Dispelling the Anthropic
Shadow
Teruji Thomas (Global Priorities Institute, Universityof Oxford)

Global Priorities Institute | September 2024
GPI Working Paper No . 20-2024

Please cite this working paper as: Thomas, T. Dispelling the Anthropic Shadow.
Global Priorities Institute Working Paper Series, No. 20-2024. Available athttps://globalprioritiesinstitute.org/dispelling-the-anthropic-shadowteruji-thomas

Dispelling the Anthropic Shadow
Teruji Thomas*

Abstract
There are some possible events that we could not possibly
discover in our past. We could not discover an omnicidal catastrophe, an event so destructive that it permanently wiped out
life on Earth. Had such a catastrophe occurred, we wouldn’t
be here to find out. This space of unobservable histories has
been called the anthropic shadow. Several authors claim that theanthropic shadow leads to an ‘observation selection bias’, analogous to survivorship bias, when we use the historical record to
estimate catastrophic risks. I argue against this claim.

1 Introduction
Estimating the probability of catastrophic events is a difficult business.
We don’t have much to go on when the catastrophes would be of anovel kind, arising from hypothetical social or technological developments. On the other hand, for some types of catastrophes we have along geological record to consult, along with other forms of data andthe results of scientific modelling. For example, when it comes to asteroid impacts, we can use geological dating techniques and the observed
distribution of crater sizes to inform our estimates of future risks.
A curious thing about this historical data, however, is that there
are some possible data points that we could not possibly observe. We
could not find in the historical data an omnicidal catastrophe, an event
so destructive that it permanently wiped out life on Earth. Had sucha catastrophe happened, we wouldn’t be here to find out. Ćirković,* Global Priorities Institute, Faculty of Philosophy, University of Oxford. I especially thank Andreas Mogensen and Toby Ord for helpful comments.

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Sandberg, and Bostrom (2010) call this space of unobservable histories
the anthropic shadow.
Some striking claims have been made about the significance of theanthropic shadow. For example, Ćirković et al. claim that, becauseof the anthropic shadow, a straightforward treatment of the historical
record will lead to systematically underestimating the chances of potentially omnicidal events. In the extreme case, they write,
we should have no confidence in historically based probability estimates for events that would certainly extinguishhumanity…(1497)
Tegmark and Bostrom (2005a) elaborate this line of thought in an earlier ‘brief communication’ published in Nature:
Given that life on Earth has survived for nearly 4 billion
years (4 Gyr), it might be assumed that natural catastrophic
events are extremely rare. Unfortunately, this argument isflawed because it fails to take into account an observationselection effect…, whereby observers are precluded fromnoting anything other than that their own species has survived up to the point when the observation is made. If it
takes at least 4.6 Gyr for intelligent observers to arise, thenthe mere observation that Earth has survived for this duration cannot even give us grounds for rejecting with 99%
confidence the hypothesis that the average cosmic neighbourhood is typically sterilized, say, every 1,000 years. The
observation-selection effect guarantees that we would findourselves in a lucky situation, no matter how frequent thesterilization events.
To avoid any straightforward appeal to the historical absence of omnicidal events, Tegmark and Bostrom develop a more complicated method
based on modelling the formation times of habitable planets. More recently, Snyder-Beattie, Ord, and Bonsall (2019) estimate the lifespanof the human species and check how robust this estimate is with respect
to different evolutionary hypotheses. They reason:[I]f human existence required a 10 million year (Myr) period of evolution free from asteroid impacts, any humanobservers will necessarily find in their evolutionary history2

a period of 10 Myr that is free of asteroid impacts, regardless of the true impact rate. Inferring a rate based on those
10 Myr could therefore be misleading, and methods must
to be used to correct for this bias.
In this paper I argue for a deflationary position: the existence of
the anthropic shadow is essentially irrelevant to estimating risks. There
are several interesting points that come up along the way, but an initial
reason for skepticism is easy to state. According to standard forms ofBayesianism or evidentialism, what we ought to think depends on whatevidence we actually have. The fact that we could not easily have had
different evidence is not important in itself. So, even if we could noteasily have had evidence of past omnicidal events, our actual evidence
that there were no such events should make us think that the rate ofthem is low.
The core of this paper, sections 2–4, analyses a stylized example
close to the one in Ćirković et al. Section 5 extends my analysis applies
to the models used by Tegmark and Bostrom and by Snyder-Beattie etal., while section 6 concludes. As I mentioned, Ćirković et al. focus
on potentially omnicidal events. For present concreteness, let us saythat a ‘potentially omnicidal event’ is one that has a 10% chance of
permanently ending life on Earth. Then, the upshot of my analysis isessentially as follows.
(A) The fact that life has survived so long is evidence that the rate ofpotentially omnicidal events is low.
(B) Given the fact that life has survived so long, historical frequencies
provide evidence for a true rate rather higher than the observed
rate.
(C) These two effects cancel out, so that, overall, the historical record
provides evidence for a true rate close to the observed rate.
Thus, contrary to claims about the anthropic shadow, the historical
record is (in the stated sense) a reliable guide to the rate of potentiallyomnicidal events.
On my reading, the authors quoted above are too focused on (B).
Based on (B), the suggestion is that using the historical rate as an estimate of the true rate may lead to a bad underestimate. However, I argue
that (A) is true and undermines this suggestion. Effectively, focusing3

