He presents a bunch of arguments that SSA leads to insane conclusions there that I don't see you engaging with. (I can be more specific if that would be helpful.)
Having read about half of the Joe Carlsmith article so far, I have to say it feels like his approach seems to be going in a direction I don’t fully agree with. I'll continue reading it, but would certainly appreciate if you could point me to the specific objections you'd like me to address.
I believe the correct way to model the SB problem, and anthropics in general, is that we exist according to a process:
1. A universe is created
2. Observers are "placed" in this universe, as a result of the universe's inherent processes
3. You are "assigned" to one of these observers at random
Under this paradigm, the result is mathematically well defined and none of the issues that I've seen him list so far seem to exist. If you insist on declaring some kind of reference class, then it would be the class of observers - people who are capable of asking questions about anthropic reasoning. But it's not like you wouldn't also need to define this exact same class of observers in the SIA, so there's no special disadvantage to using it with the SSA.
I'll read in more detail and give another comment later, but this is what I saw in the short amount of time I had to look at it right now.
So, suppose you don't know how many observers will come to exist, but you seem to be early and have control over how many observers come to exist. (A classic example is "you are Adam or Eve, and can unilaterally decide not to have kids, meaning only two people will ever exist, or to reproduce, in which case there will at least be hundreds of billions".) Do you agree that SSA says you should be much more surprised to be "early" if lots of people come to exist? If so, do you think you can cause a coin to come up heads by committing to keep the population low iff it does? (This is described as "telekinesis" by Carlsmith.)
I'm really appreciating the Carlsmith series - thanks for the link. It's helping me see several cases I hadn't considered, though I haven't been convinced all of them are relevant. I have too many thoughts on the example you gave to fit into one comment, so I'll do my best to cover the important bits.
Upfront, I think the right way to model things is not necessarily SIA or SSA - it's to figure out the model that actually matches the process you care about, and then model that, approximating where necessary. In the absence of testable predictions, this is our only hope. This is not a universally applicable approach, and when it isn't applicable I don't necessarily take that to mean we should take a different inapplicable approach just because we can.
In the case you gave, the issue is that if you are Adam or Eve your decision is by construction nonrandom. As such, it is impossible to model it as a random variable, and any function of that variable cannot be truly random.
Of course, if we were anyone besides Adam then we could model Adam/Eve's decision as a random variable and the math would be fine. But if you *are* Adam then that's not a reasonable approach, and once the universe you're in and/or the number of people within it stop being random, probability stops being meaningful.
I also reject the idea that this failure somehow makes the SIA more reasonable. Applying the SIA still requires you to assume the existence of people who don't exist, and that simply doesn't match the process we're trying to model.
This may just be a case where our tools are too limited to give us the answers we want. Other such cases exist - consider the measure problem in cosmology. Either way, definitely something I will continue to think about, thanks again for the recomendation.
>>"Upfront, I think the right way to model things is not necessarily SIA or SSA"
Carlsmith gestures at this at the end, with his discussion of UDASSA, which I think is a step in the right direction despite the paradoxes. Personally, I think something like this--but with more of the deflationary attitude you took in your Tegmark post, when it comes to thinking about the output of Turing machines as "real"--feels promising to me.
On Sleeping Beauty, I have both halferish and thirderish instincts: I think halferism is right that we're trying to predict *the world* not the *observer moment*; possible worlds with lots of observers that never actually exist shouldn't count towards our expectation.
On the other hand, Radford Neal's Fully Non-Indexical Conditioning, and a calculation inspired by it done by ksvanhorn, convinced me that you can still get 1/3 as a reasonable answer even without the metaphysically dubious stuff that SIA seems to carry with it.
An interesting takeaway from ksvanhorn's calculation is that you can model it so that SB should predict any number from 1/2 to 1/3 as an asymptote, as a function of how much information SB has observed and how the probability distributions governing the info she can observe differs between the two awakenings.
Finally, on the weird metaphysics, I also don't like that SSA has some bad consequences too: aside from the determinism stuff mentioned in the comments above, there's also Carlsmith's observation that SSA has a bit of a vibe of, "since this is a world where I exist, and we're doing our expectation only over people in actual worlds, I *must* exist"---which strikes me as equally bad as the "all possible people exist" of SIA.
I think one of the things that appeals to me about UDASSA or something like it is that it is borrows a bit of both: you're picking over worlds first, via Turing machines, but then also picking over observers within worlds. The complexity penalty on observers means that we're still primarily picking based on worlds: adding more observers to a world doesn't help much if they don't emerge "naturally" from a simpler representation of the world; but we can still consider alternate worlds with different observers.
Sorry to add to an already long digression, but one more thought occurs to me:
One reason I think people's intuitions on SIA vs SSA for anthropic reasoning get pulled in different directions is that: if SSA is thinking about uncertainty over what world you're in, and SIA over what observer you are, then depending on the framing of the problem we may feel inclined to think that our uncertainty is more a matter of world vs observer: I think this is why "expect like a thirder, believe like a halfer" feels right for SB: predicting a coin feels like a "world-like" question.
But when it comes to questions about why the world is the why it is and fine tuning etc, it's much harder to even know how to factor our uncertainty into these components.
Thanks for the thoughtful engagement! I'm glad you've backed off the "this is simple; SSA is straightforwardly right" angle.
