Note that "Being counterfactually dead is just like counterfactually making a different observation" can also be derived from the Ghosts are as good as gone principle! By reinterpreting the decision tree you can treat death like a normal observation.
If there are two theories: there is finite number of actual observers in the universe and there is infinite number of them which includes all possible observers - then SIA strongly prefer infinite version.
However, we can use "I exist" for update only once. Now if someone asks second question - which of two theories is more likely, there is one civilization per galaxy on average and there are 100 civilizations per galaxy, there is no update from SIA. So SIA is a used ticket. However, we can still used SSA if we define exact reference class like: here is two galaxies and one is 10 times larger than another, there I am more likely to be?
tl;dr: in situations without duplicates:
Basic anthropic reasoning
Anthropic reasoning is applied to many things, such as trying to figure out where the specific life-compatible properties of the universe come from, estimating the probability of alien life, estimating the chance of humanity dooming itself and whether we escaped past dangers through fluke or because the risks were low.
That’s a lot of things for a theory to cover! Here I’ll dig deeper into anthropic probabilities and write up some past results and new ones.
The first post will focus on “basic” anthropic problems. These are problems where we have no uncertainty as to which agent we are (and therefore there’s only a single copy of us). So Great Filter, origin of life, Doomsday argument, Sailor’s child: in. Sleeping Beauty: out.
The subsequent post will deal with the much more subtle cases where identical copies do exist.
Because the universe looks like it’s infinite in extent, we expect that there will be infinitely many copies of us across the universe. So a basic anthropic problem has to be local: it only applies to, e.g. the observable universe, which is too small to be likely to contain any duplicates.
The thesis of this post is that in basic anthropic problems, anthropic reasoning is very simple: it updates in the same way as making standard non-anthropic Bayesian updates. This kind of updating matches up with SIA (the Self-Indication Assumption). However, as the end of the post will show, SIA generally doesn’t have the counter-intuitive consequences that people commonly assume it does (at least, not in basic problems); for instance, it doesn’t particularly favour large populations. A lot of the counter-intuitive arguments in anthropics (such as the Presumptuous Philosopher) really come from the choice of prior.
We can establish the Bayesian-SIA equivalence in a couple of lines:
Of course, showing that standard Bayesian updating and SIA are the same thing does not demonstrate that they are correct. To show that they actually are correct, I’ll first define what a non-Bayesian anthropic update would look like. Then I’ll show mild conditions that demonstrate you can’t deviate from Bayesian updating in your own life, even when facing issues of death or counterfactual death. Then I’ll extend that result to events that could have destroyed our civilisation or life on Earth. To conclude the anthropic update argument, I’ll extend it to events leading up to the creation of life.
The end of the post will show how SIA actually behaves in basic anthropic situations (it’s not as weird as you may think), and show how the choice of prior determines so much of anthropic reasoning.
Being counterfactually dead is just like counterfactually making a different observation
The fundamental intuition is that it doesn’t matter if the counterfactual you in another branch of the universe is dead, or just made a different observation. In both cases, those histories are dead to you: they don’t exist, they aren’t the current you, so you can write them off equally.
This section will aim to establish that result (conditional on some reasonable premises) and hence justify the intuition.
What we mean by a non-Bayesian anthropic update
Here is an anthropic argument I’ve heard and made:
This seems to be saying that
It seems to be implying that the risk rate in the Cold War was higher than our survival indicates – we just got very lucky. In standard Bayesian updating, “luck” can be located in two places: we could have got lucky because we survived a very risky situation, or we could have got lucky because the situation was less risky than we thought. An intuitive example: if we roll five 6s in a row, we could be lucky on a fair die, or we could be lucky enough to have a loaded die.
Bayesian updating can use the rest of the evidence and the prior to compute how much luck went into each factor: we compute
The “Cold War Danger” argument is that
Generalising this, I’ll define an anthropic non-Bayesian update as an update that gives values
Note that, importantly: updating on non-anthropic evidence is standard Bayesian updating. We will be investigating whether there can be any extra anthropic updating beyond the standard Bayesian updating, in anthropic situations.
