[This is the ninth piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics.

You should start here if you only want to read one piece on the PPA, but if you only want to read two pieces, this is the second one to read.

The previous piece is "The Fermi Paradox, Fine-Tuning, and the SSA."

TLDR:

• The SSA's bias for Doomsday only applies if your reference class only includes your own species. The more more species in your universe/reference class, the less Doomsday effect you see
• The Basic Model of anthropics models the universe as a process progressing randomly until certain relevant subprocesses emerge spontaneously. An intelligent species like humans would be an example of one such process. It has 3 components:
  • (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.
• The previous arguments in this series tell us many things about the Basic Model
  • Diversity has no effect on the SSA, SIA or PPA
  • The SIA acts as an improper prior over longevity but not propensity
  • The SIA gives an undue bias towards large propensity
• The PPA treats propensity as an epistemic parameter and longevity as aleatoric, avoiding both Doomsday and presumptuousness

The next piece, "Deriving the PPA," begins the more technical portion of this series.]

SSA biases only affect small universes

One frustration for me when I started thinking harder about anthropic philosophy was the slow discovery that many of the worst conclusions of various anthropic models, at least as I understood them, were actually greatly exaggerated. Not all of them, of course, but there were quite a few cases where things treated like foregone conclusions (like the SSA implying Doomsday) often disappeared with certain choices of prior. 

As I've references in this series before, I believe a lot of this comes from the unfortunate tendency to discuss anthropics in terms of small thought experiments, almost always forcing us to choose between exactly one of two universes. This by itself is not a problem - I employed the same tools in several parts of this series - but it runs the real risk of oversimplifying past the point where we are actually saying something applicable.  

Consider, for instance, the Doomsday Argument. Typically, this is presented as a choice between some large universe and some small universe , each with a 50% chance a priori. Every observer, through their birth order, is assigned some number (like your brith rank) and, upon learning your own number, the SSA has you update towards a universe that is more likely in proportion to the ratio between and . For instance, if is 10 and is 20, then finding out we are individual 4 means we are twice as likely to exist in universe :

(If you're coming here straight from the intro, you may wish to take a quick peak at this earlier piece on mosaic plots.)

This is famously countered by the SIA, where L starts out twice as likely and therefore produces no bias when you learn your rank:

The problem with this entire setup, though, is that it assumes our current species is the only one that exists. 

Suppose instead that there were two species within our universe (and our reference class) - one blue and one red - and each was independently assigned to be either “long” or “short”. Our distribution under the SSA would look like this:

Now look what happens when you learn you are the 4th member of the blue species:

The ratio of short to long has evened out considerably from 2:1 to about 1.43:1. Add in an orange species, and the ratio becomes even closer to even at about 1.27:1:

In the limit, as you expand your universe and/or reference class to include more individuals, the shortness bias disappears.

This effect, by the way, does not rely on learning your color in the previous example. Simply knowing that there were more species in consideration is enough to asymptotically eliminate Doomsday bias from the SSA. By the time you get to the kinds of scales that seem likely by our current understanding of physics (see this previous piece), the SSA’s main problematic features - Doomsday and temporal inconsistency - begin to asymptotically vanish.^[1] 

The Basic Model

Let’s re-introduce some terminology before we continue. As in the series intro, we’ll call the length of your species - whether it is “long” or “short” - its “longevity,” and we’ll use terms like “shortness bias” to refer to biases in longevity. When we talk about the “size” of the universe, using terms like “largeness bias,” we will instead be talking about the number of species that emerge. 

The previous section showed us that as our lower bound for the size of universe increases, the SSA’s shortness bias decreases with it. The SIA, once you learn your rank in the species, has neither a shortness nor longness bias, but it does have a largeness bias - however big your evidence suggests the universe must be, the SIA insists it must be much much large. In particular, if “infinitely large universe” has even the tiniest bit of positive probability in your prior, then you must discount all other possibilities.

Back when we introduced these terms int he intro, we talked about putting both of them into a single model, which we called the “Basic Model” of anthropics. 

The idea was to imagine the universe as a giant random process, with each independently-emerging species as a subprocess. The exact number of subprocesses (the actual size of the universe) would also be random, but the expected number of subprocesses - the general tendency of the universe to independently create life - would not be random, but rather an epistemic unknown which we called the universe’s “propensity for life.” We illustrated with a toy thought experiment involving a grid of unknown size and spontaneously emerging chains of blue pixels.

The the subprocesses themselves are modeled as completely random, since by the time you get to the granularity of determining a species’ longevity, what you are measuring isn’t really the product of unknown physical laws, so much as randomness within the laws that are already there. The specifics of quantum mechanics or the speed of light, for instance, may have affected the likelihood of humans existing, but probably don’t play largely into how long humanity lasts once it’s here.^[2]

Then, we throw in an extra variable to cover all the other facts about our species not related to longevity or propensity, and we get a model with three independent components:

• (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^[3]: Everything else. When a process starts, what do the observers experience? What do they look like? etc.