If you're still interested in talking about it, I'd enjoy getting to hear a bit more about your model of the telekinesis problem. In particular, I'm confused about "it doesn't make sense to model your own actions as a random variable." Notably, the telekinesis result still works if you hook up a machine that sterilizes you iff the coin comes up heads. After you've hooked up the machine, but before you flip the coin, does it seem right to say that whether you'll be sterile after the flip is random? Put another way, I don't think you need to make any random decisions to get telekinesis; you only need to tie the existence/non-existence of future people to a random outcome.
ETA: Also, am I right that if you're Adam, and *Eve* is like "check this out; I can make the coin come up heads by telekinetically committing to only have kids if it doesn't" you would say it makes sense to predict heads with ~100% probability?
Well, I still think SSA is straightforwardly right in the uncomplicated Sleeping Beauty problem, but I see how I overstated my case. It's also clear though that, for all I thought I read, I clearly had not covered all my bases. Those are the hazards of writing outside my core expertise, I suppose.
Anyway, to your point, I think the best I can do at this point is think aloud. Since you've stated that Eve's actions are actually independent of her choice let's simplify for a moment and remove her from the equation entirely. Let's say you have a universe that has at least N people. After the Nth person, a coin is flipped and then if it heads, no more people are created. If it's tails, then we create another 10000N people.
Conditioned on the knowledge you are one of the first N people, should you assign very low credence that the coin came up heads? I think that, to the extent it makes sense to assign probability to observers at all, the answer would have to be yes.
And if the answer is "No," which it could be, then I think the solution is that the problem is undefined, or at least incorrectly defined by either SSA or SIA.
One thing that keeps hovering in the back of my mind for all of this, however, is that the concept of probability itself is not always this super well-defined, deterministic thing. We're using probabilities here to represent ratios of outcomes of certain processes, and the issue we're running into is how to fit a situation (you are randomly selected among observers) into a process where we can then compute ratios. It's entirely possible that there is no meaningful way to do that.
Which I guess makes some of what I wrote in my essay wrong, or at least only right conditional on the reasonableness of selecting observers randomly.
Seems like we're converging. I don't think it makes sense to allocate probability to observers, and I think both SSA and SIA are confused in thinking that "indexical probability" is a thing. My position (which I attempt to argue in my blog post, linked in my root comment) is that there's no truth about which instance you are across the multiverse, and what we actually allocate is something more like caring than like probability mass.
Hmm... I've always felt that 1/3 is the only correct answer. Every time I read through arguments for 1/2, it has just sounded like a logical fallacy to me. But this helps somewhat I think. (although I may need to read through it again more slowly to fully follow some of your logic).
My argument for 1/3 boils down to 2 main facts:
#1: It doesn't make sense (at least in the SB scenario) to believe one thing but bet as if a different thing is true. That seems like cognitive dissonance, or hypocracy to me. Even if there are cases where there would be a disconnect, I don't see any reason to think that would apply here. For example, whether someone is willing to pay $100 to possibly win a billion dollars seems very irrelevant.
#2, and more importantly: the probability of any single occurrence of an event happening is, by definition, always equal to the long-term expected value of [ # of times it occurs / total # of trials ].
You don't seem bothered by #2 at all, but I'm not sure why:
> The first thing I’ll say is that I disagree the Sleeping Beauty problem corresponds to either of those scenarios, and the reason I disagree is that in the SB problem the coin isn’t being flipped over and over again - it’s being flipped once.
This just seems like a gross violation of the basic foundations of probability to me. When I say that the odds of heads coming up is 1/2 what that means is literally that "if I flip the coin a large enough number of times, the ratio of heads to total flips will approach 1/2". That's a basic axiom, I don't see how there is any way to define probability without that.
Also, there are at least 2 major theories of physics which point strongly towards a multiverse where different possible outcomes all happen at once: quantum mechanics, and inflation. In both cases, it's difficult to get any sensible explanation for the experimental results without concluding that we live in a multiverse. Quantum mechanics in particular seems like a "smoking gun" to me for this. So even if you did only 1 trial, you can be sure that was repeated in many other universes where the opposite outcome happened.
The main thing I do agree with in your arguments against 1/3'ers is that just making up any nonsense and saying "maybe it's possible there are many many people in some alternate universe" should have little to no persuasive power or influence on what odds you give something. But this is not the fault of the SIA, it's just a basic fact about Bayesian probability: it's only ever on a solid footing when you have a good idea what the prior distribution is. If it's just some wishy-washy possibility someone made up and there is no rigorous way to compare the a priori likelihood of that and something else then Bayesian probability is useless. You should never use it in that case. It can only be used in a rigorous framework (such as Many Worlds) where you know exactly what the prior distribution looks like. Anything else is always going to be pseudo-science... and IMO that applies just as well to SSA as it does to SIA, if not more.
>> "It doesn't make sense (at least in the SB scenario) to believe one thing but bet as if a different thing is true"
It's a minor point, but I think this is wrong; every bet is implicitly a bet on, not just the matter under discussion, but also the resolution criteria of the bet, the ability of the counterparty to pay, the value of the winnings at the time of resolution, etc.
Usually these are taken to be either trivial considerations, or at least modelable separately, but there are absolutely cases where accounting for these effects means your beliefs and what you're willing to bet can come apart: a nice, fun example is this (https://ericneyman.wordpress.com/2025/03/24/will-jesus-christ-return-in-an-election-year/) discussion of why some people on Polymarket are buying "yes" for the market, "will Jesus Christ return this year?"