Predictable loss or inaccuracy arguments
Dutch book arguments can’t be used in anthropics because in one branch, there is no “you” to have the gain or loss [2] .
Joyce’s accuracy argument does work in anthropic settings. If an agent deviates from Bayesian updating, then you can set up a situation where another agent is more accurate in every branch of the probability tree.
However, those arguments don’t get to the core of the intuition of what’s going on; and one could always say that they force behaviour, not beliefs. So let’s lay them aside for now.
Different Divergent Roads
Consider the following setup:
Immediately I concluded that the booby-trapped road must have been road B, the one I didn’t take. Because if road A had been booby-trapped, I wouldn’t be there to talk with my friend. But consider instead:
I would similarly have concluded that it must have been road B – if road A had been blocked, I would not have made it through to talk with my friend either.
I would also have concluded that it must have been road B – if road A had been thus painted, a version of me with no memory of fluorescent pink would not be standing talking to my friend either.
We’ll aim to show that those three situations are identical from a probability perspective and should be treated identically.
Counterfactually, I may have seen death coming
In all three models – deadly, blocked, coloured – let the road with the odd property be called the tampered road, and the other road the untampered road. Assume that with probability
Also assume that death is not instantaneous from the booby trap: the counterfactual me gets to see their death coming.
At 8:00 I started my journey; at 12:00, if I took an untampered road, I would have emerged from the woods.
If I took the tampered road, then at 10:00 exactly I would have started to explode/hit the roadblock/seen the garish pink. So at 10:00 me and my counterfactual version both exist and know precisely which road is tampered.
Since I am not yet dead or counterfactually dead at 10:00, I must follow standard Bayesian updating, which is identical in each Divergent road setup. And the information at 10:00 is everything I see which distinguishes tampered from non-tampered. Thus, once I’ve seen that the road is non-tampered, I will update on further observations in a non-anthropic way [3] .
To get this result in its full generality, we need two assumptions:
So, by assumption 1, I should make standard Bayesian updates the same way in anthropic and non-anthropic variants of the same situation. By assumption 2, I should do this, whether I found out about the anthropic nature of the problem before or after getting other evidence of which world I am in.
Assumption 1 is probably the most load-bearing assumption in this post. So consider what denying it might entail. It would mean that it is very important for my current probabilities that I know exactly how my counterfactual self would have died. And their epistemic state before that. Such as whether they developed suspicions on the way; had they seen the booby-trap and wondered about it before triggering it, this changes my probability. Maybe there was enough evidence lying about that they could have deduced the existence of the booby-trap, ahead of time, if they had been fully rational. I would have to start thinking about how information propagates through a dying or exploding brain: is there any part of me that could be considered a rational agent, before disintegration, that could have picked up on me dying? It would entail conversations like this:
Me: “Hey folks, sorry to say this, but we are doomed!”
Crime scene investigator: “Actually, I found that the booby-trap from ten years ago would have just been visible from the previous turn in the road you didn’t take.”
Visiting biologist: “On top of that, recent research has shown that neurones maintain their connections for a pico-second even in the face of destructive explosions.”
Me: “Phew! Ignore what I said before, everything is probably fine.”
Moving back through the generations
Let’s jump back to before most of us were born: 27 October 1962, during the Cuban missile crisis. There, Vasily Arkhipov prevented his submarine from firing a nuclear torpedo and probably stopped a nuclear war.
Consider the counterfactual where the torpedo was fired and nuclear war began. Some people at least would have been alive on the 28 October while knowing that the war had started. We can compare their epistemic state with the real versions of themselves on 28 October with no nuclear war. By the same argument as above, their probability updates are standard Bayesian.
Now, suppose I met one of these older people (I have met several). By Aumann’s agreement theorem, if we are two rational agents with the same prior (and neither of us has any doubts as to which agents we are) then we cannot agree to disagree. In particular, they’d know that they would agree with every update I ever made, so I cannot have made anything but standard Bayesian updates.