on (B) neglects the base-rate provided by (A). I must admit, however,that I find the exact position of Ćirković et al. somewhat difficult todecipher. So, while I will try to indicate the specific points at which
I disagree with other participants in this literature, the primary aim of
this paper is to lay out the true story as clearly as I can, in a way thatwill forestall any further confusion.

2 Supervolcano
My analysis will be based on a toy model, a small elaboration of theone considered by Ćirković et al. While this toy model is unrealistic invarious ways, it should allow us to see whether the anthropic shadow
has any general significance. Here is the set-up:
Supervolcano. There is a sequence of Earthlings (one ineach generation until extinction). Before each generation,
including the first, there is chance P of a supervolcanic
eruption, and, for each eruption, a chance Q that life survives, and therefore a chance 1 − Q that the eruption isfatal, ending life on Earth.
Let Jack be the nth Earthling, for some large number n.
He knows Q = q but wants to estimate P . Checking thegeological record, he is able to observe F , the frequency oferuptions so far. He finds F = f , with no fatal eruptions.
Should Jack become confident that P is close to f ?
My answer: Yes, at least if n is sufficiently big. Indeed, Jack’s situation
is in all important respects the same as the situation of someone who
flips a coin many times and thereby becomes confident that the chanceof heads is close to the observed frequency.1
In contrast, Ćirković et al. seem to think that the frequency will
be a significant underestimate of P . I will reconstruct their position insection 4. For now, some comments about the case Q = 0 may be instructive. This is the case where any eruption is fatal. So Jack, knowing
1 More formally, the claim is that, for any ϵ > 0 and any p < 1, if n is large enough,Jack should end up assigning probability at least p to the hypothesis that P is within
ϵ of f . My general convention is to use capital letters for random variables and small
letters for their candidate values.

4that he is the nth human, can be sure even without checking the geological record that there were no eruptions at all: F = 0. On my view, heshould therefore be confident that P is close to 0. In apparent contrast,
this case leads Ćirković et al. to the view I quoted in the introduction,that ‘we should have no confidence in historically based probability estimates’ for omnicidal events. Of course, 0 is the lowest possible value
for P , so, even on my view, this observed frequency will probably be anunderestimate. But this has nothing to do with the anthropic shadow.
Consider witnessing a coin land tails 1,000 times in a row. One should
then be confident that the chance of heads is close to 0, even though,for the same reason, this will probably be an underestimate. There is, I
claim, nothing different going on in the Q = 0 version of Supervolcano.

Two Misleading Analogies
To get an initial sense of where Ćirković et al. are coming from, andwhy I disagree, it might be helpful to contrast Supervolcano with somesuperficially similar examples. Here is one:
Fishing. Sarah goes fishing in a pond, using a net with sixinch holes. The fact that she only catches fish longer thansix inches doesn’t provide much or any evidence about theprevalence of smaller fish in the pond.2
There is an apparent analogy between Fishing and Q = 0 Supervolcano.
Just as Sarah could not have found a small fish in her net, Jack could nothave found a fatal eruption in Earth’s past. So, one might think, just as
Sarah has little or no evidence about the prevalence of small fish, Jack
has little or no evidence about the prevalence of fatal eruptions. This
analogy seems to be the basic reason for thinking that that anthropic
shadow is important. However, this analogy may lead us astray.
Consider Fishing more carefully. An initial point is that, for theexample to work, Sarah’s evidence must include the fact that her net
has big holes. If the net has big holes but she herself is agnostic about
their size, then finding only big fish will give her evidence that there
are no small fish. So the case does not work merely on the basis that
Sarah could not have caught any small fish; it requires this fact to bepart of her evidence. The only question we have to ask is what her total
evidence supports.
2 This is a version of the example that opens Bostrom (2002).