The explanation he lands on is: "no" bettors are likely to want their money back to bet on other things before the year is out, and so will sell their "no" contracts to free up money--the people buying yes aren't betting on "will Jesus return at the end of the year", they're "really" betting on, "will the people betting against Jesus returning at the end of the year want to cash out their position before the year ends"--their bet on Jesus doesn't actually reflect their beliefs on Jesus, it reflects their beliefs on Jesus + their beliefs on the market for betting on Jesus.
I think you can argue that SB is in a similar situation: she's not betting on her belief that the coin is heads, she's betting on her belief the coin is heads + her belief that she has two opportunities to bet if the coin is tails.
For #1 - The reason is because you are betting on a different value than what we are actually trying to guess on. The issue is that each wake up is not equally likely. Take a look at the urn interpretation - if you are in a universe with one person, you have a 100% of being that person, whereas if you are in a universe with two people, you have a 50% of being either one.
Remember that expectation weights each value by the odds of achieving that value. You are treating each wake up as of equal odds, which is fine if you’re sleeping beauty waking up twice and getting paid equally each time. But if you’re asking why our universe is fine-tuned - as in, if you are modeling our world and not just playing the SB game - then you don’t get to wake up more than once, so you have to weight the odds of each wake up by the fact that there are more people you might be.
For #2 - I think I tried to fit in too much and wound up under explaining a lot.
Let’s say I have a fair coin and assign 0 to heads and 1 to tails. The average of multiple trials will converge to 0.5. A single trial will, with 100% probability, always be either 0 or 1. In the urn example, where you flip one coin and then place in balls of a single color, if you do a single trial (that is, if you flip a single coin), there is a 50% chance you will draw a green ball, and a 50% you will draw a red ball. Every single time.
If you flip two coins before pulling a ball out of the urn, then your odds of pulling out a green ball will be (1/4 * 1) + (1/2 * 1/3) + (1/4 * 0) = 5/12, since there is a 25% chance you got two heads, a 50% chance you got one heads (leaving 1/3 of the balls green), and a 25% chance you got no balls. As you perform more flips, the number of balls in the urn will be higher, and your odds will converge to 1/3 relatively quickly. But if there is only one flip, your odds of green are always 50%.
As for the multiverse and quantum mechanics, you are partially right, but it all depends on where you’re drawing the line for what defines a universe. If the question is “why are we in this branch of the quantum waveform,” then it makes sense to model each branch as a separate universe. But if it’s “is god necessary to explain our existence?” then any multiverse posited by physics probably ought to be considered a single “universe,” with “god” and other hypotheses relegated to separate universes as well.
I actually remembered another reason to be careful about separating your epistemic probability from your betting probability:
One way to think about how you'd end up a thirder is: ok, heads and tails are equally likely, so there's a 1/2 chance that heads Monday/heads happens. It's also a 1/2 chance that each of Tuesday/heads, Tuesday/tails happens since each of Monday/Tuesday happens with probably 1 conditional on tails.
So you have three possibilities, each equally likely to happen, so 1/3 to each; thus 1/3 chance that it's Monday/heads, and since that's the only case where heads occurs, 1/3 chance of heads.
But note how the calculation started: by observing p(H)=p(T)! If you now think p(H)=1/3, shouldn't you go back and redo the calculation with the correct value? In which case, p(MH)=1/3, p(MT)=p(TT)=2/3, so you have three possibilities where two of them are each twice as likely as the third, so the probability of that one should be 1/5.
So p(H)=1/5, actually! But wait, that means we used the wrong p(H) in our calculation....
It seems like, somehow our conclusion that p(H)=1/3 isn't meant to be used in the same way we initially used p(H)=1/2 to get the ball rolling; somehow that probability assignment remains privileged in a way 1/3 (and 1/5 and...) aren't.
gah wrote a long comment but substack deleted it when i accidentally swiped off.
I'm a thirder who believes in SIA, but I acknowledge that I have no idea what to do when infinity gets involved. I used to be a double halfer like you seem to be, but I was convinced over. Obviously I disagree that P(A)=P(A | H)=1, because A should be "this very awakening occurs", which will lead to the standard P(H | this awakening) Bayes equation that I'm sure you've seen thirders use before in their papers.
The interesting thing with Sleeping Beauty problem is that indeed anthropic reasoning is very odd and strange, and it has wide implications. I'll give one of the standard thirder vibes-based interpretation instead of getting too bogged down in the Bayes:
If heads, woken up once. If tails, woken up a million times in a row, mind erased each time, of course.
I can vividly imagine waking up, unsure of the day, and being asked whether I think the coin came up heads, and thinking of the potential thousands, tens of thousands, hundreds of thousands of people with my same conscious experience that came before or will come after me that were asked the same question. I consider what they felt -- were they unsure? Groggy? tired? did one of them see a butterfly? did a hundred of them see the same bug in the corner of the room that I see? And I think thinking of them each as individual perspectives with individual, unique experiences is best. If I’m just one “observer-moment” plucked out of that huge sea of near-duplicates, the mere fact that I find myself awake at all is way better explained by the scenario that provided a million of these unique experiences than a scenario which provides a single one.
There are a bunch of things here that I think are incorrect:
1) You assume that your evidence in the sleeping beauty is that you're awake. But that's not your evidence. Your evidence is "I'm awake now." Given that now might be day 2, in which case it's only true on tails, that should update you in favor of tails (see here for a parallel case https://benthams.substack.com/p/weintraub-disproves-the-main-halfer?utm_source=publication-search). This is what you're supposed to update on, not you being likelier to wake up on heads.