Some Observations

There are a few things you should know about the Basic Model and how it behaves under the SSA and SIA. These are things we cover in more detail in the technical portions of this series that follow.

The first is that, as we previously discussed, the diversity term is basically only there to be thrown out. So long as diversity has no a priori effect on longevity (which, in our simplified model, it doesn’t), neither the SSA nor the SIA induce any additional diversity bias one way or the other. It can be safely ignored.

The second is that the SIA cannot really handle infinite propensity - having it as a hypothesis either forces you to insist the universe is infinitely large, or, more likely, just gives you an invalid distribution; this is part of the presumptuous philosopher effect. In the case of infinite longevity, however, the SIA can function as an improper prior, starting essentially undefined but snapping into a proper distribution as soon as we gather some (possibly incomplete) evidence about our rank in our species.

The reason an improper prior only helps with longevity but not propensity is that, at least at the moment, we have no way to learn the propensity of the entire multiverse, which means that we can never update out of the improper regime into something that actually lets us make predictions. Nor, as we discussed, could we ever expect to make such an update. With longevity, we have a well-defined concept like birth rank which can be defined a priori for all observers before our universe is actually sampled; to get an equivalent metric for species index, you would need to acquire some sort of global information about all species in the universe like "ranked distance to largest star" or even just "Nth species ever created," and even that would not localize you within the multiverse.

As for the SSA, we already mentioned that large propensity means that it loses its shortness bias, and in the limit of infinite propensity the SSA’s longevity distribution looks exactly like the SIA’s. But the SSA only handles longevity well if you already have reason to believe your universe's minimum propensity is high. While I did argue that high minimum propensity is probably accurate, if you don’t agree, then under the Basic Model the SSA will have both a smallness bias and a shortness bias. The SIA’s biases are not correlated in this fashion, so learning your rank only eliminates longevity biases without touching the largeness. 

Neither approach, therefore, can simultaneously handle all longevity and propensity values to our satisfaction. But the previous results do suggest a way forward: what we need is a distribution that treats longevity just like the SIA/infinite-propensity SSA, but doesn’t unduly bias us towards one propensity or the other. 

Which, you presumably already know, the PPA does handle both approaches.

The PPA, In Summary

Since introducing the PPA way back in the series intro, we haven’t actually spoken about it much. If you recall, the PPA operated by dividing the variables in our universe into two components: an aleatoric “process” portion and an epistemic “parameter” portion. In later pieces, we connected this to the concept of “parametric models” in Bayesian statistics.

The motivation for this dichotomy was the “argument from repetition,” where we showed that if you took, say, the Sleeping Beauty problem and sampled it over and over again, then whether you wound up deriving the SIA or the SSA depended, respectively, on whether your probability was aleatoric or epistemic. To create an epistemic unknown, we imagined the scientists in the Sleeping Beauty problem calculating the googolth digit of pi instead of using a coin. This was a value that would never change over repetitions, but was nonetheless unknown to us.

The PPA, we said, was just what happens when you applied this reasoning to a more complex universe model. The Basic Model is that more complex model, where we treat propensity like the digits of pi and longevity like a coin flip. The universe (or multiverse) gets sampled, in a virtual sense, over and over again, such that all possible species longevities occur at precisely the expected prior rate; longevity predictions converge the same way they did in large-universe SSA. As for propensity, even though the actual number of species in each of these resamples varies, the average across all of them stays fixed. So propensity remains SSAish, even as longevity becomes SIAish.^[4]

And that’s about it, at least until we get more technical in the next couple pieces. There’s an extra term I’ll discuss there to correct for very low-propensity universes, and we'll need to do some work with limits to make the infinite cases work how we want them to. Aside from that, though, we've covered most of the key ideas. You can think of the PPA, as I originally did, as nesting the SIA inside the SSA, or you can simply think of it as just taking the SSA to infinite propensity and then backfilling that structure to finite propensity. I think both approaches take you to the same place.

The next piece, "Deriving the PPA," begins the more technical portion of this series.

1. ^{^}

   There is an important requirement here, which is that you must assume all species longevities are drawn from the same distribution. Or, to be precise, that they are drawn from distributions with the same mean.

2. ^{^}

   That said, one might still wonder if it makes more sense to think of the average longevity of all species as a parameter fixed by the laws of physics. I will discuss this in a piece I plan to publish within the coming weeks. This can be added to the model, and it does have a Doomsdayish effect, but I also believe there are good reasons not to make any part of longevity a parameter.

3. ^{^}

   Diversity has no epistemic/aleatoric label because it does not effect our model which one you assign it

4. ^{^}

   The PPA actually does have a real but almost negligible presumptuousness from two effects:

   1. The existential term: The PPA's down-weighting of universes where observers are unlikely to exist (as in the previous piece) causes a small asymptotically disappearing pull towards larger universes
   2. Since randomness within the universe is handled in an SIAish way, there is a tiny increase in presumptuous round this. In the model we'll use in the next sections, this will add exactly 1 additional species.