Epistemic status: Extremely uncertain

In this post I argue against the claim that we should take into account the Anthropic Shadow effect when estimating the risk of catastrophic events.

If we wish to estimate the probability of some catastrophic event occurring in the next 100 years, the first thing that we are likely to try is to count how many times the catastrophe has already occurred in the past^{[1]}. Anthropic Shadow is the idea that this naive approach will lead to an underestimate of the true risk, due to observer selection effects. The lower the probability of human survival following the catastrophe, , the greater the degree of anthropic bias in our estimate.

I have spent some time trying to understand the logic underpinning the Anthropic Shadow effect, and I am becoming increasingly convinced that it has significant flaws. My personal view would now be that the naive approach to catastrophic risk estimation is probably ok after all. (My all-things-considered view is that a lot of smart people uncritically espouse the Anthropic Shadow argument, so there is a high chance that I am missing something!)

## Outline of my argument

The original Anthropic Shadow argument can be summarized as follows (for a toy model), in two steps:

Step 1:Consider a planet where some catastrophe occurs randomly, at an average rate of times per million years. Whenever it occurs, it causes the permanent extinction of all observers with probability (defining as in the original paper). If observers still exist after 1 million years, we should expect them to have fewer than catastrophes in their past, on average. In mathematical notation:

It is easiest to see why this is true when . Observers then necessarily have catastrophes in their past.

Step 2:We are necessarily observers. Therefore, if we base our estimate of catastrophe frequency on the frequency of catastrophes in our past, our estimate will be biased downwards.

This argument seems intuitively plausible. The first step is certainly true. But does the second step really follow from the first? In the next section I will consider a similar, non-anthropic, toy model for which an analogous argument fails, despite holding some of the same intuitive appeal. I will then spend the rest of the post exploring how the 'anthropicness' of the Anthropic Shadow effect might explain why the two models should be treated differently. Ultimately, I fail to do this convincingly, and conclude that the Anthropic Shadow argument probably does not work either.

More precisely, I am only able to get the Anthropic Shadow argument to work if I make two additional assumptions, not stated explicitly in the original paper:

- There are a very large number of observer-containing planets (or in the case of universe level catastrophes, such as vacuum decay, there are a very large number of observer-containing universes).
- A very specific choice of observer reference class must be made, when applying the anthropic part of the argument.

The first assumption is not so bad. If Anthropic Shadow occurred only under that assumption, we should still take it seriously. But the second assumption is very problematic. In the book Anthropic Bias, Nick Bostrom argues that we should only respect anthropic conclusions if they are relatively robust to the choice of observer reference class. It is on this basis that he rejects the Doomsday Argument (a less popular anthropic argument that also says we underestimate extinction risk). I am arguing here that Anthropic Shadow suffers from the same problem, and should probably be treated with similar suspicion to the Doomsday Argument.

Finally, I consider SIA (a largely reference class independent method of anthropic reasoning) to show that the Anthropic Shadow effect does not occur under that approach to anthropic reasoning either.

## The non-anthropic analogy

As above, consider a planet where some catastrophe occurs randomly, at an average rate of times per million years. But this time, there is no chance of it causing extinction. Instead, whenever it occurs there is a probability of it permanently changing the colour of the sky from blue to green. Inhabitants of this planet could still make an analogous argument to Anthropic Shadow:

Step 1: If the sky is still blue after 1 million years, we should expect there to be fewer than catastrophes in the past, on average. In mathematical notation,It is easiest to see why this is true when . A blue sky then necessarily implies catastrophes in the past.

Step 2: We see a blue sky after 1 million years. Therefore, if we base our estimate of catastrophe frequency on the frequency of catastrophes in our past, our estimate will be biased downwards.

The only meaningful difference between this argument and Anthropic Shadow is the ommission of the word 'necessarily' in the second step. But this argument is *wrong*. The colour of the sky gives these observers no additional information about the risk of catastrophe, on top of what they already have from the historical record. It is true that if you simulated history on this planet a large number of times, then observers with a blue sky after 1 million years would have biased estimates of catastrophe frequency, on average. But, perhaps counter-intuitively, that does *not *mean that they should take the colour of the sky into account when making their estimate.

