PPA 5: Improper Priors and the Infinite SIA

[This is the fifth 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 "Parametric Modeling and Types of Uncertainty."

TLDR:

• Improper priors allow you to use a prior that is not a true distribution.
  • They can be updated and, with enough new information, be made proper
  • Until there is enough information to become proper, improper priors cannot be used to make predictions
• The SIA over universes with potentially infinite observers can be thought of as an improper prior that updates to proper once you learn enough about your identity

The next piece is "Indexical Models Don't Have Diversity Biases."]

Not all priors make predictions

[If you already know what an improper prior is and why the SIA functions as one, then you may wish to skip to the "SIA as an improper prior” section]

One thing that everyone knows is that it is not possible to make a uniform selection over the set of all integers. If you tried to make a uniform distribution over this range, you would have to assign each number a probability of 1/, which is just equal to zero, and adding up an infinite number of zeros just gets you zero, not 1.

I have seen this fact used to argue that, since identities are selected uniformly, neither the SIA nor the SSA can actually construct a probability distribution over infinite observers. At minimum, I myself have erroneously made claims of this sort in the past.

In reality, however, there are many different ways that the distribution of observers can be infinite, and at least a couple of tricks we can do to get valid results. The more technical parts of this series will go through a whole bunch of these. For now, what I want to talk about is the concept of an improper prior⁠.

In a sentence, an improper prior is a kind of prior that lets you treat certain non-distributions - like uniform selection over infinity - as if they were real distributions for the purposes of performing Bayesian updates.

Consider what a prior actually is. If start with some event, such a coin landing on heads, then a prior is a mathematical object that lets us:

1. Update when we get new evidence. For instance, we may observe 10 tails in a row and update away from a 50% chance of heads
2. Make predictions about what will happen next, e.g. that the coin will land heads with 50% probability

An improper prior is, informally, just a special kind of prior that let's us to the first thing, but doesn’t let us do the second thing, at least not until we've gathered enough evidence to make our prior proper again.

Another technically-not-correct-but-still-useful way to think about an improper prior is that it’s a stand-in for when we know so little about a distribution that it isn’t even rational to make a specific guess. In the uniform-over-infinity case, for instance, an improper uniform prior is simply telling us that we know there’s a number that was selected, but couldn’t even begin to guess what it might be. Then, if we later learn the number is, say, no more than a million, the improper prior still lets us update to “uniform between one and a million” in the normal way. 

SIA as an improper prior

What this means for anthropics is that, under the right circumstances, we will be able to use the SIA (and PPA) as improper priors. There will be situations where, before we learn anything, these models won’t allow us to make predictions without data, but which will suddenly become valid distributions as soon as we collect some data. 

To take a really simple example, let’s imagine an “infinite Sleeping Beauty problem” under the SIA. It proceeds like this:

• On heads, Sleeping Beauty is woken up once a day for exactly 7 consecutive days
• On tails, Sleeping Beauty is made immortal and woken up once every day for an infinite number of days

Without knowing the day, this simply can’t be constructed as a proper distribution. With some definitional kung fu, we could probably make some kind of pseudo-distribution where we conclude with 100% certainty that the coin went tails, but there is no formalism where we could give any actual positive probability to any particular day.

But it still works fine as an improper prior! If Sleeping Beauty wakes up and learns it’s the fourth day of the experiment, for instance, then she just updates on that and voilà! She has a nice 50-50 chance of existing in an infinitely long universe:

Interestingly, while the SIA is often the one thought of as worse at infinity, this improper prior trick cannot be made to work on the SSA. You can see this for yourself by trying to imagine how we would redraw these mass charts for the SSA; the bottom universe would be sliced into infinitely many pieces, but when you conditioned on, say, day 4 that slice would have zero width. The best you could maybe do, with some limit math, would be to say that P(Heads) = 1, which would be completely degenerate if you ever learned the day was greater than 7.

The next piece is "Indexical Models Don't Have Diversity Biases"

1. Alternative intuition: any number you pick will always have infinitely more numbers above it than below it, so you will always pick something in the bottom 1/ of the distribution.
2. One example - the set of observers could be unbounded, but finite in expectation. In this scenario, the SSA would have no trouble, but the SIA wouldn’t be able to build a true proper distribution. Stuart Armstrong has pointed this out.
3. The Wikipedia article gives a good technical explanation. Improper priors are possible because Bayes’ Rule doesn’t technically require that the prior be normalized in order to perform an update.
4. We’re still making an assumption even in the improper case, though. For instance, the decision to be uniform in N instead of, say, uniform in log(N)
5. Since the PPA has as a denominator, we will need to redefine it in terms of limits to get this result.