[This is the tenth 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 "Putting it All Together."

This piece starts the more technical portion of this series.

TLDR:

• We make formal the repetition argument from the intro this series, showing that repeated sampling of the universe specified by some parameter creates an SIA instance nested inside an SSA instance.
• The argument from repetition cannot eliminate parameters that often lead to empty universes, so we add an "existential term" and call the result the PPA.
• The PPA can be expressed concisely as:
  
  when and 0 otherwise.

The next piece is "The Basic Model."]

The Argument from Repetition

In a previous piece, we discussed how repeating species draws under the SSA causes it to converge to something a bit like the SIA.

Here we’re going to formalize that. We’ll start with some universe prior  over universes , and then make that indexical by adding in another set of observer assignment variables , such that assigning to value corresponds to the logical predicate “I am observer ” We then define a joint distribution over and . Each universe is also associated with a set of observers, corresponding to the observers who actually exist in that universe.

The two most common indexical distributions are:

• The SSA:  if , and 0 otherwise
• The SIA:  if , and 0 otherwise

where is the number of observers in universe .

To represent what happens when universes are redrawn, we will add an additional parameter , from which we can draw universes . Our universe prior will be extended into a joint distribution over both and . We will sometimes refer to as “epistemic” and as “aleatoric” or the “process variable.”

Then we can define the “argument from repetition”: 

Argument from Repetition (AFR): Define to be the indexical distribution formed from the following sampling procedure:

• Sample from  
• Sample aleatoric universes from
• Select uniformly from the disjoint union

This is just describing the basic procedure we previously used when talking about parametric models and large-universe SSA: for each possible value of , imagine arbitrarily many samples, put all the observers from these samples in a pile, and then select one at random. If we applied this to the Sleeping Beauty problem, for instance, we would flip a bunch of coins and get a sequence (say, H H T) and then select uniformly from an associated set of universe-observer pairs (e.g. , just as we did in the intro to this series. (In the Sleeping Beauty problem, is trivially a singleton since we don't have any epistemic unknowns in the problem setup).

Since for a fixed each universe is sampled in proportion to its likelihood but takes up space in proportion to its size, we have that, in the limit,

.

In other words, AFR converges to the SIA over the conditional universe prior . We’ll write this as and say that, as N approaches infinity, 

.^[1]

Existential Consistency

The one problem with the AFR approach is that it doesn’t do well for which have a high chance of producing no observers. Fortunately, from a model-design perspective, this problem is easy to fix. 

Consider the following situation, which we’ll call God’s Calculation:

To decide if he will create life in the universe, God computes the googolth digit of pi. If it is even, then he creates Adam and Eve. If it is odd, he flips ten trillion coins, and if every single one comes up tails, he creates Adam and Eve. If a single one comes up heads, he creates nothing.

You wake up and discover you are Adam.

Both the SSA and SIA agree in this situation: you should assume that the googolth digit of pi is probably even. But AFR does not: it confidently declares that the digit has . 

The problem is that because we assume the aleatoric parts of the process repeat endlessly, it doesn’t really matter how many flips it takes for God to create life; we have to assume he eventually would.

I don’t have a prefect philosophical answer for this, but I’m leaving that for future work.  For now, though, the problem is very easy to fix mathematically. What we can do is add an adjustment factor that scales each SIA instance by the likelihood that any observers would even exist in that universe, multiplying the probability of each parameter by . This effectively separates each epistemic region of our probability space into a “life exists” and “life doesn’t exist,” then eliminates the “life doesn’t exist” portion.

The one thing I will say to justify this decision is that it is in fact equivalent what we would get if we replaced our prior with the prior  before we built our indexical distribution. In other words, the existential adjustment is equivalent to non-indexically conditioning on at least one observer existing, and only then building the AFR out of that distribution. Formally:

.

Whether or not that counts as philosophical justification I leave to the reader.^[2]

Either way, when we reconsider God’s Calculation with this new adjustment, the “odd digit” universe gets scaled by , effectively eliminating it from the running exactly as in the SSA and SIA. The implications of this are mostly related to fine-tuning - thanks to this adjustment, the PPA can reproduce the arguments we discussed here.

The Parametric Process Assumption

This, at last, gives us the PPA:

Parametric Process Assumption (PPA): The PPA is the indexical distribution defined as 

when , and 0 otherwise.

We can think of this as having three layers:

• Epistemic: - What are the laws of physics?
• Existential: - How likely is life to exist under these laws of physics?
• Aleatoric: - Who are you likely to be under these laws of physics?

We can make this more self-contained by observing that the SIA is defined (for ) as 

Thus, the PPA can be written as:

when , and 0 otherwise.^[3]

Finally, we note that the PPA can actually be reproduce both the SSA or the SIA with the right prior. If you consider all unknowns to be aleatoric, then can only take on a single value, so the entire distribution reduces to the SIA. Similarly, if you consider all unknowns to be epistemic (so that the aleatoric randomness is always fixed), the SIA portion is just a uniform distribution over observers in , so the distribution reduces to the SSA. We saw this same effect in the intro to this series, where modifying the Sleeping Beauty problem to select exclusively between epistemic and aleatoric probabilities caused us to switch between the SSA and SIA.

The next piece is "The Basic Model"

1. ^{^}

   Convergence here is convergence in total variation. Claude pointed this out on a proofread

2. ^{^}

   Also, I’m salty about this observation because I didn’t make it. Opus provided it unsolicited even though I didn’t ask for original insights, just proofreading. Now I have to add this stupid footnote. The exact identity written here is mine though.

3. ^{^}

   This also relies on the simplification that , which I also did not notice until it was pointed out by Claude during proofreading.