It is clear that we need to be more precise in order to unpick what is going on. To prove my claim that this non-anthropic analogy argument is false, I take a Bayesian approach. is no longer a fixed unknown quantity. Instead, it is a random variable, reflecting our subjective uncertainty in catastrophe frequency. We now wish to update our prior distribution on , given the evidence. To make this update, the only quantity that need concern us is the likelihood:

Values of with higher likelihood will then have their probability boosted in appropriate proportion by the Bayesian update.

We can now prove that the non-anthropic analogy is false by considering the likelihood, as follows:

The second factor above is now independent of , and so has no effect on the Bayesian update. The first term is independent of the colour of the sky, so the colour of the sky can have no effect on the Bayesian update. Therefore, the colour of the sky is irrelevant, and the inhabitants of this planet should stick with their naive estimate of catastrophe frequency.

## Where does the 'anthropicness' in 'Anthropic Shadow' come in?

We have now shown that the anthropicness of the Anthropic Shadow argument (or the word 'necessarily' in Step 2) must be essential. In hindsight this should not be surprising. 'Anthropic' is in the name after all! But I still found this non-anthropic analogy very helpful to think about. It had not been clear to me when I first read the Anthropic Shadow paper what role the anthropicness was playing. The argument did seem intuitively plausible to me, but I can now see that it was intuitively plausible for the wrong reasons. That same intuition would have led me astray in the non-anthropic analogy above. (Of course, this might say more about me than it does about the argument.) But now that this has been clarified, can we explain, in similar Bayesian language to that used above, how the anthropic shadow argument actually works?

It will be helpful to first explain *why* the non-anthropic analogy argument fails. When the inhabitants of the planet see a blue sky, they know that they belong to a category of observers who will underestimate catastrophe frequency on average. Why shouldn't that concern them? We can understand why by splitting the likelihood up in a different way:

Split this way, it is now much easier to see where our non-anthropic analogy argument went wrong. The second factor above is where the bias referred to by our argument comes in. This factor favours , as the argument said it should. *But our argument went wrong by ignoring the first factor*. The first factor favours smaller . The two opposite effects must exactly cancel, so that in the end we only need to consider the naive likelihood: , as already demonstrated.

This point is very important, so I will summarise it again in less mathematical terms:

The planet's inhabitants can consider the bias in their historical record, given their blue sky, if they wish, but they must then also consider that the colour of the sky gives them information about as well. The sky is more likely to be blue when is smaller. Overall, they will end up with the same conclusions as if they had just ignored the colour of the sky completely.

The next question we need to ask is: why *doesn't* the same thing apply in the anthropic case? Why shouldn't we also consider the fact that we are more likely to exist today when the catastrophe rate is smaller, precisely cancelling out the effect of anthropic bias in our historical record?

Lets consider the same equation in the anthropic case:

To make the Anthropic Shadow argument work, all we need to do is justify why it is ok to ignore the first factor in the anthropic case. That will then leave us with only the anthropic-bias-adjusted second factor. Next I will consider two possible ways that you could try to do this, and why I haven't found either of them to be satisfactory.

## Possible solution 1: You should reason as if your existence is guaranteed (as long as it is logically possible)

One way to justify the Anthropic Shadow argument would be to simply assert that , as a general approach to anthropic reasoning. In other words, we should never be surprised about the fact that we exist. After all, we could never have observed non-existence.

This solution works, but it is *very weird*. Suppose that you play Russian roulette with one of two guns (chosen randomly), one that fires 999/1000 times and one that fires 1/1000 times. If you pull the trigger and survive, do you really have no information about which gun you picked up? It feels to me like you should be fairly confident that you picked up the second one.