But even if we had different priors, then, as long as those priors have the same support [4] , Aumann’s agreement theorem says that I cannot disagree with the knowledge state they would have been in if they’d used my prior instead of theirs [5] . That hypothetical agent would know they’d agree with every update I ever made, so I similarly cannot have made anything but standard Bayesian updates.
So let’s add:
Back to the beginning of life
The argument of the previous section can be chained generation by generation, going back as far as we identify an agent that could potentially be rational. It doesn’t stretch beyond that; in particular it doesn’t stretch back to the beginning of life itself, since there is no observer in the counterfactual branch where life isn’t created.
So let’s introduce:
The probability of an updating machine existing is not zero. If such a machine existed, maybe in orbit around the planet looking down, then it exists in the counterfactual branch where life doesn’t exist and hence its Bayesian updates are reliable and non-anthropic.
By the same argument as before, I (or the first potentially rational agent in existence) cannot disagree with the hypothetical updating machine – we couldn’t disagree with it if it were a rational observer, so how could we disagree with it if it were an automated machine that produced exactly the same estimates as the rational observer.
We can simplify:
Same argument: I don’t actually need to agree on the prior, I just need the likelihood ratios. Let’s simplify further; we don’t actually need the likelihood ratios, the raw observations will do:
As long as this tape exists in both branches, it forces me to avoid extra anthropic components in my updating. Let’s simplify again:
Now, much of that recording has almost certainly been lost to time. But not certainly, and there is much we could do to recover it. We could do a deep dive into the structure of the solar system and the universe. There may be light beams from that time that have gone round black holes and are heading back to Earth. Maybe new theories of physics or mathematics will allow us to peer back through chaos to the origins of life.
And maybe we are wrong about physics. If they existed, wormholes and faster-than-light travel could allow us to actually peer back and see life’s beginning. Maybe the universe is cyclical in space, and we will just naturally get to see that beginning by looking far enough into the distance.
So add the following two assumptions:
We’ve pushed back to the beginning of life itself; now there seems no longer any room for counterfactuals with extra anthropic effects. We have to be standard Bayesians the whole way through, without room for extra anthropic effects beyond this.
Conclusion
In basic anthropic problems, assume the following holds:
Then the correct update should be the standard Bayesian:
In basic situations, this is equivalent to:
Selection issues
Notice that if you are agent
That’s because your knowledge could have reached you in a biased and selective way. If you are born American, you should not be surprised that you see Americans all around you, and you shouldn’t conclude “almost everyone is American”.
There’s a similar issue with the Sailor’s Child problem. If the sailor was moving between Amsterdam and Boston, and if I went to Amsterdam and found the sailor’s child there, I would put a
However, if someone else searched Amsterdam and Boston for me, and then presented me with a sailor’s child from Amsterdam, then I would stay 50-50 on the coin (I expect to be presented with a child, I’m presented with a child, each city is equally likely – no update). And the Amsterdam child would agree with me again. Because, though they start with
SIA doesn’t have the implications that you probably think it does
It’s been said that SIA increases the probability of large numbers of observers existing. This is not true. SIA increases the probability of you specifically existing and making the observations you have made. Sometimes, as a side effect, this increases the probability of large numbers of observers existing. Sometimes it does not. Sometimes it decreases this probability.
A lot depends on the prior. In anthropic reasoning, you generally get to update on your own existence only once; and the examples are often contrived so that you don’t get further updates from observations.
A single update means that you’re exceptionally vulnerable to the choice of the prior; and many of the strange intuitions about anthropic updates come from selecting specific priors. Let’s look at three of them.
SIA dramatically increases observers
Well, life being common is perfectly compatible with your observation, while life being rare has only one chance in a trillion. So the update is a trillion to one against life being rare. If we started with an equal prior on both theories, SIA has effectively doubled the expected number of inhabited worlds in the universe.