4) You note SIA has issues with infinity, but so does every view. On SSA, it's similarly unclear how to partition probabilities once there are infinite people. Now, regarding the maximum number problem, there are two ways to go:
a) you could accept that theories on which a larger portion of possible people exist are likelier to be true, not higher numbers overall. Them. SIA will just take for granted the number of possible people you independently thought there were, and instruct you to say that many people is actual.
b) you could accept the number of people is the most possible--too many to be a set.
If you wake up regardless of what happens, then P(A) (the probability you wake up) will be 1 and the probability of waking up in a specific world, such as P(A|H) is also 1. The math just means "there's no world where you don't wake up."
(To be clear, I'm not defending that statement. For example, one could argue that it's equivocating between waking up the first time vs waking up the second time. But I think the math is fine, if you agree with the premise that "waking up" is an event that happens in both timelines.)
Well, not really. 1 / (# of time points) would, by definition, be your probability of waking up at a given time.
To Realifin’s comment, the probability of waking up the first time would not be the probability of waking up, it would be, tautologically, the probability of waking up the first time. Your probability of waking at all would be the sum of all the probabilities of waking.
The idea here is that in all scenarios, there is a number of times you wake up, and though you may not know which one you are, you know you must be one of those. If you select over possible observers, then by mathematical necessity (as explained in the warehouse section) you will have probability of being in a time you *don’t* wake up. This doesn’t really make sense, because there’s no physical analog to the statement “the existence of your non-existence.”
I think one way to think about this is, from the perspective of SB herself, "how many time points are there" is... not obviously well-defined; this is a point that noted BB-antagonist Ape in the Coat regularly makes.
If, from SB's perspective, the two awakenings are identical, to some extent *for her* there is really only one awakening.
Externally, there remain of course two different wakings, but maybe we should think of what's happening as SB losing a shared time frame of reference with the external world: two moments of external time map to one moment of internal time.
In some ways, this is analogous to SB being in a quantum superposition or something.
I think two (compelling) arguments against this are: it's implausible for the two awakenings to be *actually* identical, and it's implausible for her reference frame to lose all contact with external ones even while external observers can agree on assigning her moments to their reference coordinates.
So in practice, I'm a thirder. But the calculation done by ksvanhorn I referenced in an earlier comment makes precise the fact that "how distinguishable the two awakenings are" is a parameter that affects SB's probability, ranging from 1/2 if indistinguishable to 1/3 in the limit of SB gaining more information that can in principle distinguish the two awakenings.
I think this lends support to the halfer side that, defining the number of awakenings is not trivial, and availing oneself of the external reference frame to do so is a move that, while almost certainly justifiable in "practice", still depends on a bunch of unexamined assumptions.
I'll also note, that if you take the "SIA and possible people" framing, it suggests that "how many possible people there are" may be reference frame dependent?! To me, this makes it feel like, to the extent that SIA is right, it's right for a reason that isn't well-articulated by the simple justifications given for it.
I would love to see an implementation of this in python to really FEEL the difference between thirders and halfers. I find the Monty Hall problem much more intuitive when presented in the necessarily constructive manner of code.
I had considered adding something like that. Here's an attempt I haven't actually run, so apologies for any typos. You can see that in some sense the process is switched. In SIA, the observers exist first, in SSA the coin flip happens first:
(It seems the formatting is a little mangled, actually. I may see if I can post elsewhere and link)
I see! Perhaps it’s not that the observers necessarily exist first, but the question boils down to whether to apply the principle of indifference to the possible *worlds* or the possible *awakenings*, yes?
On 2), I think this is why UDASSA seems promising: if your addition of new people is truly *arbitrary*, ie, doesn't emerge as a consequence of some simpler assumption, then you need to impose a complexity penalty on those people, that limits how much extra probability you put on those people existing. I haven't found it, but there's some LW posts discussing this, where I think someone argued that using a simple model, you only get a constant advantage to just adding more people, if you use Solomonoff priors to penalize adding more people.
Thanks for writing this! Some of it resembles my stance on the topic, particularly around "expect like a thirder, believe like a halfer." (https://utopiandreams.substack.com/p/anthropic-reasoning)
I'm curious if you encountered/read Joe Carlsmith's essays arguing for SIA > SSA.
https://joecarlsmith.com/2021/09/30/sia-ssa-part-1-learning-from-the-fact-that-you-exist
He presents a bunch of arguments that SSA leads to insane conclusions there that I don't see you engaging with. (I can be more specific if that would be helpful.)
Having read about half of the Joe Carlsmith article so far, I have to say it feels like his approach seems to be going in a direction I don’t fully agree with. I'll continue reading it, but would certainly appreciate if you could point me to the specific objections you'd like me to address.
I believe the correct way to model the SB problem, and anthropics in general, is that we exist according to a process:
1. A universe is created
2. Observers are "placed" in this universe, as a result of the universe's inherent processes
3. You are "assigned" to one of these observers at random
Under this paradigm, the result is mathematically well defined and none of the issues that I've seen him list so far seem to exist. If you insist on declaring some kind of reference class, then it would be the class of observers - people who are capable of asking questions about anthropic reasoning. But it's not like you wouldn't also need to define this exact same class of observers in the SIA, so there's no special disadvantage to using it with the SSA.
I'll read in more detail and give another comment later, but this is what I saw in the short amount of time I had to look at it right now.