Another way to see how weird this solution is, is to consider planetary and cosmological fine tuning. It is well known that various conditions on our planet, and universe, seem fine tuned to allow the existence of life. One popular explanation of this is anthropic. In the case of our planet, it is obvious how this works. There are a very large number of planets in the universe and it makes sense that some of them are, by chance, suited to life. Of course we would have to find ourselves on one of these. At the cosmological level, the same solution could work, but usually it is stated that this requires some kind of multiverse.

Taking the approach to anthropic reasoning would be equivalent to saying that the multiverse, or the existence of large numbers of planets, is not necessary to explain fine tuning. Under this approach, the fine tuning of the cosmological constants would *not* constitute evidence of a multiverse. This is because our existence should never be surprising anyway.

This solution seems too far-fetched to me.

In the original draft of this post, this section concluded here. However, Jonas Moss left some helpful comments below challenging these arguments. It was interesting to learn that their intuitions on the Russian roulette thought experiment were different to mine. They initially defended the idea that you really do have no information about which gun is which, after firing a gun once and surviving. Here is my more sophisticated challenge to that idea, which does not purely rely on an appeal to intuition:

Suppose that you really do have no information after firing a gun once and surviving. Then, if told to play the game again, you should be indifferent between sticking with the same gun, or switching to the different gun. Lets say you settle on the switching strategy (maybe I offer you some trivial incentive to do so). I, on the other hand, would strongly favour sticking with the same gun. This is because I think I have extremely strong evidence that the gun I picked is the less risky one, if I have survived once.

Now lets take an outside view. We imagine an outside observer watching the game, betting on which one of us is more likely to survive through two rounds. Obviously they would favour me over you. My odds of survival are approximately 50% (it more or less just depends on whether I pick the safe gun first or not). Your odds of survival are approximately 1 in 1000 (you are guaranteed to have one shot with the dangerous gun).

This doesn't prove that the approach to formulating anthropic probabilities is wrong, but if we are ultimately interested in using probabilities to inform our decisions, I think this suggests that an alternative approach is better.

We could also imagine applying a similar chain of reasoning in the more relevant case of planetary catastrophes. If there were two planets in the solar system, and we somehow knew that one of them experienced catastrophes at a much higher rate than the other, which one should we decide to live on? Should we stay on the one we evolved on, with relatively few catastrophes in our past, or should we switch to the other? If we try to take the Anthropic Shadow effect into account then we could end up making a bad decision here.

## Possible solution 2: There are a large number of planets containing life

This solution is inspired by the analogy with the anthropic explanation of fine tuning.

Suppose that instead of one observer-containing planet experiencing a catastrophe at a rate of times per million years, there are a very large number of such observer-containing planets. If the number of planets is sufficiently large, then it is true that:

You could now try to argue for the Anthropic Shadow effect along the same lines as the anthropic explanation of fine tuning. It seems plausible to assert that in general

If this does not seem obvious to you, recall that this is precisely how the anthropic explanation of fine tuning works (and if you are still not convinced, you might believe in SIA instead, to be discussed in the Appendix). With this assumption it now looks like we almost have the result we want.

But there is a problem with this solution as well. The statement above is one thing, but what we actually need to show is this:

Or if we cannot show this, we at least need to show that the LHS is independent of . But crucially, the validity of this equation depends upon our choice of reference class (the set of observers that you consider yourself to be a sample from). It is well known that anthropic conclusions are often very sensitive to the choice of reference class, and that certainly applies here. If we take our reference class to be all the observers who exist at 1 million years into their planet's history, then it is true that the above equation holds and that the Anthropic Shadow argument does work. But this is a very specific, arbitrary, choice. It is also not immediately clear how to generalise this choice to the real world, outside of this toy example. Should our reference class be the observers who exist 4.5 billion years into their planet's development, or 3.7 billion years into life's development, or 100,000 years into the development of language possessing observers?

To see that Anthropic Shadow only occurs with this very specific reference class, we can imagine choosing a more general one instead. For example, we might choose the reference class of all observers who exist at 1 million years or prior, or the reference class of all observers who ever exist on one of these planets. We then have to consider the possibility that we might have existed at a different time. The probability of us finding ourselves at any particular point in history is equal to the proportion of observers in our reference class who live at that point in history.