Yes, but... Let’s look at the prior again. Life being everywhere is a single statement – life exists on all planets. Life being rare is actually a trillion different statements – “life exists only on planet 0, life exists only on planet 1, etc...”. So when you observe which planet you are born on, you are selecting the only available statement on the “life is everywhere” side, and one of a trillion different ones on the “life is rare” side.
The prior was set up to overweight “life is everywhere”. Let’s fix that.
SIA slightly increases observers
Let’s try a different prior:
This means that there are
Upon updating that we are born on planet N, we rule out exactly half of those statements (those which don’t put life on planet N). The probability of life on any planet M not equal to N remains 50%. The expected number of planets with life goes up from half-a-trillion, to half-a-trillion-and-a-half. SIA has increased the expected number of inhabited worlds in the universe by exactly one half-world.
SIA dramatically decreases observers
Yet another prior:
We live on a world with oxygen and water; so
Overall, we’ve gone from expecting roughly fifty billion inhabited worlds, to expecting roughly ten billion. SIA has divided the number of expected inhabited worlds by five.
The Presumptuous Philosopher’s error
This is the error in the Presumptuous Philosopher thought experiment. If you compare theories merely by the number of observers in them, SIA’s impact is undefined. I consider priors like Prior 1 to be particularly devious, because it bakes in a large increase but feels reasonable.
Prior 2 is the “life in one area of the galaxy is independent of life elsewhere” prior; theory
So beware anthropic arguments where the prior is doing all the work. In particular, there are reasons to reject priors exactly like Prior 2 – clearly there are some correlations, life on Earth does mean that, e.g., stars exist, which increases life’s odds on other planets. But we shouldn’t go straight to priors like Prior 1: there are many plausible scenarios where correlations are weak, so there are many priors in between Prior 1 and Prior 2, which should be included as well.
And remember that we make other observations about the universe as well. An anthropic update of trillion, trillion, trillion is only about 120 bits of information. This paragraph contains more than 120 bits of information. And we’ve made far more observations than the length of this paragraph. So what we have observed about the universe should generally overwhelm purely anthropic reasoning.
More prior effects: Doomsday, God, and simulations
The Doomsday argument
Because it’s so famous, it’s worth looking at the Doomsday argument directly. One phrasing of it is:
This argument is known to fail, given SIA. Thus it fails on standard Bayesian updating. But to best grasp how it might fail, to attack the intuition behind it, consider the following variant:
Now, once we have crossed the lake, it is true that we were only briefly in the first 1% of it. So the Doomsday/Edge Of The Lake arguments are retrospectively valid: 99% of all existing humans won’t be in the first 1%; 99% of the lake crossing distance is not in the first 1%.
But that doesn’t help us estimate the lake size ahead of time. If we are equally unsure as to whether the lake is 200 meters, 10 km, or 100 km across, then we can’t row 50 meters and conclude that it’s probably the smaller lake, after all, because if we were in the larger ones, we’d be in the very atypical first 1% of them. But all three candidate lakes are longer than 50 meters, so “I have rowed 50 meters and not yet arrived” has likelihood
Now, people might object that distance across the lake is a standard physical variable to measure, while existing is more complicated. So we can present:
All three arguments are based on the same intuitive typical/atypical intuition and the
Re-instating SSA with prior manipulation
The arguments of this post only demonstrate that in basic anthropic situations, one should update in a standard Bayesian way. They say nothing about the prior.
We can still get SSA (the Self-Sampling Assumption), if we feel like it, by selectively downgrading worlds with more observers in them. To do that, split the prior into deterministic worlds (a stochastic world is just a probabilistic mix of deterministic worlds), divide each world by
However, this is a case of “use weird priors, get weird results.”
Furthermore:
God and fine-tuning
I won’t go into details here about applications of anthropic reasoning to argue for the existence/non-existence of a God. I feel that most of the God-related arguments also depend on the prior. Compare what prior probability people are giving to a universe that supports our kind of life versus their prior on the existence of a God capable and willing to intervene to create our kind of life. That prior is generally doing most of the work in the argument [6] .