So, suppose you don't know how many observers will come to exist, but you seem to be early and have control over how many observers come to exist. (A classic example is "you are Adam or Eve, and can unilaterally decide not to have kids, meaning only two people will ever exist, or to reproduce, in which case there will at least be hundreds of billions".) Do you agree that SSA says you should be much more surprised to be "early" if lots of people come to exist? If so, do you think you can cause a coin to come up heads by committing to keep the population low iff it does? (This is described as "telekinesis" by Carlsmith.)
I'm really appreciating the Carlsmith series - thanks for the link. It's helping me see several cases I hadn't considered, though I haven't been convinced all of them are relevant. I have too many thoughts on the example you gave to fit into one comment, so I'll do my best to cover the important bits.
Upfront, I think the right way to model things is not necessarily SIA or SSA - it's to figure out the model that actually matches the process you care about, and then model that, approximating where necessary. In the absence of testable predictions, this is our only hope. This is not a universally applicable approach, and when it isn't applicable I don't necessarily take that to mean we should take a different inapplicable approach just because we can.
In the case you gave, the issue is that if you are Adam or Eve your decision is by construction nonrandom. As such, it is impossible to model it as a random variable, and any function of that variable cannot be truly random.
Of course, if we were anyone besides Adam then we could model Adam/Eve's decision as a random variable and the math would be fine. But if you *are* Adam then that's not a reasonable approach, and once the universe you're in and/or the number of people within it stop being random, probability stops being meaningful.
I also reject the idea that this failure somehow makes the SIA more reasonable. Applying the SIA still requires you to assume the existence of people who don't exist, and that simply doesn't match the process we're trying to model.
This may just be a case where our tools are too limited to give us the answers we want. Other such cases exist - consider the measure problem in cosmology. Either way, definitely something I will continue to think about, thanks again for the recomendation.
>>"Upfront, I think the right way to model things is not necessarily SIA or SSA"
Carlsmith gestures at this at the end, with his discussion of UDASSA, which I think is a step in the right direction despite the paradoxes. Personally, I think something like this--but with more of the deflationary attitude you took in your Tegmark post, when it comes to thinking about the output of Turing machines as "real"--feels promising to me.
On Sleeping Beauty, I have both halferish and thirderish instincts: I think halferism is right that we're trying to predict *the world* not the *observer moment*; possible worlds with lots of observers that never actually exist shouldn't count towards our expectation.
On the other hand, Radford Neal's Fully Non-Indexical Conditioning, and a calculation inspired by it done by ksvanhorn, convinced me that you can still get 1/3 as a reasonable answer even without the metaphysically dubious stuff that SIA seems to carry with it.
An interesting takeaway from ksvanhorn's calculation is that you can model it so that SB should predict any number from 1/2 to 1/3 as an asymptote, as a function of how much information SB has observed and how the probability distributions governing the info she can observe differs between the two awakenings.
Finally, on the weird metaphysics, I also don't like that SSA has some bad consequences too: aside from the determinism stuff mentioned in the comments above, there's also Carlsmith's observation that SSA has a bit of a vibe of, "since this is a world where I exist, and we're doing our expectation only over people in actual worlds, I *must* exist"---which strikes me as equally bad as the "all possible people exist" of SIA.
I think one of the things that appeals to me about UDASSA or something like it is that it is borrows a bit of both: you're picking over worlds first, via Turing machines, but then also picking over observers within worlds. The complexity penalty on observers means that we're still primarily picking based on worlds: adding more observers to a world doesn't help much if they don't emerge "naturally" from a simpler representation of the world; but we can still consider alternate worlds with different observers.
Sorry to add to an already long digression, but one more thought occurs to me:
One reason I think people's intuitions on SIA vs SSA for anthropic reasoning get pulled in different directions is that: if SSA is thinking about uncertainty over what world you're in, and SIA over what observer you are, then depending on the framing of the problem we may feel inclined to think that our uncertainty is more a matter of world vs observer: I think this is why "expect like a thirder, believe like a halfer" feels right for SB: predicting a coin feels like a "world-like" question.
But when it comes to questions about why the world is the why it is and fine tuning etc, it's much harder to even know how to factor our uncertainty into these components.
Thanks for the thoughtful engagement! I'm glad you've backed off the "this is simple; SSA is straightforwardly right" angle.
If you're still interested in talking about it, I'd enjoy getting to hear a bit more about your model of the telekinesis problem. In particular, I'm confused about "it doesn't make sense to model your own actions as a random variable." Notably, the telekinesis result still works if you hook up a machine that sterilizes you iff the coin comes up heads. After you've hooked up the machine, but before you flip the coin, does it seem right to say that whether you'll be sterile after the flip is random? Put another way, I don't think you need to make any random decisions to get telekinesis; you only need to tie the existence/non-existence of future people to a random outcome.
ETA: Also, am I right that if you're Adam, and *Eve* is like "check this out; I can make the coin come up heads by telekinetically committing to only have kids if it doesn't" you would say it makes sense to predict heads with ~100% probability?
Well, I still think SSA is straightforwardly right in the uncomplicated Sleeping Beauty problem, but I see how I overstated my case. It's also clear though that, for all I thought I read, I clearly had not covered all my bases. Those are the hazards of writing outside my core expertise, I suppose.
Anyway, to your point, I think the best I can do at this point is think aloud. Since you've stated that Eve's actions are actually independent of her choice let's simplify for a moment and remove her from the equation entirely. Let's say you have a universe that has at least N people. After the Nth person, a coin is flipped and then if it heads, no more people are created. If it's tails, then we create another 10000N people.