For simplicity, since we are already using a toy model, lets assume that each planet has the same population, and that this population is constant over time. Then, in general, in the limit of a large number of planets, we would have:

For Anthropic Shadow to go through as originally argued, we need the right hand side to be independent of , but it is clearly not. In general, it is going to be a very complicated beast, with a strong dependence on the chosen reference class. We can ask instead whether this likelihood will consistently lead to an underestimate of catastrophic risk. In that case we might say that the Anthropic Shadow argument is still qualitatively correct, if not quantitatively (in a reference class insensitive sense). But no, the numerator is now once again proportional to the likelihood of a particular planet surviving for 1 million years, which was exactly the factor we saw that we needed in order to negate the anthropic bias in our historical record. There is not a lot we can say about the complicated RHS in general, but one of the few things we *can* say is that that numerator by itself will always cancel the anthropic bias in our historical record.

What about the denominator? It's true that there is some much more complicated dependence down there as well. In fact, the denominator is going to penalise small , especially if the reference class is wide. This is because the denominator is much larger if life on more planets lasts longer. This means that instead of cancelling the effect of anthropic bias in our historical record, this factor could actually make it even worse! But it would not be right to call this Anthropic Shadow. All we have done here is rediscover the Doomsday Argument. This says that the future is likely to be short, because if it were big then it would be unlikely for us to find ourselves so early on in history. This is precisely the same effect which is being captured by the denominator in the above expression.

To sum up, there is certainly a lot going on in the above equation, but there is nothing that you could meaningfully call Anthropic Shadow. If you were going to sum it up in words, it would best be described as: "naive estimate of catastrophe frequency + Doomsday Argument". Even the special reference class where we saw the Anthropic Shadow effect does hold could be re-interpreted in these terms. You can think of the Anthropic Shadow effect in the many-planet case as the Doomsday Argument applied to catastrophe probability estimation where the reference class is all observers across all planets who live at the same point in history as us.

## Conclusion

In conclusion, I can only make sense of the Anthropic Shadow argument if two assumptions are made:

- There are a large number of planets (or in some cases, universes) containing life.
- Our reference class should only contain observers who live at the 'same' point in history as us. (Although it is not clear to me how to define 'same' outside of toy models, since this reference class must include alien life.)

In the book 'Anthropic Bias', Nick Bostrom says the following after reviewing some successful applications of anthropic reasoning:

"I wish to suggest that insensitivity (within limits) to the choice of reference class is exactly what makes the applications just surveyed scientifically respectable. Such

robustnessis one hallmark of scientific objectivity."

He goes on to make an analogy with Bayesian statistics: scientifically respectable conclusions should be insensitive (within limits) to your choice of prior. It is on this basis that he refutes the Doomsday Argument, and similar anthropic paradoxes (Adam+Eve paradox), because of their strong sensitivity to reference class choice. In his words:

"These arguments will fail to persuade anybody who doesn't use the particular kind of very inclusive reference class they rely on -indeed, reflecting on these arguments may well lead a reasonable person to adopt a more narrow reference class. Because they presuppose a very special shape of the indexical parts of one's prior credence function, they are not scientifically rigorous."

It seems to me that with its strong dependence on reference class choice, Anthropic Shadow belongs with the Doomsday Argument and Adam+Eve paradox in this category of non-rigorous anthropic argument. It differs from Doomsday and Adam+Eve in that its reference class is extremely exclusive, rather than extremely inclusive, but that is not important. What matters is that it is still extremely sensitive to the reference class. If anything, the reference class dependence is worse for Anthropic Shadow, because it is not even clear how you would define the required reference class outside of toy models.