In particular, any existing agent would be able to construct a model of a God that would make their existence most likely; the Smulons from the planet Smuooga can imagine a Smu-God that values pure abstract Smuoorulity as the highest value, and thus will create Smulons. What is necessary for the argument to work is to show that a God that creates humans specifically has a shorter description than humans do (and hence is a priori more likely than all the alternative Smu-Gods). That may be possible, but that work has to be actually done, rather than using a prior that assumes it’s true.
Simulations
Simulation arguments also seem very prior dependent. First of all, there’s the very issue of defining what a simulation is: in a Tegmark Type 4 Multiverse, every agent can be seen as a simulation, or, equally, as a real agent.
But even setting that aside, in the original simulation argument, Nick Bostrom contrasted three situations:
Notice that in 1., 2., and a small part of 3., the universe is as it seems: we can go around looking at it, and observational evidence tells us what it’s like.
But if we assume we might be simulations, then everything falls apart. Our observations no longer tell us much about the base reality. We might say that being “ancestor simulations” means that we can port our simulated observations to base reality. But we can be pretty sure that, if we ever did find a way of doing simulations, any ancestor simulations (or quasi-ancestor simulations) would not be drawn statistically neutrally from our own ancestors. So we expect that our simulations would not be in an accurate reality. Now we have to invert that: if we were quasi-ancestor simulations, what could we really deduce about base reality? Even a small statistical deviation between us and our simulators can make the simulators’ world much more likely, or much less likely, than ours appears to be, messing up computations in the simulation argument.
To do it properly, we’d need to take some sort of universal prior, identify conscious entities within that, identify which are simulations, and work backwards to figure out which entities are most likely to simulate humans like us. And then define what counts as “close enough” for those entities to count as doing ancestor simulations, and update on that. Again, the prior is going to have to do much of the work.
The multiverse and beyond
I haven’t mentioned one of the most important anthropic related areas: the potential existence of the multiverse. But an infinite multiverse definitely has multiple copies of agents, so we’re no longer in basic situations.
In the next post, I’ll bring back agent copies, and see what that implies. First of all, I’ll show that no anthropic probability theory (not even SIA) can maintain sensible numbers across duplicate creation. And show that SIA is incompatible with standard algorithmic priors, and can’t apply to our universe. Then I’ll propose D-SIA, a distribution version of SIA that fixes a lot of these issues.
This is because enlarging the reference class by a factor of
We could repair those by positing that Dutch book arguments would apply to altruistic agents (or agents that care about something in the universe other than themselves), if gains/losses come from the outside source of value. Then we might complete the argument by saying that Dutch books determined probability for not-completely-selfish agents. And noting that it would be odd for the probabilities of selfish agents to be different because they are selfish. ↩︎
It would be somewhat entertaining if “no further anthropic updates” were untrue. We might get situations where the booby trap would have caused me to die over the next five days. Then would I stick with standard Bayesian odds for five days, and then suddenly jump to different odds at the exact moment where my counterfactual self would have died? In these cases, I might have strong reasons to investigate the level of medical care a counterfactual version of me would have got. ↩︎
Our priors have different support if there’s a world
Take their prior and posterior, re-derive their likelihood ratios, apply these likelihood ratios to my prior instead. The likelihood ratios are derived from the properties of the worlds and the observations, not from the prior. ↩︎
For what it’s worth, my feeling is that fine-tuning argument/counter-arguments tend to neglect the possibility that the fine-tuned values are less unlikely than they may seem. Consider for instance the theory of cosmic inflation, which tries to explain why the universe is much more homogeneous than it should be. Cosmic inflation needs its own fine-tuning, but it’s still much, much, much more likely than the alternative, that the inhomogeneities spontaneously or miraculously vanished. So who knows what future theory will resolve current fine-tuning requirements.
Since much of the argument depends on the ill-defined prior anyway, I’m adding my “actually, the fine-tuned values are forced by physics we haven’t discovered yet” to the mix. An unlikely argument, but one strongly boosted by observation. We’re down to 20-30 free parameters in the Standard Model; it’s very plausible we could go down even further. ↩︎