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To answer this question, we note that Sarah has two key pieces ofevidence: the fact that the net has big holes, and the fact that she only
catches big fish. Given the first fact (and Sarah’s other background evidence), the second fact provides no further evidence about the prevalence of small fish. And, importantly, we naturally imagine that thefirst fact itself has little or no bearing on the prevalence of small fish.
Overall, then, Sarah has little or no evidence about the prevalence ofsmall fish.
So far, so good. Note, however, that we could have imagined thedetails of the case in a different way. Suppose that the net was carefully
designed so that it could catch all of the fish in the pond, and that Sarah
knows this. Then the first fact—the fact that the net has big holes—isevidence that there are no small fish in the pond. On this alternative
way of filling in the details, Sarah does end up with strong evidence that
there are no small fish. Note that, in both versions of the case, Sarah’s
evidence could not have been different, in the sense that Sarah could
not have caught any small fish because her net has big holes. So thedifference in the verdicts illustrates why, as I said in the introduction,‘The fact that [Sarah] could not easily have had different evidence is notimportant in itself ’.
Turning back to Supervolcano, Jack, like Sarah, has two key piecesof evidence: the fact that he is the nth Earthling, and the fact that
Fn = 0, meaning that there were no fatal eruptions in the first n generations. Again, given the first fact, the second provides no further evidence about the value of P . But we still have to ask whether the firstfact tells us anything about P .3 My basic contention is that it does tell
us something important: it is evidence that P is close to zero. In thisrespect, Supervolcano with Q = 0 is analogous to the second version of
Fishing, not the first. This conclusion falls out of my analysis of Supervolcano in the next section. For now I just emphasize that the analogy
with Fishing may well be misleading.
Similar points can be made with respect toSurvivorship Bias. David seeks out a cohort of ten centenarians who follow a Mediterranean diet. Everyone inthis cohort has lived 100 years without suffering a fatal
3 I made a similar point in the introduction, where I said that we have to attend to(A) as well as (B). The difference is that here I am focusing on certainly, rather thanpotentially, omnicidal events, corresponding to Q = 0.

6heart-attack. But this fact provides David with little or noevidence that a Mediterranean diet is good for one’s heart.4
Like Jack and Sarah, David has two key pieces of evidence: the fact thatthe people in his cohort are centenarians, and the fact that they have allgone 100 years without fatal heart-attacks. Given the first fact, the second fact provides no further evidence about the healthiness of their diet.
But we must still ask whether the first fact tells us anything of use. It
doesn’t, the way the scenario is described: David sought out centenarians, and presumably would have found some even if the Mediterranean
diet were quite unhealthy. But we need not think that Supervolcano
is analogous to Survivorship Bias in this respect. We can imagine analternative scenario, in which David chose the members of his cohort
at random when they were still young, and they all happen to live to100. Then the fact that everyone in the cohort is a centenarian would
be good evidence about the merits of the Mediterranean diet. Neither
Survivorship Bias nor this variation on it is exactly like Jack’s situation,and until we look more closely we cannot tell which scenario provides
a better guide.

3 Analysis
I am claiming that Jack’s situation in Supervolcano is not in any important respect different from the situation of someone observing a sequence of coin-flips and using the observed frequency of heads to estimate the chance. One could, with a little care, give essentially the same
analysis in both cases. However, there are some superficial differences
between these cases that I think are worth examining closely. My strategy is to consider two intermediate cases that isolate these differences,and to argue that each difference is irrelevant.
Here is the first intermediate case:
The Martians. On Mars there is a parallel sequence ofgenerations, but no extinction risk. Jill is the nth Martian,Jack’s contemporary. Jill learns all about the situation on
Earth. She wants to estimate the chance P of an eruption
4 In general, survivorship bias is the bias that arises from treating people who sur-

vived an event as if they were typical of the larger group. Ord (2020, 84), for one,says that the anthropic shadow leads to a form of survivorship bias.

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before each generation. She learns that eruptions have occured with frequency f , but there have been no fatal eruptions.
The notable difference between Jill’s situation in The Martians and Jack’s
in Supervolcano is that Jill’s existence, unlike Jack’s, does not dependon the past absence of fatal eruptions. As a result, histories with fatal
eruptions are in the anthropic shadow from Jack’s point of view, but notfrom Jill’s, so one might think that any effect of the anthropic shadow
would show up in a difference between Jack’s credences and Jill’s.
On the other hand, an interesting feature of Supervolcano is thatthe total number of people who ever exist—the ‘number of observers’—
depends on the absence of fatal eruptions. At some points Ćirković
et al. write as if this is the crucial feature of the case.5 And it is not
something that makes a difference between Jack’s situation and Jill’s. So
I will compare The Martians with a more everyday case involving coinflips.
Barking Dog. Jacob is bored at home and decides to measure the bias of his favourite coin. That is, he wants toestimate P , the chance that it lands heads. He flips it ntimes and determines the frequency F of heads, finding
F = f . Jacob’s dog, Gemma, is normally calm, but gets
excited by portraiture: each time she sees heads, there is aknown chance 1−q that she barks. As it happens, Gemma
does not bark at all during the sequence of flips.
Barking Dog is analogous to The Martians, with an eruption before thekth generation being analogous to the coin landing heads on the kth
flip, and a fatal eruption being analogous to Gemma’s barking. But(it is natural to imagine) Gemma’s barking makes no difference to thenumber of observers.
Barking Dog is a minor variation on
Ordinary Coin. Jacob is bored at home and decides tomeasure the bias of his favourite coin. That is, he wants
to find out P , the chance that it lands heads. He flips itn times and determines the frequency F of heads, finding
F = f . There are no other relevant considerations.
5 See their flow-chart on p. 1501. Readers familiar with the ‘doomsday argument’will also anticipate some connection here.