## Mini Appendix: What about SIA?

The Self-Indication Assumption (SIA) is an alternative, popular, approach to anthropic reasoning. It rejects the equation we saw before:

and instead claims that the LHS is more likely in worlds with more observers. Its appeal comes from the fact that it is largely reference class independent, and that it avoids the Doomsday Argument and Adam+Eve conclusions already mentioned (although it comes with unattractive conclusions of its own). I won't discuss it much more here except to say that SIA doesn't contain the Anthropic Shadow effect either. It doesn't matter what reference class we use to demonstrate this, so the simplest way to see it is to return to the final example considered in Possible Solution #2, and consider our final likelihood equation. SIA has the effect of removing the denominator in the likelihood expression, leaving only the numerator, which we already saw had the required form to precisely negate the anthropic shadow effect.

^{^}We are here considering events where we don't have good reason to think the base risk per century has changed much over time.

Edit:I don't endorse the arguments of this post anymore!Your example with the sky turning green is illuminating, as it shows there is nothing super special about the event “the observer exists” in anthropic problems (at least some of them). But I don't think the rest of your analysis is likely to be correct, as you're looking at the wrong likelihood from the start.

In the anthropic shadow problem, as in most selection problems, we are dealing with two likelihoods.

The first is the likelihood from the bird's-eye view. This is the ideal likelihood, with no selection at all, and the starting point of any analysis. In our case, the bird's-eye view likelihood at the time t is

P(St=n,Yt=y∣θ),where y∈{0,1} equals 1 if the sky has turned green within time t and St is the number of catastrophic events up to time t, and θ is some parameter (corresponding to n in your post). From the bird's-eye view, we observe every St regardless of the outcome of Yt, and your Bayesian analysis is correct. But we do not have the bird's-eye view, as we only observe the Sts associated with Yt=0!

The second likelihood is from the worm's-eye view. To make your green-and-blue sky analogy truly analogous to the anthropic shadow, you will have to take into account that you will never be in a world with a green sky. In our case, we could suppose that worms cannot live in a world with a green sky, making Yt=0 a certainty. That entails conditioning on the event Yt=0 in the likelihood above, yielding the conditional likelihood

P(St=n∣θ,Yt=0).The likelihood from the bird's-eye view and the likelihood from the worm's-eye view are not the same, they do not even have the same signature. We find that the worm's-eye view likelihood is

P(St=n∣θ,Yt=0)=P(St=n∣θ)(1−q)n∑ti=0P(St=i∣θ)(1−q)i∝P(St=n∣θ)(1−q)n,where q is the (independent) probability of the sky turning green whenever a catastrophic event occurs.

The posterior from the bird's-eye view is

p(θ,0∣n)=P(St=n∣θ)(1−q)n∫P(St=n∣θ)(1−q)ndθp(θ)=P(St=n∣θ)∫P(St=n∣θ)p(θ)dθp(θ)and is independent of Yt, as you said. However, the posterior from the worm's-eye view is

p(θ,0∣n)=P(St=n∣θ)(1−q)n∑ti=0P(St=i∣θ)(1−q)i∫P(St=n∣θ)(1−q)n∑ti=0P(St=i∣θ)(1−q)ip(θ)dθp(θ)=P(St=n∣θ)∑ti=0P(St=i∣θ)(1−q)i∫P(St=n∣θ)∑ti=0P(St=i∣θ)(1−q)ip(θ)dθp(θ)≠p(θ∣n,0).As you can see, the factor(1−q)i can't be canceled out in the integrating factor.

By the way, the likelihood proportional to P(St=n∣θ)(1−q)n is not always hard to work with. If we assume that St is binomial with success probabiltiy p, one can use the binomial theorem to show that the integrating constant is 1−pq, yielding the normalized pmf P(St=n∣p)=(tn)(p(1−q))n(1−p)n−k(1−pq)t.

Thank you for your comment! I agree with you that the difference between the bird's-eye view and the worm's eye view is very important, and certainly has the potential to explain why the extinction case is not the same as the blue/green sky case. It is this distinction that I was referring to in the post when asking whether the 'anthropicness' of the extinction case could explain why the two arguments should be treated differently.