Conditioned on the knowledge you are one of the first N people, should you assign very low credence that the coin came up heads? I think that, to the extent it makes sense to assign probability to observers at all, the answer would have to be yes.
And if the answer is "No," which it could be, then I think the solution is that the problem is undefined, or at least incorrectly defined by either SSA or SIA.
One thing that keeps hovering in the back of my mind for all of this, however, is that the concept of probability itself is not always this super well-defined, deterministic thing. We're using probabilities here to represent ratios of outcomes of certain processes, and the issue we're running into is how to fit a situation (you are randomly selected among observers) into a process where we can then compute ratios. It's entirely possible that there is no meaningful way to do that.
Which I guess makes some of what I wrote in my essay wrong, or at least only right conditional on the reasonableness of selecting observers randomly.
Seems like we're converging. I don't think it makes sense to allocate probability to observers, and I think both SSA and SIA are confused in thinking that "indexical probability" is a thing. My position (which I attempt to argue in my blog post, linked in my root comment) is that there's no truth about which instance you are across the multiverse, and what we actually allocate is something more like caring than like probability mass.
Hmm... I've always felt that 1/3 is the only correct answer. Every time I read through arguments for 1/2, it has just sounded like a logical fallacy to me. But this helps somewhat I think. (although I may need to read through it again more slowly to fully follow some of your logic).
My argument for 1/3 boils down to 2 main facts:
#1: It doesn't make sense (at least in the SB scenario) to believe one thing but bet as if a different thing is true. That seems like cognitive dissonance, or hypocracy to me. Even if there are cases where there would be a disconnect, I don't see any reason to think that would apply here. For example, whether someone is willing to pay $100 to possibly win a billion dollars seems very irrelevant.
#2, and more importantly: the probability of any single occurrence of an event happening is, by definition, always equal to the long-term expected value of [ # of times it occurs / total # of trials ].
You don't seem bothered by #2 at all, but I'm not sure why:
> The first thing I’ll say is that I disagree the Sleeping Beauty problem corresponds to either of those scenarios, and the reason I disagree is that in the SB problem the coin isn’t being flipped over and over again - it’s being flipped once.
This just seems like a gross violation of the basic foundations of probability to me. When I say that the odds of heads coming up is 1/2 what that means is literally that "if I flip the coin a large enough number of times, the ratio of heads to total flips will approach 1/2". That's a basic axiom, I don't see how there is any way to define probability without that.
Also, there are at least 2 major theories of physics which point strongly towards a multiverse where different possible outcomes all happen at once: quantum mechanics, and inflation. In both cases, it's difficult to get any sensible explanation for the experimental results without concluding that we live in a multiverse. Quantum mechanics in particular seems like a "smoking gun" to me for this. So even if you did only 1 trial, you can be sure that was repeated in many other universes where the opposite outcome happened.
The main thing I do agree with in your arguments against 1/3'ers is that just making up any nonsense and saying "maybe it's possible there are many many people in some alternate universe" should have little to no persuasive power or influence on what odds you give something. But this is not the fault of the SIA, it's just a basic fact about Bayesian probability: it's only ever on a solid footing when you have a good idea what the prior distribution is. If it's just some wishy-washy possibility someone made up and there is no rigorous way to compare the a priori likelihood of that and something else then Bayesian probability is useless. You should never use it in that case. It can only be used in a rigorous framework (such as Many Worlds) where you know exactly what the prior distribution looks like. Anything else is always going to be pseudo-science... and IMO that applies just as well to SSA as it does to SIA, if not more.
>> "It doesn't make sense (at least in the SB scenario) to believe one thing but bet as if a different thing is true"
It's a minor point, but I think this is wrong; every bet is implicitly a bet on, not just the matter under discussion, but also the resolution criteria of the bet, the ability of the counterparty to pay, the value of the winnings at the time of resolution, etc.
Usually these are taken to be either trivial considerations, or at least modelable separately, but there are absolutely cases where accounting for these effects means your beliefs and what you're willing to bet can come apart: a nice, fun example is this (https://ericneyman.wordpress.com/2025/03/24/will-jesus-christ-return-in-an-election-year/) discussion of why some people on Polymarket are buying "yes" for the market, "will Jesus Christ return this year?"
The explanation he lands on is: "no" bettors are likely to want their money back to bet on other things before the year is out, and so will sell their "no" contracts to free up money--the people buying yes aren't betting on "will Jesus return at the end of the year", they're "really" betting on, "will the people betting against Jesus returning at the end of the year want to cash out their position before the year ends"--their bet on Jesus doesn't actually reflect their beliefs on Jesus, it reflects their beliefs on Jesus + their beliefs on the market for betting on Jesus.
I think you can argue that SB is in a similar situation: she's not betting on her belief that the coin is heads, she's betting on her belief the coin is heads + her belief that she has two opportunities to bet if the coin is tails.
That last paragraph is a great answer to his question
For #1 - The reason is because you are betting on a different value than what we are actually trying to guess on. The issue is that each wake up is not equally likely. Take a look at the urn interpretation - if you are in a universe with one person, you have a 100% of being that person, whereas if you are in a universe with two people, you have a 50% of being either one.
Remember that expectation weights each value by the odds of achieving that value. You are treating each wake up as of equal odds, which is fine if you’re sleeping beauty waking up twice and getting paid equally each time. But if you’re asking why our universe is fine-tuned - as in, if you are modeling our world and not just playing the SB game - then you don’t get to wake up more than once, so you have to weight the odds of each wake up by the fact that there are more people you might be.
