[This is the third 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.
Non-indexical reasoning (what we're calling the "Updateless Double Halfer" method, or UDH) states that you should not assign explicit probability to the statement "I am a particular observer"
Unlike the SSA or SIA, this approach is not robust to observations of randomness independent of all other variables.
Radford Neal's Full Nonindexical Conditioning (FNC) states that as you observe more randomness, the UDH converges to the SIA
As we discussed in the last piece, this series will be primarily focused on indexical models of anthropics, where “who you are” is modeled explicitly with a variable, and the existence of this variable causes a distortion in the probabilities you assign to world states. These distortions, applied unilaterally, inevitably seem to cause all sorts of paradoxes. The appeal of non-indexical models is that, by virtue of not adding these variables, they produce no distortions to the prior.
There was a point where I found this sort of non-indexical reasoning very tempting, mostly because it really did seem, at first, to resolve many problems of anthropics. In particular, Ape in the Coat’s series on the topic did a lot to convince me that no indexical model could be truly self-consistent.
Unfortunately, after spending weeks in the thrall of non-indexical double-halferdom, I ultimately concluded that the position was simply untenable. Yes, it solves many problems, but only by pretending they don’t exist. If probabilities actually could not be used to self-localize, then much of modern engineering would be impossible. What is the entire field of robotics, if not probabilistic localization of limbs and bodies? Or rocketry, or even a modern AI agent figuring out the specifics of its software environment? All of these things involve collecting evidence not just about the world itself, but also the specific angle from which you are observing it.
Full Non-Indexical Conditioning
Strangely, despite being a nominally “halfer” position, UDH is actually a thirder position, at least when applied in any realistic scenario. A version of this was first observed by Radford Neal, who called it “Full Nonindexical Conditioning”, or FNC. Its math is explained well both by Neal and ksvanhorn, but I wanted to give it my own spin in this section because (1) I think it’s much easier to understand visually, and (2) my impression is that many people come away from FNC thinking something fishy has happened, and I want to try to shine some light on that.
The problem I think is that neither of these sources state the FNC in the plainest possible terms. The truth is all the FNC merely says is that if you start with UDH, then observe a bunch of unrelated random events, your distribution over worlds will converge to the SIA’s. This is strange, both because unrelated independent random events presumably shouldn't cause you to update anything, and also because it suggests you should have just used indexicals anyway.
This becomes easier to swallow when you understand why FNCish effects don’t happen in normal indexical distributions like the SSA or SIA. We'll start with the Sleeping Beauty problem, but add the twist that when Sleeping Beauty wakes up on either day, she is also given a coin labelled “A” on one side and “B” on the other.
Our intuition, borne out by all indexical models, is that flipping this coin any number of times should not change the probability that the original coin flip - the one determining the number of wake-ups - has landed heads or tails. We can illustrate this in the SSA by drawing the distribution after a single flip of the AB coin: we simply split the universe distribution by all possible outcomes of this experiment.
Before the AB coin is flipped, we have this, the vanilla SSA Sleeping Beauty distribution:
Then, assuming all versions of the Beauty flip the AB coin, we get this:
Here, the “Tails” world has been split into 4 parts instead of two, because each wake-up might see different results. T:BA, for instance, refers to the world where, after a tails flip, the Beauty sees B on Monday and A on Tuesday.
Suppose now that the Beauty sees “A”. On heads, she must have seen this on Monday. On tails, she could have seen it on Monday, or on Tuesday, but in either case she has learned nothing about the other day. This induces the following update:
If you were to move the "Tue + A" box up and to the right and then stretch everything a tiny bit (remember that it is the proportions that matter, not the shape or placement) you would get exactly the original SSA.
A similar approach also shows that the SIA is invariant to these sorts of updates. The original looks like this:
and after observing the AB coin land A, it looks like this:
Now we can try it with the UDH. We start here, recalling that being non-indexical means that "T + Monday + Tuesday" is one event meaning "Monday and Tuesday wake-ups both happen on tails," since our distribution has no notion of "today is Monday":
Adding the AB coin then makes our distribution look like this:
Each world remains a solid horizontal bar, because we don’t differentiate between observers. That means when we update on, say, seeing coin A, we can only eliminate worlds where no one saw a coin land on A:
We’ve cut out half of the probability mass for H, but only a quarter of the mass for T; T is now 50% more likely than H. This happens because T offers more opportunities for A coins to exist, and they only need to exist once to stop us from eliminating one of the world.
And the distribution only gets more lopsided the more times we flip the AB coin. Suppose we flip it again and get B:
This plot's already ugly and will only get uglier if we keep flipping the AB coin, but you get the idea. In the limit, it will converge to the point where tails is twice as likely as heads, i.e. the SIA.[1]
This is, on one hand, an illustration of how strange non-indexical reasoning can be; updating on unrelated information is a very odd thing to ask. But it’s also a sign that that trying to escape indexical reasoning is futile. You can only ever gather evidence from your own perspective, and that will shape your understanding of the world whether you want it to or not. You might as well accept that so you can focus on building the correct model.
One last thought to close this out: while I think that non-indexical methods are misguided as an approach to anthropics, I don’t think that they are, broadly, problematic. In the context of decision theory, non-indexical approaches can sometimes be the only way to consistently make rational choices. But this doesn’t make them rational in anthropic situations. I don’t want to say too much about this because I have an entire (much shorter) follow-up series planned on this topic, but I believe there’s a deep connection between updateless methods of decision making (specifically, something like the FDT/UDT) and indexical methods in anthropics, but it's not as straight-forward as simply simply accepting UDH. Stay tuned for more on that.