But I'm not sure I agree that you are handling the worm's-eye case in the correct way. I could be wrong, but I think the explanation you have outlined in your comment is effectively equivalent to my 'Possible Solution #1', in the post. That is, because it is impossible to observe non-existence, we should treat existence as a certainty, and condition on it.

My problem with this solution is as I explained in that section of the post. I think the strongest objection comes from considering the anthropic explanation of fine tuning. Do you agree with the following statement?:

"The fine tuning of the cosmological constants for the existence of life is (Bayesian) evidence of a multiverse."

My impression is that this statement is generally accepted by people who engage in anthropic reasoning, but you can't explain it if you treat existence as a certainty. If existence is never surprising, then the fine tuning of cosmological constants for life cannot be evidence for anything.

There is also the Russian roulette thought experiment, which I think hits home that you should be able to consider the unlikeliness of your existence and make inferences based on it.

I wouldn't say you treat existence as certainty, as you could certainly be dead, but you have to condition on it when you're alive. You have to condition on it since you will never find yourself outside the space of existence (or blue skies! ) in anthropic problems. And that's the purpose / meaning of conditioning; you restrict your probability space to the certain subset of basic events you can possibly see.

Then again, there might be nothing very special about existence here. Let's revisit the green sky problem again, but consider it from a slightly different point of view. Instead of living in the word with a blue or a green sky, imagine yourself living outside of that whole universe. I promise to give you a sample of a world, with registered catastrophes and all, but I will

notshow you a world with a green sky (i.e., I will sample worlds until the sky turns out blue). In this case, the math is clear. You should condition on the sky being green. Is there a relevant difference between the existence scenario and this scenario?Maybe there is? You are not guaranteed to see a world at all in the "existence" scenario, as you will not exist if the world turns out to be a blue-sky world, but you are guaranteed an observation in the "outside view" scenario. Does this matter though? I don't think it does, as you can't do an analysis either way if you're dead, but I might be wrong. Maybe this is where our disagreement lies?

I don't find the objection of the Russian roulette persuasive at all. Intuition shouldn't be trusted in probability, as e.g. the Monty Hall experiment tells us, and least of all in confusing anthropic problems. We should focus on getting the definitions, concepts, and math correctly without stopping to think about how intuitive different solutions are. (By the way, I don't even find the Russian roulette experiment weird or contra-intuitive. I find it intuitive and obvious. Strange? Maybe not. Philosophical intuitions aren't as widely shared as one would believe.)

> "The fine tuning of the cosmological constants for the existence of life is (Bayesian) evidence of a multiverse."

> My impression is that this statement is generally accepted by people who engage in anthropic reasoning, but you can't explain it if you treat existence as a certainty. If existence is never surprising, then the fine tuning of cosmological constants for life cannot be evidence for anything.

I don't know if that's true, though it might be. I suppose the problem about fine-tuning could be sufficiently different from this one to warrant its own analysis.

I think that's a good summary of where our disagreement lies. I think that your "sample worlds until the sky turns out blue" methodology for generating a sample is very different to the existence/non-existence case, especially if there is actually only one world! If there are many worlds, it's more similar, and this is why I think anthropic shadow has more of a chance of working in that case (that was my 'Possible Solution #2').

I find it very interesting that your intuition on the Russian roulette is the other way round to mine. So if there are two guns, one with 1/1000 probability of firing, and one with 999/1000 probability of firing, and you pick one at random and it doesn't fire, you think that you have no information about which gun you picked? Because you'd be dead otherwise?

I agree that we don't get very far by just stating our different intuitions, so let me try to convince you of my point of view a different way:

Suppose that you really do have no information after firing a gun once and surviving. Then, if told to play the game again, you should be indifferent between sticking with the same gun, or switching to the different gun. Lets say you settle on the switching strategy (maybe I offer you some trivial incentive to do so). I, on the other hand, would strongly favour sticking with the same gun. This is because I think I have extremely strong evidence that the gun I picked is the less risky one, if I have survived once.