For #2 - I think I tried to fit in too much and wound up under explaining a lot.
Let’s say I have a fair coin and assign 0 to heads and 1 to tails. The average of multiple trials will converge to 0.5. A single trial will, with 100% probability, always be either 0 or 1. In the urn example, where you flip one coin and then place in balls of a single color, if you do a single trial (that is, if you flip a single coin), there is a 50% chance you will draw a green ball, and a 50% you will draw a red ball. Every single time.
If you flip two coins before pulling a ball out of the urn, then your odds of pulling out a green ball will be (1/4 * 1) + (1/2 * 1/3) + (1/4 * 0) = 5/12, since there is a 25% chance you got two heads, a 50% chance you got one heads (leaving 1/3 of the balls green), and a 25% chance you got no balls. As you perform more flips, the number of balls in the urn will be higher, and your odds will converge to 1/3 relatively quickly. But if there is only one flip, your odds of green are always 50%.
As for the multiverse and quantum mechanics, you are partially right, but it all depends on where you’re drawing the line for what defines a universe. If the question is “why are we in this branch of the quantum waveform,” then it makes sense to model each branch as a separate universe. But if it’s “is god necessary to explain our existence?” then any multiverse posited by physics probably ought to be considered a single “universe,” with “god” and other hypotheses relegated to separate universes as well.
I actually remembered another reason to be careful about separating your epistemic probability from your betting probability:
One way to think about how you'd end up a thirder is: ok, heads and tails are equally likely, so there's a 1/2 chance that heads Monday/heads happens. It's also a 1/2 chance that each of Tuesday/heads, Tuesday/tails happens since each of Monday/Tuesday happens with probably 1 conditional on tails.
So you have three possibilities, each equally likely to happen, so 1/3 to each; thus 1/3 chance that it's Monday/heads, and since that's the only case where heads occurs, 1/3 chance of heads.
But note how the calculation started: by observing p(H)=p(T)! If you now think p(H)=1/3, shouldn't you go back and redo the calculation with the correct value? In which case, p(MH)=1/3, p(MT)=p(TT)=2/3, so you have three possibilities where two of them are each twice as likely as the third, so the probability of that one should be 1/5.
So p(H)=1/5, actually! But wait, that means we used the wrong p(H) in our calculation....
It seems like, somehow our conclusion that p(H)=1/3 isn't meant to be used in the same way we initially used p(H)=1/2 to get the ball rolling; somehow that probability assignment remains privileged in a way 1/3 (and 1/5 and...) aren't.
gah wrote a long comment but substack deleted it when i accidentally swiped off.
I'm a thirder who believes in SIA, but I acknowledge that I have no idea what to do when infinity gets involved. I used to be a double halfer like you seem to be, but I was convinced over. Obviously I disagree that P(A)=P(A | H)=1, because A should be "this very awakening occurs", which will lead to the standard P(H | this awakening) Bayes equation that I'm sure you've seen thirders use before in their papers.
The interesting thing with Sleeping Beauty problem is that indeed anthropic reasoning is very odd and strange, and it has wide implications. I'll give one of the standard thirder vibes-based interpretation instead of getting too bogged down in the Bayes:
If heads, woken up once. If tails, woken up a million times in a row, mind erased each time, of course.
I can vividly imagine waking up, unsure of the day, and being asked whether I think the coin came up heads, and thinking of the potential thousands, tens of thousands, hundreds of thousands of people with my same conscious experience that came before or will come after me that were asked the same question. I consider what they felt -- were they unsure? Groggy? tired? did one of them see a butterfly? did a hundred of them see the same bug in the corner of the room that I see? And I think thinking of them each as individual perspectives with individual, unique experiences is best. If I’m just one “observer-moment” plucked out of that huge sea of near-duplicates, the mere fact that I find myself awake at all is way better explained by the scenario that provided a million of these unique experiences than a scenario which provides a single one.
Are you familiar with Wei Dai's paradox? (https://www.lesswrong.com/posts/QmWNbCRMgRBcMK6RK/the-absolute-self-selection-assumption?commentId=QyYS5mWQrTLQMp4w4)
(Also discussed here: https://utopiandreams.substack.com/p/anthropic-reasoning)
I'm curious if you have a sense of how SIA deals with it.
There is a pretty big distinction between Lewisian halfling and double halfing.
Double halfing (which this post seems to argue for) doesn’t support the SSA and its conclusions like the DA or anthropic shadow.
There are a bunch of things here that I think are incorrect:
1) You assume that your evidence in the sleeping beauty is that you're awake. But that's not your evidence. Your evidence is "I'm awake now." Given that now might be day 2, in which case it's only true on tails, that should update you in favor of tails (see here for a parallel case https://benthams.substack.com/p/weintraub-disproves-the-main-halfer?utm_source=publication-search). This is what you're supposed to update on, not you being likelier to wake up on heads.
2) We don't need a warehouse of souls but instead just uncertainty about which of the possible people you are https://benthams.substack.com/p/elga-proved-13?utm_source=publication-search and see https://benthams.substack.com/p/precisely-defining-the-self-indication?utm_source=publication-search
3) I think betting does favor SIA pretty strongly but the situation is complicated https://benthams.substack.com/p/money-pump-arguments-destroy-non?utm_source=publication-search
4) You note SIA has issues with infinity, but so does every view. On SSA, it's similarly unclear how to partition probabilities once there are infinite people. Now, regarding the maximum number problem, there are two ways to go:
a) you could accept that theories on which a larger portion of possible people exist are likelier to be true, not higher numbers overall. Them. SIA will just take for granted the number of possible people you independently thought there were, and instruct you to say that many people is actual.
b) you could accept the number of people is the most possible--too many to be a set.