8The difference between Ordinary Coin and Barking Dog is not philosophically interesting, but perhaps it is not instantly obvious that inboth cases, Jacob should end up confident that P is close to f .

Ordinary Coin vs Barking Dog
In this section I explain the fairly simple mathematical point that both
Ordinary Coin and Barking Dog lead to the same result: Jacob should
end up confident that the chance matches the observed frequency (asusual: assuming the number of flips is sufficiently large). I begin byrecalling how this works in Ordinary Coin.
Let’s suppose Jacob has some prior probability measure Pr over various hypotheses. Specifically, Pr represents Jacob’s credences before heobserves the sequence of coin flips; it can take into account the othersorts of background evidence that one would typically have prior to witnessing a sequence of flips. The exact form of Pr will not matter, as longas it is suitably ‘agnostic’, not ruling out any relevant hypothesis about
P or F .
As a general Bayesian norm, on gaining new evidence E , Jacobshould update the probability he assigns to any other hypothesis A from
Pr(A) to Pr(A | E ). According to Bayes’s Theorem,
Pr(A | E ) =Pr(E | A)Pr(A).
Pr(E )The key point is that the probability he assigns to A gets multiplied by
Pr(A | E )= Pr(E | A) × (a factor that doesn’t depend on A).
Pr(A)In other words, learning E shifts probability towards hypotheses thatmake E likely, in proportion to how likely they make E . If Pr(E | A)is higher than Pr(E | B ), we can say that E supports A rather than B .
And, whatever agnostic prior Jacob starts with, if Pr(E | A) is sufficiently
many times larger than Pr(E | B ), Jacob should end up thinking that A
is far more likely than B .6
Now, in our particular case, the evidence that Jacob gains is thatF = f . This evidence supports certain hypotheses about the value of
6When considered as a function of A, L (A) := Pr(E | A) is often called the likelihood function. I will use this terminology to some extent when we get to section 5,
but, to be clear, I use the word ‘likely’ in the usual way as a synonym for ‘probable’.

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Figure 1: A coin with unknown bias P is flipped n times, and the frequency F = 0.4
of heads is observed. The ‘likelihood’ L ( p) := Pr(F = 0.4 | P = p) is given by thebinomial distribution, peaking at p = 0.4. Plotted here is L ( p) as a percentage of thepeak value L (0.4), for n = 20 (solid line), n = 60 (dashed), and n = 400 (dotted).
As n increases the peak becomes narrower, meaning that the evidence more strongly
confirms that P is close to 0.4.

P . To find out which hypotheses it supports most strongly, we should
consider the conditional probability Pr(F = f | P = p). That is, whatvalues of P would make the frequency f particularly likely?
Since Jacob doesn’t initially have any special information about theflips to begin with, the probability he assigns to each flip landing heads,
given that P = p, should be p. It follows from some elementary combinatorics that the probability that F = f is given by the binomial
distribution:7
Pr(F = f | P = p) =n!
p f n (1 − p)n− f n .
( f n)!(n − f n)!

Holding f fixed, think of this as a function of p. It equals zero for
p = 0, 1 (unless f = 0, 1) and peaks at p = f . The peak gets narrower
as n increases. The effect, then, is that learning F = f shifts probability
towards P = p, and this effect gets stronger as n gets bigger. See Figure
1.
That is the sense in which learning F = f provides Jacob with evidence that P ≈ f , and the sense in which the evidence gets stronger as
n gets larger. Whatever agnostic prior Jacob started with, if n is large
enough, Jacob should become confident that P ≈ f .
7 Note that f n is just the observed number of heads. The probability of any particn!
ular sequence of flips with f n heads is p f n (1 − p)n− f n , and there are ( f n)!(n−f n)! suchsequences.

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Now let us turn to Barking Dog. In this case too, Jacob should
become confident that P ≈ f . The dog is irrelevant. Why so?
The chance P determines how likely any given number of heads is,and the number of heads (along with Q ) determines how likely Gemma
is to bark. But if we fix the number of heads, P plays no further role indetermining how likely Gemma is to bark. More precisely, given that
there were a total of f n heads, the probability that Gemma never barks
is q f n , regardless of any hypothesis about P :Pr(NO BARk | F = f , P = p) = q f n .
So, once Jacob knows F = f , NO BARk provides no further evidence
about the value of P . On learning that F = f and that Gemma did
not bark, Jacob’s confidence that P ≈ f should end up exactly the sameas in the previous case, Ordinary Coin.

