# The questions and classes of SSA

Important summary: **For SSA, the reference class to be used is defined in the question asked.**


I [said](https://www.lesswrong.com/posts/nxRjC93AmsFkfDYQj/anthropic-probabilities-answering-different-questions) that [SSA](https://en.wikipedia.org/wiki/Self-sampling_assumption) derives from the following question:

* Q: What proportion of universes, where versions of me exist, will be large?

And it's sort of true, that sort of is the SSA question. But only when talking about probabilities of universes, not when updating on information.

For example, consider the [sleeping beauty problem](https://en.wikipedia.org/wiki/Sleeping_Beauty_problem). If the coin is tails ($T$), Sleeping Beauty ($SB$) is awoken twice, on Monday and Tuesday. On heads ($H$), she is only awoken on Monday.

Then if the "large" universe is the tails universe, and "versions of me" cover both Monday and Tuesday (reference class $R_{SB}$), then the answer to that question will be "$1/2$". So, writing $P_{SB}$ for the notationally unwieldy $P^{SSA}_{R_{SB}}$, we have:

* $P_{SB}(T)=1/2$,

the standard halfer/SSA answer.

Now suppose that every version of Sleeping Beauty is (eventually) told which day it is. Suppose she is told it is Monday. What is the correct question now? It seems like it should be:

* QM: What proportion of universes, where versions of me exist on Monday, will be large?

However, the answer to that question is... also $1/2$. The fact of it being Monday has not updated anything at all! This is surprising, because if the question had asked about Tuesday, it would have updated to $1$ on the universe being large (ie tails).

## Sliding reference classes

What's happening is that the reference class has changed between question Q and question QM. For QM, the reference class is the set of all Sleeping Beauties who are eventually told that it is a Monday (call this $R_{SBM}$). For that, we still have:

* $P_{SBM}(T | \textrm{Monday}) = P_{SBM}(T)=1/2$.

What about $P_{SB}(T | \textrm{Monday})$, using the original reference class, which is equal to $1/4$? Well, the best question I could come up with that captures that is:

* Q': What proportion of universes, where a randomly selected version of me exists on a Monday, will be large?

Randomly selected meaning "randomly selected from the versions of me that exist in the universe".

This question doesn't feel as natural as either Q or QM. And it highlights the importance of the reference class, the set of "versions of me". So let's look in more details at the reference class.

## What are reference classes for?

There are several purposes for reference classes. We can use them in [decision theory](https://arxiv.org/abs/1110.6437), when considering the class of agents that will reach the same decision as us. Or we might use them in ethics, when considering the class of agents whose preferences we might care about. The "Rawlsian veil of ignorance](https://plato.stanford.edu/entries/original-position/#VeiIgn)" gives a sort of reference class of all humans.

But in probability, reference classes are a lot simpler. They tend to go along the lines of "if all I know is that I'm a member of $R$, I should expect that I am a typical member of $R$".

This justifies a lot of "[Outside view](http://www.overcomingbias.com/2007/07/beware-the-insi.html)" reasoning. If I'm planning for a project, and don't have a track record of being exceptionally timely, then I should expect that I will plan as well as any other human, ie [badly](https://en.wikipedia.org/wiki/Planning_fallacy). Or if I'm about to become a parent, then I should expect I'd behave about the same level as typical parents of my class and background.

Or if me and a hundred other people are each locked in a hundred numbered rooms (numbered $0$ to $99$), then I should expect that there's $75\%$ chance my room number is less than $75\%$.

Notice that in that last example, I don't have to be similar to the other people at all for the argument to go through. The other 'people' could be aliens, robots, rocks, or dead, and the reasoning would still go through.

**Now, when we ask a question like Q', the reference class to be used is defined in the question asked.**

## Minimal reference class

But are there some forms of "natural" reference classes, universal in some senses? I can think of two "natural" reference classes that can be used for SSA. The first is the reference class $R_\infty$ of everything whatsoever that ever existed. The second is the narrow class $R_s$ of beings subjectively indistinguishable from you. Any reference class $R$ between those two would have the problem that there exist reference class problems with natural reference classes $R'$ with $R'$ neither a subset nor a superset of $R$. So $R$ doesn't fit naturally with reference class reasoning in general.

And there are arguments to use $R_s$, since that is the only reference class where "all I know is that I'm a member of $R_s$" is always true.

So, onwards with SSA and $R_s$. Right?

## Consistent updating

The problem with the minimal reference class is that it is changing - two copies of me with the same subjective experiences can diverge (or, more rarely, converge). This means that using this reference class will result in SSA being time-inconsistent, in the same way that [FNC is](https://www.lesswrong.com/posts/jH3NfxoNgKTh9r5KW/anthropics-full-non-indexical-conditioning-fnc-is). Indeed, in terms of its probability about the universe, FNC essentially **is** SSA with $R_s$.

If we want time consistency, a desirable trait, we could instead talk about an agent's subjective experience at a particular point in time - call this experience at this time $s_t$. Then the class $R_{s_t}$ would be all agents who were ever in the subjective state $s_t$. This provides a one-off update to the prior probabilities of all worlds. But in order to maintain time-inconsistency across duplication, we have to do some extra work, such as allowing SSA to provide for a distribution of agents in $R_{s_t}$, rather than just selecting them with equal probability (if one agent becomes two, then the probabilistic weight of that one agent has to be the same as the sums of the weights of the two descendants). But those challenges are doable.

Ok, that's fine, but at point in an agent's subjective experiences should they pick? Well, what is the point of time consistency anyway? If we're time inconsistent, we open ourselves up to being [dutch-booked](https://plato.stanford.edu/entries/dutch-book/). On the other hand, on any given single probability decision, $R_s$ it the better reference class to use.

Therefore it makes sense to set $s_t$ to be as late as possible before it becomes relevant, ie just before major anthropic decisions need to me made.