Now lets take a birds-eye view, and imagine an outside observer watching the game, betting on which one of us is more likely to survive through two rounds. Obviously they would favour me over you. My odds of survival are approximately 50% (it more or less just depends on whether I pick the safe gun first or not). Your odds of survival are approximately 1 in 1000 (you are guaranteed to have one shot with the dangerous gun).

This doesn't prove that your approach to formulating probabilities is wrong, but if ultimately we are interested in using probabilities to inform our decisions, I think this suggests that my approach is better.

On the fine tuning, if it is different, I would like to understand why. I'd love to know what the general procedure we're supposed to use is to analyse anthropic problems. At the moment I struggle to see how it could both include the anthropic shadow effect, and also have the fine tuning of cosmological constants be taken as evidence for a multiverse.

Here's a rough sketch of how we could, potentially, think about anthropic problems. Let Pt be a sequence of true, bird's-eye view probability measures and Qt your own measures, trying to mimic Pt as closely as possible. These measures aren't defined on the same sigma-algebra. The sequence of true measures is defined on some original sigma-algebra Σ, but your measure is defined only on the sigma-algebra {A∩{ω where the sky is blue at time t}}.

Qt(A)=Pt(A∣ where the sky is blue at time t).Now, the best-known probability measure defined on this set is the conditional probability

This is, in a sense, the probability measure that most closely mimics Pt . On the other hand, the measure that mimics Pt most closely is Qt(A)=Pt(A), hands down. This measure has a problem though, namely that maxQt(A)<1, hence it isn't a probability measure anymore.

I think the main reason why I intuitively want to condition on the color of the sky is that I want to work with proper probability measures, not just measures bounded by 0 and 1. (That's why I'm talking about, e.g., being "uncomfortable pretending we could have observed non-existence".) But your end goal is to have the best measure on the data you can actually observe, taking into account possibilities you can't observe. This naturally leads us to Qt(A)=Pt(A) instead of Qt(A)=Qt(A)=Pt(A∣ where the sky is blue at time t).

I think that makes sense!

There is another independent aspect to anthropic reasoning too, which is how you assign probabilities to 'indexical' facts. This is the part of anthropic reasoning I always thought was more contentious. For example, if two people are created, one with red hair and one with blue hair, and you are one of these people, what is the probability that you have red hair (before you look in the mirror)? We are supposed to use the 'Self-Sampling Assumption' here, and say the answer is 1/2, but if you just naively apply that rule too widely then you can end up with conclusions like the Doomsday Argument, or Adam+Eve paradox.

I think that a complete account of anthropic reasoning would need to cover this as well, but I think what you've outlined is a good summary of how we should treat cases where we are only able to observe certain outcomes because we do not exist in others.

The roulette example might get to the heart of the problem with the worm's-eye view! From the worm's-eye view, the sky will always be blue, so P(skycolor=green)=0, making it impossible to deal with problems where the sky might turn green in the future.

In the roulette example, we're effectively dealing with an expected utility problem where we condition on existence when learning about the probability, but not when we act. That looks incoherent to me; we can't condition and uncondition on an event willy-nilly: Either we will live in a world where an event must be true, or we don't. So yeah, it seems like you're right, and we're effectively treating existence as a certainty when looking at the problem from the worm's-eye view.

As I see it, this strongly suggests we should take the bird's-eye view, as you proposed, and not the worm's-eye view. Or something else entirely; I'm still uncomfortable pretending we could have observed non-existence.

I discuss different arguments against anthropic shadow in my new post, may be it would be interesting for you https://forum.effectivealtruism.org/posts/bdSpaB9xj67FPiewN/a-pin-and-a-balloon-anthropic-fragility-increases-chances-of

EDIT:Added comments on the Russian roulette case

My intuition is that the arguments

and

Are both valid and don't actually conflict. They are entitled to both decrease how likely they take the catastrophes to be(due to no catastrophe changing the color of the sky), but they should also think that they are more likely than their historical record indicates.

This is because, if you are in a world where X is true your being in that world should increase how likely you think being in a world where X is true, but it should also make you think you have a data set biased in favor of events correlated with X being true.