Barking Dog vs The Martians
Now I will explain why I think Jill in The Martians, like Jacob in Barking
Dog, should become confident that P ≈ f .
As I said in section 3, the salient difference between the cases is that,in The Martians, the total number of observers is a matter of chance (itdepends on whether and when there is a fatal eruption), whereas inBarking Dog we naturally imagine that the total number of observers is
independent of anything that happens with the coin.
I put it this way, in terms of what we ‘naturally imagine’, because
our analysis of Ordinary Coin and Barking Dog did not involve any
explicit premise about the total number of observers, and it seems odd
to claim that our ordinary chance reasoning involves a hidden premiseof this type. In fact, our analysis of these cases flowed entirely fromthe premise that, conditional on P = p, Jacob should initially assign
probability p to the coin’s landing heads:
Pr(HEADS | P = p) = p.

(*)

The worry must be that the analogous premise does not apply to Jill.
There is, it turns out, a reason to worry here (though not, I claim,
a good one). In talking about Jacob’s priors Pr, I meant prior to observing the frequency and (therefore) prior to learning whether Gemma barks.
But we can imagine a more radical ‘prior circumstance’ in which Jacob
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also lacks all self-locating evidence: though he has all the requisite background knowledge in third-personal terms, he does not know which of
the many people in the world he is; he does not know I am Jacob.8
Let Pr◦ represent the credences of this self-ignorant version of Jacob.
What probability should he assign to each coin-flip landing heads? One
might think that the answer is the same as before:Pr◦ (HEADS | P = p) = p.

(**)

Unfortunately, (*) and (**) are not generally compatible.
To say more, let’s assume that although Jacob does not initially know
I am Jacob, he does know I exist and also I am a person (as well as Jacob
exists and Jacob is a person); beyond this, he lacks self-identifying information. Then, he should think that I am Jacob is relatively likely conditional on there being few people, and relatively unlikely conditionalon there being many. Learning I am Jacob should therefore shift probability towards worlds with relatively few people.9 And so, if the resultof the coin-flip is correlated with the total number of people, learning
I am Jacob will change the probability he assigns to heads; it will tend
to favour whichever result is correlated with there being fewer people.10
Under these circumstances, Jacob can align his credence in heads withthe known chance either (**) before learning I am Jacob or (*) afterwards,
but not both. In particular, if something like (**) is correct, then Jill
8 Cf. Lewis (1979). There is a complication I will gloss over here. We should
imagine that Jacob doesn’t know other indexicals like It is now 5 o’clock. Because ofthis, the relevant thing is not ultimately the number of observers but the number of‘observer-moments’ (people at times).
9 For example, suppose Jacob’s credences are initially split between two possible
worlds: world 1 contains 10 people and world 2 contains 100. Assuming thathe gives roughly equal probability to different hypotheses about his own identity
in each world, his credence in world 1 is split roughly equally between 10 different self-locating hypotheses, and roughly equally among 100 for world 2. ThenPr◦ (I am Jacob | world 1) ≈ 1/10 whereas Pr◦ (I am Jacob | world 2) ≈ 1/100. So
I am Jacob provides support for world 1 over world 2 by a factor of 10.
10 This is a version of the ‘doomsday argument’ to the effect that doomsday eventsare more likely than one would naively think (see Thomas, 2021, for discussion).
Compare also the famous case of Sleeping Beauty (Elga, 2000). There as here, purely
self-locating evidence (‘It’s Monday’) bears on third-personal hypotheses about theworld (‘Heads’ versus ‘Tails’) because different third-personal hypotheses leave open
different self-locating possibilities. (The ‘double-halfer’ view about Sleeping Beauty essentially denies that purely self-locating evidence bears on third-personal hypotheses
even in this case.)12

should not satisfy (*), and so the analysis of her case will be different
from Jacob’s.
Now, we could block this line of thought by removing the assumption that the radically ignorant version of Jacob knows I exist and I am aperson. That is, we could stipulate that the Pr◦ that appears in (**) does
not already incorporate these pieces of evidence. We could hold that
one or the other of these pieces of evidence bears on the total numberof people (and therefore on the probability of heads) in just the rightway so as to reconcile (**) with (*).11
My own view of the matter is simpler: we should endorse (*) andnot worry too much about whether (**) is true.12 The compelling principle is that one should align one’s credences with the known chancesin normal situations. As to the bizarre situation of having no idea whoor where or when one is in the world, let the chips fall where they may.
It would be unfortunate—and from a theoretical point of view, it iscompletely unnecessary—for our ordinary reasoning about chances tobe hostage to an analysis of the downstream effects of coin-flips on the
total number of people, or hostage to speculation about the evidential
weight of one’s own personhood and existence.
While this is bound to be one of the points of controversy, (*) is
certainly defensible both in Jacob’s case and in Jill’s, and from there theanalysis proceeds in the same way: they should both end up confident
that P is close to f .