[This is the third 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 "Mosaic Plots for Visualizing Anthropic Distributions."
TLDR:
The next piece is "Parametric Modeling and Types of Uncertainty."]
As we discussed in the last piece, this series will be primarily focused on indexical models of anthropics, where “who you are” is modeled explicitly with a variable, and the existence of this variable causes a distortion in the probabilities you assign to world states. These distortions, applied unilaterally, inevitably seem to cause all sorts of paradoxes. The appeal of non-indexical models is that, by virtue of not adding these variables, they produce no distortions to the prior.
There was a point where I found this sort of non-indexical reasoning very tempting, mostly because it really did seem, at first, to resolve many problems of anthropics. In particular, Ape in the Coat’s series on the topic did a lot to convince me that no indexical model could be truly self-consistent.
Unfortunately, after spending weeks in the thrall of non-indexical double-halferdom, I ultimately concluded that the position was simply untenable. Yes, it solves many problems, but only by pretending they don’t exist. If probabilities actually could not be used to self-localize, then much of modern engineering would be impossible. What is the entire field of robotics, if not probabilistic localization of limbs and bodies? Or rocketry, or even a modern AI agent figuring out the specifics of its software environment? All of these things involve collecting evidence not just about the world itself, but also the specific angle from which you are observing it.
Full Non-Indexical Conditioning
Strangely, despite being a nominally “halfer” position, UDH is actually a thirder position, at least when applied in any realistic scenario. A version of this was first observed by Radford Neal, who called it “Full Nonindexical Conditioning”, or FNC. Its math is explained well both by Neal and ksvanhorn, but I wanted to give it my own spin in this section because (1) I think it’s much easier to understand visually, and (2) my impression is that many people come away from FNC thinking something fishy has happened, and I want to try to shine some light on that.
The problem I think is that neither of these sources state the FNC in the plainest possible terms. The truth is all the FNC merely says is that if you start with UDH, then observe a bunch of unrelated random events, your distribution over worlds will converge to the SIA’s. This is strange, both because unrelated independent random events presumably shouldn't cause you to update anything, and also because it suggests you should have just used indexicals anyway.
This becomes easier to swallow when you understand why FNCish effects don’t happen in normal indexical distributions like the SSA or SIA. We'll start with the Sleeping Beauty problem, but add the twist that when Sleeping Beauty wakes up on either day, she is also given a coin labelled “A” on one side and “B” on the other.
Our intuition, borne out by all indexical models, is that flipping this coin any number of times should not change the probability that the original coin flip - the one determining the number of wake-ups - has landed heads or tails. We can illustrate this in the SSA by drawing the distribution after a single flip of the AB coin: we simply split the universe distribution by all possible outcomes of this experiment.
Before the AB coin is flipped, we have this, the vanilla SSA Sleeping Beauty distribution:
Then, assuming all versions of the Beauty flip the AB coin, we get this:
Here, the “Tails” world has been split into 4 parts instead of two, because each wake-up might see different results. T:BA, for instance, refers to the world where, after a tails flip, the Beauty sees B on Monday and A on Tuesday.
Suppose now that the Beauty sees “A”. On heads, she must have seen this on Monday. On tails, she could have seen it on Monday, or on Tuesday, but in either case she has learned nothing about the other day. This induces the following update:
If you were to move the "Tue + A" box up and to the right and then stretch everything a tiny bit (remember that it is the proportions that matter, not the shape or placement) you would get exactly the original SSA.
A similar approach also shows that the SIA is invariant to these sorts of updates. The original looks like this:
and after observing the AB coin land A, it looks like this:
Now we can try it with the UDH. We start here, recalling that being non-indexical means that "T + Monday + Tuesday" is one event meaning "Monday and Tuesday wake-ups both happen on tails," since our distribution has no notion of "today is Monday":
Adding the AB coin then makes our distribution look like this:
Each world remains a solid horizontal bar, because we don’t differentiate between observers. That means when we update on, say, seeing coin A, we can only eliminate worlds where no one saw a coin land on A:
We’ve cut out half of the probability mass for H, but only a quarter of the mass for T; T is now 50% more likely than H. This happens because T offers more opportunities for A coins to exist, and they only need to exist once to stop us from eliminating one of the world.
And the distribution only gets more lopsided the more times we flip the AB coin. Suppose we flip it again and get B:
This plot's already ugly and will only get uglier if we keep flipping the AB coin, but you get the idea. In the limit, it will converge to the point where tails is twice as likely as heads, i.e. the SIA.[1]
This is, on one hand, an illustration of how strange non-indexical reasoning can be; updating on unrelated information is a very odd thing to ask. But it’s also a sign that that trying to escape indexical reasoning is futile. You can only ever gather evidence from your own perspective, and that will shape your understanding of the world whether you want it to or not. You might as well accept that so you can focus on building the correct model.
One last thought to close this out: while I think that non-indexical methods are misguided as an approach to anthropics, I don’t think that they are, broadly, problematic. In the context of decision theory, non-indexical approaches can sometimes be the only way to consistently make rational choices. But this doesn’t make them rational in anthropic situations. I don’t want to say too much about this because I have an entire (much shorter) follow-up series planned on this topic, but I believe there’s a deep connection between updateless methods of decision making (specifically, something like the FDT/UDT) and indexical methods in anthropics, but it's not as straight-forward as simply simply accepting UDH. Stay tuned for more on that.
The next piece is "Parametric Modeling and Types of Uncertainty"
Specifically, for