I think the anthropic nature of anthropic shadow comes in due to if X is, you exist. Then it doesn't indicate that it is more likely, only that it is possible because unlike the sky you couldn't be in a world where you didn't exist.

I think the Russian roulette can be untangled if we frame it differently, I think you definitely have information that the gun is the 1/1000 because you are more likely to be an observer moment in a universe where it was that gun rather than the other. But you shouldn't be surprised that you exist, if you learn that it was the 999/1000 gun, you should be suppressed that it was that you are in a universe where your existence is unlikely rather than likely.

"the arguments ... Are both valid and don't actually conflict. They are entitled to both decrease how likely they take the catastrophes to be(due to no catastrophe changing the color of the sky), but they should also think that they are more likely than their historical record indicates. "

I agree with this. Those are two opposing (but not contradictory) considerations for them to take into account. But what I showed in the post was: once both are taken into account, they are left with the same conclusions as if they had just ignored the colour of the sky completely. That's what the bayesian calculation shows. The two opposing considerations precisely cancel. The historical record is all they actually need to worry about.

The same will be true in the anthropic case too (so no anthropic shadow) unless you can explain why the first consideration doesn't apply any more. Pointing out that you can't observe non-existence is one way to try to do this, but it seems odd. Suppose we take your framing of the Russian roulette example. Doesn't that lead to the same problems for the anthropic shadow argument? However you explain it, once you allow the conclusion that your gun is more likely to be the safer one, then don't you have to allow the same conclusion for observers in the anthropic shadow set-up? Observers are allowed to conclude that their existence makes higher catastrophe frequencies less likely. And once they're allowed to do that, that consideration is going to cancel out the observer-selection bias in their historical record. It becomes exactly analogous to the blue/green-sky case, and then they can actually just ignore anthropic considerations completely, just as observers in the blue/green sky world can ignore the colour of their sky.

These arguments don't cancel out. The argument

Is identical in form to,

However, there is the other argument

These can't perfectly cancel out or we could never know anything, the net result of these two arguments must still be a decrease in how likely we take catastrophic risks. I do think it's because we can't discover we don't exist that is the relevant distinction.

If I can only exist in a world with a blue sky then the fact that the sky is blue should not make me think it is more likely for worlds to have a blue sky. This is why the argument, that life is likely to exist throughout the universe because it happened here doesn't work. If I can only exist in a world with life then the fact that life exists in this world shouldn't influence how likely I think it is to be in another world, other than me knowing it is possible.

I responded to your comment on the other post I am happy to continue chatting in DMs if you like.

I am very confident that the arguments do perfectly cancel out in the sky-colour case. There is nothing philosophically confusing about the sky-colour case, it's just an application of conditional probability.

That doesn't mean we can never learn anything. It just means that if X and Y are independent after controlling for a third variable Z, then learning X can give you no additional information about Y if you already know Z. That's true in general. Here X is the colour of the sky, Y is the probability of a catastrophic event occurring, and Z is the number of times the catastrophic event has occurred in the past.

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In the Russian roulette example, you can only exist if the gun doesn't fire, but you can still use your existence to conclude that it is more likely that the gun won't fire (i.e. that you picked up the safer gun). The same should be true in anthropic shadow, at least in the one world case.

Fine tuning is helpful to think about here too. Fine tuning can be explained anthropically, but only if a large number of worlds actually exist. If there was only one solar system, with only one planet, then the fine tuning of conditions on that planet for life would be surprising. Saying that we couldn't have existed otherwise does not explain it away (at least in my opinion, for reasons I tried to justify in the 'possible solution #1' section).

In analogy with the anthropic explanation of fine-tuning, anthropic shadow might come back if there are many observer-containing worlds. You learn less from your existence in that case, so there's not necessarily a neat cancellation of the two arguments. But I explored that potential justification for anthropic shadow in the second section, and couldn't make that work either.

Thanks for the post! I found the below point quite informative too.