The Martians vs Supervolcano
Now we get to the pair of cases that seems to speak most directly tothe original worry about the anthropic shadow. The main difference
between Jack and Jill is that Jill could have discovered a fatal eruptionin Earth’s past; Jack could not have, since a fatal eruption would haveprecluded his existence. Should this make a difference to their credences
about P ?
I claim it shouldn’t. The overall point is one I already mentioned in
the introduction: Jack and Jill have essentially the same evidence about
11 In the literature, the relevant assumption about the evidential bearing of I exist

is known as ‘the self-indication assumption’; the status of I am a person is relevant todiscussions of one’s ‘reference class’. See Bostrom (2002) for these ideas.
12 I develop this view more formally in Thomas (2021), and draw out its consequences. Gallow (forthcoming) defends a similar view from a different angle.

13

the value of P , and the allegation that Jill but not Jack could have had
different evidence is irrelevant. But let’s dig a bit deeper.
Let me first dispel the thought that Jack can know a priori that there
were no fatal eruptions (his very existence depends upon it!) whereas
Jill has to find it out. This is not right (and even if it were right, it isn’t
clear why it would matter). Just like Jill, Jack’s basis for thinking there
were no fatal eruptions through the nth generation is his knowledge ofthe basic set-up and the fact that there is an nth Earthling. It’s true that
Jack himself is the nth Earthling, but this is not something he knowsa priori. Indeed, a priori, he might have lived on Mars; he might havebeen Jill!
To go further, we can fill in the details of the case so that Jack andJill end up with nearly all the same evidence. For example:
1. They both know the basic chance set-up governing eruptions on
Earth, and they both know the value of Q , although neither ofthem knows the true value of P .
2. They both know the same things about the history of the universeup through the present time, from a third-personal perspective.
They both know the frequency of eruptions, they both know that
Jack and Jill exist, and they both know that Jack and Jill are thenth Earthling and the nth Martian, respectively.
3. They also have a lot of self-locating evidence in common. For
example, each of them knows I exist and I am Jack or Jill.
There is one main piece of evidence that we cannot insist they both share.
Jack knows the merely self-locating I am Jack, whereas Jill knows I am
Jill. So, if we think there is an important difference between Jack’s evidence and Jill’s, it must come down to the idea that, even once we havetaken into account all their considerable shared evidence, the merely
self-locating I am Jack has some further bearing on the value of P that
I am Jill does not.13
I do not see what this bearing would be. In the previous section, I
pointed out that merely self-locating evidence (‘I am Jacob’) can have a
13 Perhaps there are some other sorts of evidence that Jack and Jill cannot be stipulated to share, such as some essentially private evidence about phenomenal experiences.
I will gloss over this possibility, though again, the question would be whether (given all
the evidence they do have in common) this private evidence has some further bearing
on the value of P .

14bearing on the probability of third-personal hypotheses. This happens
when the rival third-personal hypotheses leave open different numbersof self-locating hypotheses. For example, if Jacob knows he’s a person
but not which person, then learning I am Jacob shifts probability towards worlds with fewer people. In the present case, however, the evidence shared by Jack and Jill leaves open exactly two self-locating hypotheses, I am Jack and I am Jill. So here there is no reason to thinkthat learning one of these hypotheses should change Jack’s views about
the chances.
Indeed, the claim would have to be that, given all the shared evidence listed above, I am Jack is more likely conditional on some valuesof P than it is conditional on others. But, again, conditional on eachpossible value of P , the shared evidence leaves open exactly two possibilities: I am Jack and I am Jill. The most natural view is that these two
possibilities are equally likely. Thus, I am Jack has the same probability
conditional on each value of P .
Now, a subjective Bayesian might think that, even if this is the mostnatural view, it is permissible for Jack to distribute his priors in a different way, so that learning I am Jack does provide some further evidence about P . In Thomas (2021) I argue against this thought: it isrationally required in cases like this to treat I am Jack and I am Jill as
equally likely.14 But even if we agree with the subjective Bayesian, we
haven’t identified anything about the situation that distinguishes Jack’s
position from Jill’s. If, when all is said and done, Jill must be confident
that P ≈ f , then so must Jack.

4 Ćirković et al.
That completes my argument that Jack should become confident thatthe chance of an eruption is close to the observed frequency. It is justlike an ordinary coin-flip case, and, in that sense, the anthropic shadow
is irrelevant.
Why, then, do Ćirković et al. suggest that attending to the historical
record will lead to underestimating the chances? The key example thatthey use to support this claim is described as follows:
For instance, suppose Q = 0.1 and P = 0.5, correspond14 See also the principle of ‘indifference’ defended by Elga (2004), and, for dissent,
Weatherson (2005).

