For simplicity, we're going to only deal with the species level diversity, meaning each species gets it's own draw (hence the subscript in ). For instance, perhaps would indicate that we have two arms, two legs, and exist on a planet like Earth. The specifics matter very little. In fact, does not need to be only aleatoric - the laws of mathematics and physics are likely responsible for our distribution over , and we could in theory treat our prior over this distribution as an epistemic parameter. As we will prove (and as we saw before), this won’t affect our model. The same irrelevance result would also apply to any randomness an observer observers that doesn't directly give them evidence about propensity or longevity.

An observer is uniquely identified within a universe as - the th member of the th species. We’ll use to indicate the set of observers within species . We won't worry about anything an observer learns not related to information about propensity or longevity.

Poisson Universe

One assumption we will use going forward is that (in the species-only model) is a rate parameter to a Poisson Point Process. For those of you unfamiliar, a Poisson Process is a random process where, with some constant rate, we assume that at any moment/location a particular event can happen. Specifically, we assume that there is some event that happens at any time with a given rate, such that this event can happen independently at any time. This is basically the simplest form of assuming that life is distributed uniformly throughout our universe/multiverse - in expectation it can happen anywhere.^[1]

Life emergence is obviously not equally likely once you condition on a particular patch of spacetime looking a particular way, but the approximation is justified because there is no universal canonical coordinate system to the universe, meaning a priori no particular “region” is more likely than another to contain life. This doesn’t quite hold temporally, since there is a universal notion of order to time, but it does work across spatial dimensions and sub-universes in a multiverse, so we’re going to use this approximation anyway. Future work can explore alternatives, but I suspect they will contain no interesting insights.

The Predictions

In the intro to this series, we listed all of the main failure modes of existing anthropic models, claiming that the PPA did not suffer from these failures. Our plan is to go through these one at a time, proving each one under the Basic Model not just for the PPA, but also for the SSA and SIA. 

Independence Results

Before we dig in, though, we have to establish a couple independence results. Using an indexical distribution replaces our prior with a completely different distribution, so it’s important to understand how the relationships between these variables are affected when we go indexical:

1. Under the SSA, SIA, and PPA, diversity updates independently of both propensity and longevity 
2. Under the SIA and PPA, longevity updates independently of propensity
3. Under the SSA, propensity and longevity are positively correlated when you learn indexical information about either one. In the limit of high propensity, this effect disappears and longevity updates do not affect propensity.

An important complexity here is that when we say “diversity” or “longevity” (but not propensity!) we are making indexical statements - we are referencing the values these variables take within our own species, whichever one that may be. Under the SSA, for instance, learning that propensity is low will also cause you to specifically update towards your own species having low longevity. 

The proofs of statements 2 and 3 are in the appendix of this piece; statement 1 follows the same roadmap.

If you don't want to read that, you can get an idea for how a proof would look by observing that all three anthropic assumptions can be written in the following form:

if and 0 otherwise

Where takes on the following values:

• SSA: 
• SIA: 
• PPA: 

(In the PPA we remove because it is a constant)

Then when we write out something like, say, for some anthropic assumption , what we can do is write out the full distribution this produces, notice that does not depend on for any possible , and then move around terms until we get an equivalence to . Doing this for sufficient combinations of variables, we will get the results above. 

In particular, since in the SSA does depend on both longevity and propensity variables, this is the one spot where the trick does not work. Even so, as increases more and more probability mass goes to the cases where , which is why the SSA’s longevity estimate approaches the universe prior in the limit of high propensity.

The SSA predicts Doomsday in small universes, the SIA and PPA never do

This one’s pretty straightforward. For the SSA, we can simply observe that at , the entire problem reduces to the standard Doomsday argument, where - there’s only one species, so there’s nothing to dampen the effect. As propensity increases, so does the amount of shortness bias, and by the time gets really high, we’ve reproduced the “argument from repetition” from the previous piece. The Doomsday bias asymptotically disappears as the propensity grows.

For the SIA, lack of Doomsday follows directly from the independence result, which shows us it inherits the standard Doomsday argument behavior regardless of propensity: 

In particular, this means that

whenever .

Finally, the PPA does not predict Doomsday because conditioning on all parameters reduces the PPA to the SIA:

and the same exact logic applies.

The SIA is fully presumptuous and the PPA is negligibly so 

By presumptuousness, I mean that it forces us to assume there are an arbitrarily large number of other species in the universe besides ourselves. Although we already discussed this at length in an earlier piece, the Basic Model gives us a way to formally show our work.

As we mentioned, all three variable types in the Basic Model were independent under the SIA and PPA. The most important offshoot of this is that the distribution of species looks the same wherever it emerges. 

One thing I naively believed at one point was that the SIA’s largeness bias only applied to exact clones of yourself, or at least only to observers whose experiences are indistinguishable from your own. The idea was that if you had some subset of observers who all had the same experience, then - in other words, every universe is re-weighted according to the number of clones of that observer it contains.

However, by establishing that species variables don’t have any effect on the marginal probability of , we establish that, for any variable not correlated (through the universe prior) with  (i.e. only correlated with longevity or diversity), we have . 

You can think of this as saying that, in expectation, every possible location where a species exists has the same chance of having produced you. Therefore, the SIA forces you to re-weight by the number of possible locations, and presumptuousness applies not just to clones, but to our entire understanding of the universe.

For the PPA, we have to be a little more careful, since is an aleatoric variable but the propensity is a parameter. Both are independent of , though, so the first thing we can establish is that the PPA induces no distortion on : 

But the PPA will introduce a small distortion on . Specifically, using the fact that conditioning the PPA on gives you the SIA, we have

Now recall that our model of life emergence is a Poisson point process, which means that is Poisson distributed and we can write

to indicate that is the result of the PMF of a Poisson distribution with rate evaluated at point . This PMF represents the exact number of species emergences. 