Want to come on the podcast to discuss?
"using Bayes Rule and the fact that you wake up at least once in each scenario, meaning P(A) = P(A|H) = 1"
You've lost me here - why does the fact you wake up at least once in each scenario mean P(A) = P(A|H) ?
If you wake up regardless of what happens, then P(A) (the probability you wake up) will be 1 and the probability of waking up in a specific world, such as P(A|H) is also 1. The math just means "there's no world where you don't wake up."
(To be clear, I'm not defending that statement. For example, one could argue that it's equivocating between waking up the first time vs waking up the second time. But I think the math is fine, if you agree with the premise that "waking up" is an event that happens in both timelines.)
Thanks. That's helped me work out what I disagree with.
The probability that you wake up isn't 1. It's 1 / (how many time points you're measuring).
Well, not really. 1 / (# of time points) would, by definition, be your probability of waking up at a given time.
To Realifin’s comment, the probability of waking up the first time would not be the probability of waking up, it would be, tautologically, the probability of waking up the first time. Your probability of waking at all would be the sum of all the probabilities of waking.
The idea here is that in all scenarios, there is a number of times you wake up, and though you may not know which one you are, you know you must be one of those. If you select over possible observers, then by mathematical necessity (as explained in the warehouse section) you will have probability of being in a time you *don’t* wake up. This doesn’t really make sense, because there’s no physical analog to the statement “the existence of your non-existence.”
I think one way to think about this is, from the perspective of SB herself, "how many time points are there" is... not obviously well-defined; this is a point that noted BB-antagonist Ape in the Coat regularly makes.
If, from SB's perspective, the two awakenings are identical, to some extent *for her* there is really only one awakening.
Externally, there remain of course two different wakings, but maybe we should think of what's happening as SB losing a shared time frame of reference with the external world: two moments of external time map to one moment of internal time.
In some ways, this is analogous to SB being in a quantum superposition or something.
I think two (compelling) arguments against this are: it's implausible for the two awakenings to be *actually* identical, and it's implausible for her reference frame to lose all contact with external ones even while external observers can agree on assigning her moments to their reference coordinates.
So in practice, I'm a thirder. But the calculation done by ksvanhorn I referenced in an earlier comment makes precise the fact that "how distinguishable the two awakenings are" is a parameter that affects SB's probability, ranging from 1/2 if indistinguishable to 1/3 in the limit of SB gaining more information that can in principle distinguish the two awakenings.
I think this lends support to the halfer side that, defining the number of awakenings is not trivial, and availing oneself of the external reference frame to do so is a move that, while almost certainly justifiable in "practice", still depends on a bunch of unexamined assumptions.
I'll also note, that if you take the "SIA and possible people" framing, it suggests that "how many possible people there are" may be reference frame dependent?! To me, this makes it feel like, to the extent that SIA is right, it's right for a reason that isn't well-articulated by the simple justifications given for it.
I would love to see an implementation of this in python to really FEEL the difference between thirders and halfers. I find the Monty Hall problem much more intuitive when presented in the necessarily constructive manner of code.
I had considered adding something like that. Here's an attempt I haven't actually run, so apologies for any typos. You can see that in some sense the process is switched. In SIA, the observers exist first, in SSA the coin flip happens first:
(It seems the formatting is a little mangled, actually. I may see if I can post elsewhere and link)
```
# Halfer (`all` will be 50% H, 25% T1 or T2):
all = Counter()
for _ in range(10000):
if random.choice(['H', 'T']) == 'H':
observers = ['H']
else:
observers = ['T1', 'T2']
self = random.choice(observers)
all[self] += 1
```
--------------------
--------------------
--------------------
```
#Thirder (`all` will be 1/3 H, T1, T2)
all = Counter()
for _ in range(10000):
possible_observers = ['1', '2']
if random.choice(['H', 'T']) == 'H':
if random.choice(possible_observers) == '1':
self = 'H1'
else:
# observer 2 never wakes up
continue
else:
# Coin was tails - you can be either observer
self = 'T' + random.choice(possible_observers)
all[self] += 1
```
I see! Perhaps it’s not that the observers necessarily exist first, but the question boils down to whether to apply the principle of indifference to the possible *worlds* or the possible *awakenings*, yes?
Yes, with a “but.” Indifference to possible awakenings mathematically requires you to assign probability mass to not waking. This means:
1. You need to explain the physical analog to “an observer who exists mathematically but not in reality.”
2. I can make up worlds with arbitrarily low probability but arbitrarily more people, and those are automatically more likely.
On 2), I think this is why UDASSA seems promising: if your addition of new people is truly *arbitrary*, ie, doesn't emerge as a consequence of some simpler assumption, then you need to impose a complexity penalty on those people, that limits how much extra probability you put on those people existing. I haven't found it, but there's some LW posts discussing this, where I think someone argued that using a simple model, you only get a constant advantage to just adding more people, if you use Solomonoff priors to penalize adding more people.
> add another narcoleptic princess - let’s call her “Snow White.”
Wait, Snow White is not narcoleptic!