15

ing to a fair-coin-toss chance that a [supervolcanic] event
occurs once per [period], and that the probability of human survival following such an event is 0.1…[T]he actual
probability [the chance?] of such an event is 5.5 times our
initial estimate [the frequency?]. (1497)
As my bracketed glosses suggest, I think it is difficult to unpick what exactly is being claimed in this passage. So I will explain where the number
5.5 comes from, and what it really means, in this section. Throughout
this discussion, consider Jack’s position before he learns the frequency,
but after he knows that there were no fatal eruptions.15
I think it is useful to see what is going on from two different angles.
First: in this position, Jack should be inclined to think that there were
unusually few eruptions. That is, he should be inclined to think that, if
P = p, then the past frequency of eruptions was smaller than p. This isbecause the presence of few eruptions increases the chance that none of
the eruptions were fatal. And the lower q is (i.e., the deadlier eruptions
tend to be), the larger this discrepancy. More formally, suppose Jack
were to learn that P = p. What probability would he then assign tothere being an eruption before each generation? Not the unconditionalchance of an eruption, p, but the chance conditional on no fatal eruption.
By definition, this is the chance of a non-fatal eruption ( pq) divided by
the chance of no fatal eruption (1 − p + pq); in short,
p0 := pq/(1 − p + pq).
The same sort of calculation we did with binomial distributions in Coin
Flip leads us to this conclusion:(a) Jack should be confident, conditional on P = p, that F ≈ p0 .
Now, p0 is always smaller than p, and the ratio between them increases
as q gets smaller (further, one might say, into the anthropic penumbra).
In our particular case, if q = 1/10 and p = 1/2, then p0 = 1/11. The
discrepancy betwen p and p0 is the factor of 5.5 cited by Ćirković et al.
However, it would be wrong to conclude from this calculation that
Jack should treat the observed frequency as a 5.5-fold underestimateof the chance. The question about how Jack should treat the observed
frequency is the question of what estimate Jack should adopt for P if
he learns F = f . The answer I have given is
15 By the way, precisely analogous points apply to Jill in The Martians and to Jacobin Barking Dog.

16

(b) Jack should be confident, conditional on F = f , that P ≈ f .
It may be surprising that (a) and (b) are compatible. Here’s thepoint. According to (a), if Jack learns P = p, the fact that there were
no fatal eruptions provides evidence that there were unusually few eruptions ( p0 rather than p per period). This is because there are more likely
to be no fatal eruptions if the total number of eruptions is smaller. However, if Jack learns F = f , the fact that there were no fatal eruptions
provides no additional information about P , and does nothing to suggest that P is ‘unusually large’ (larger than f ). This is because, once wehave fixed the total number of eruptions, the underlying chance P has
no bearing on the number of fatal eruptions.
Let us look at this same issue from a different angle, which willbring out the connection with claims (A)–(C) in the introduction, andthe issue of base-rate neglect.16 Again, consider Jack’s position when heknows there were no fatal eruptions, but does not yet know the valueof F . At that point, I claim, learning F = f provides strong evidence
that P > f . More precisely, define
f0 :=

f
.
q + (1 − q) f

(This is the value of P that makes it most likely that F = f ; that is, if
p = f0 , then p0 = f .) One can check that f0 is always greater than f ,and, I claim,(c) Learning F = f would provide Jack with strong evidence thatP ≈ f0 .
For example (really the same example as before), if q = 1/10, thenF = 1/11 is strong evidence that P ≈ 1/2, 5.5 times bigger than theobserved frequency! And this evidence gets stronger as n gets larger.
16 Here is a classic example of base-rate neglect. Let D be the hypothesis that Jones
has a certain disease, and Neg the hypothesis that Jones tests negative. If Pr(Neg | D) <0.01, then a negative test provides strong evidence that Jones does not have the disease.
In fact, it ‘rules out the hypothesis that Jones has the disease, at 99% confidence’.
However, this does not mean we should become anything like 99% confident that
Jones is disease-free. It does mean that we should strongly update in that direction.
But if we started off highly confident that Jones has the disease (a high base-rate), we
might still end up quite confident that she has it.