We therefore have that

.

This is just the same Poisson distribution shifted over by one. Equivalently, we can say that under the PPA, the Basic Model’s prediction for is increased by one - our existence causes us to conclude that there is, on average, 1 extra species than predicted by our universe prior.^[2] You can also think of this as if we are predicting that there are on average species besides our own.

Note that if you didn’t use a Poisson process to model life emergence, then you could get a different level of presumptuousness. In practice, the more variance you have in your process distribution relative to a fixed parameter value, the more presumptuous the PPA will be, and vice versa.^[3] 

The PPA and SIA are temporally consistent; the SSA is not

This piece is more obligatory than anything. As we've mentioned, the PPA mostly inherits temporal consistency from the SIA by virtue of the fact that the parameters are set before the universe is actually sampled, and all dynamics after that are SIAish.

If you've read that piece and it convinced you, then you should probably save yourself some time and skip this section. 

If you wish to keep going, recall that by temporal consistency, we mean that if there is some event happening at time with known and fixed probability, then learning you are an observer created at time should not cause the probability of that event to change. All we’re going to do here is formalize this definition, then show that the PPA’s predictions are consistent with it. This is arguably gratuitous; we’re just picking a good definition and watching as the result follows trivially.

If you’re familiar with stochastic process theory you may be tempted to define temporal consistency in terms of a random process variable , where tells us the state of the entire universe at time , and the universe variable would be the set of all process variables.^[4] Then you might (incorrectly) define temporal consistency for an indexical distribution as the requirement that, for all observers and times such that , 

But this is actually just the SIA restated, since “ for all times ” is just . It also isn’t clear how we would use this to define, say, at what time the coin is flipped in the Sleeping Beauty Problem.

Instead, what we need to do is assign to each event in each particular universe a “moment” at which it resolves, then define a timing variable which tells us when the event happens.^[5] We also associate each observer with a “creation” event, such that tells us precisely when an observer came to exist. Note that any depends deterministically on . Finally, we assume that non-creation events always resolve eventually, even if they resolve to states like “the coin does not flip”.

Then we can define to be the history of the universe up to time .^[6]

This gives us:

Definition: An indexical distribution is temporally consistent if for all events , and observers ,

.

In other words, if the event happens after the observer comes into existence, then learning you are that observer cannot modify your credence in that event.

This trivially applies to the SIA since conditioning on reduces you to the universe prior, but it also applies to the PPA under the Basic Model because - which represents the laws of physics before any universe/observers are sampled - always resolves before all other events by definition. Thus, always equals 0 and all already have resolved, meaning the condition can never be violated.

Appendix: Proof of independence

First we’re going to prove that longevity and propensity update independently in the SIA and PPA. Independence results involving species count and diversity would proceed the same way. ^[7]

We’re going to use the notation to refer to the different species longevities for a particular universe. Similarly, the vector will be the diversity values, noting that we’re writing and instead of and for convenience.

The longevity of each species is independent and identically distribution according to a distribution we’ll denote . That means .

We’ll also define as the “birth rank” function that returns the rank of observer in their respective species.

Proof that

Now suppose we learn that our birth rank is . For any indexical distribution , we have:

where the first summation is over propensity variables, the second summation is over indices of species we could have, and the last two are over possible configurations of longevity and diversity for the entire universe.

Then this is equivalent to

Then, since (1) is strictly a function of for the PPA and a constant for the SIA and (2) , and are all independent of one another in the universe prior, we have that this:

where the second to last line follows from the fact that all longevities are independent and in the last line. In particular, is the same for all possible , meaning that it normalizes out and we have that, for the SSA and PPA,

.

Thus, if you learn your birth rank in the SIA or PPA, then your credence over possible longevities for your species simply reverts to the prior.

Proof that and are independent

Similarly,

and 

The other independence results ( and , and everything else) can be proven similarly.

Proof that and are not independent under the SSA

For the SSA’s lack of independence, we can start with the expression 

where  .

When , , giving us a simple bias towards smaller .

On the other hand, as increases,  also increases for any positive , meaning that more and mass gets concentrated near (you can use a Chernoff bound to get a better idea of how quickly it converges). 

Assuming we actually did have , we would have

Meaning that as gets larger, and get closer and closer to independent.

(This is also very similar to what you would get with the PPA, where , especially since approaches zero as (and, by extension, ) gets large.

The next piece will be on the using the PPA with infinite universes, and is coming soon.

1. ^{^}

   Technically, rather than a Poisson process, a better prior would be a Cox process.

2. ^{^}

   Note that technically remains the same, aside from the existential factor. But was our prior’s propensity, not the expected number in our indexical distribution.

3. ^{^}

   Specifically, the amount of presumptuousness is equal to the ratio of variance to expectation in the distribution . I knew the presumptuousness was a function of variance in some way and had planned to derive an exact relationship at some point, but I carelessly had Claude read a draft before I had computed it and it volunteered the exact result.

4. ^{^}

   Here we assume for simplicity that there’s a well defined for all events, which is probably not true for a multiverse with disjoint universes and arguably isn’t true even in our universe. None of this is especially hard to fix if you’re inclined.

5. ^{^}

   This is what's known as a stopping time.

6. ^{^}

   This is what's known as a filtration.

7. ^{^}

   I did not pass this portion through Opus except for passing in the longer equations to help with type-setting. All math is therefore my own.