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It may be surprising that (c) is compatible with (b). Indeed, one
might naively conclude from (c) that, if n is sufficiently large, Jackshould become confident that P ≈ f0 , rather than that P ≈ f . However, this conclusion would again be too hasty. For there are two key
pieces of evidence:(i) There were no fatal eruptions.
(ii) F = f .
The point is that (i) is evidence that P ≈ 0. The bigger n is, the stronger
the evidence. Since we are considering Jack’s position when he knows(i) and is yet to find out (ii), he should be confident that P is close to 0,
which is less than f . Claim (c) tells us that conditionalizing on (ii) will
shift probability towards P ≈ f0 , which is greater than f . Claim (b)tells us that the overall effect is to make Jack confident that P ≈ f . It istrue that, by considering larger and larger values of n, we can make theshift towards P ≈ f0 more dramatic. However, increasing n also makes
Jack antecedently more confident that P ≈ 0. So, overall, increasingn just makes Jack more confident that P ≈ f , not that P ≈ f0 . We
can see this overall effect very clearly if we imagine what would happen
if someone learnt (i) and (ii) in the opposite order. Learning (ii) firstwould provide powerful evidence that P ≈ f . And once one knows(ii), (i) has no further bearing on the value of P . I made exactly thispoint about Jacob’s situation in Barking Dog: once he knows F = f ,NO BARk provides no further evidence about the value of P .

5 Other Literature
So far I have focused on the model used by Ćirković et al. Here I sketch
how my analysis extends to the models used by Tegmark and Bostromand by Snyder-Beattie et al.

Snyder-Beattie et al.
Snyder-Beattie et al. try to estimate the annual extinction risk for the
species Homo sapiens. Setting anthropogenic risks aside, and focusing
on natural background risks, their model assumes an unknown, con-18

stant rate µ of extinctions per year.17 Let T be the lifespan of our
species. Then, given any candidate value µ0 of µ, the prior distributionof T is Pr(T ≥ t | µ = µ0 ) = e −µ0 t . Henceforth let tnow = 200 kyr
be an estimate for the current age of humanity. Then the ‘likelihood
function’L (µ0 ) := Pr(T ≥ tnow | µ = µ0 ) = e −µ0 tnow
gives the relative strength with which the evidence T ≥ tnow supports
different hypotheses about µ. The authors find that T ≥ tnow supports
a low value of µ ≈ 10−8 a million times more strongly than it supports
a high value of µ ≈ 7 × 10−5 . In that sense, 7 × 10−5 provides an upper
estimate for µ.18
So far, so good. But Snyder-Beattie et al. worry that such estimates
are subject to an ‘observation selection bias’ related to the anthropic
shadow. This is the passage I quoted in the introduction. Translated
into the present context, the worry is that a species like Homo sapiens
might require a long gestation time (as it were) before its members countas ‘observers’ in the sense that they are able to inspect the historical
record.19 Any human observer would necessarily find such a long period
in the human past, and the authors worry that it would be improper ormisleading to use the existence of this time-period as evidence about µ.
Let me reiterate the basic reason this worry is misguided, before
turning to the formal treatment of observation selection biases. Evenif human observers will ‘necessarily’ find a long gestation time in humanity’s past, this isn’t a matter of epistemic necessity. The need for a
gestation time is something we might discover through such empiricalsciences as geology, evolutionary biology, history, and so on. Together
with the fact that we are human observers, it implies that there was along gestation time in our past; and this empirical conclusion is perfectly good evidence that the rate of extinction is low. Again, the factthat (as a matter of nomological necessity) we could not easily have had
17To ease comparisons with the literature, I use ‘µ’ and later ‘τ’ to denote randomvariables, in exception to my general convention of using capital letters.
18 Of course, this isn’t the full Bayesian analysis; if one’s prior over values of µstrongly favoured high values, then one might still end up confident in values higher
than 7 × 10−5 .
19 They write: ‘If desired, we could more crisply define this observerhood property
as the ability for a species to collect reliable data on its own track record of survival(e.g. via fossil dating) and analyse it.’19

different evidence does not change the weight of the evidence we dohave.
In thinking this through, it may help to return again to the perspective of Jill the Martian. Consulting the historical record on Earth, sheobserves that Homo sapiens has been around for 200 kyr and that withinthat time it has reached observerhood. Given a constant-rate model ofextinction, she estimates µ in exactly the way I described above, using
the likelihood function L . But, as I’ve argued, what goes for Martians
goes for Earthlings too. That’s because merely self-locating facts like
‘I’m Jack the human’ as opposed to ‘I’m Jill the Martian’ provide no evidence about µ, given the rich background information that Jack andJill hold in common.
If these general comments are correct, they raise a puzzle: what isgoing wrong with the more formal analysis by Snyder-Beattie et al. that
purports to show observation selection bias in action? Let me sketchthat analysis along with my diagnosis.
The authors extend their model to include a random variable S giving the time from the origin of humanity to the point at which humans
become observers (we can take S = ∞ if humans never do). They define
FS (t ) = Pr(t ≥ S | T ≥ t , µ = µ0 ) to be the probability that humans become observers before time t , given that they survive until at least timet ; for the types of risks under consideration, we can assume that FS (t )does not depend on µ0 . The authors then reach modified estimates forµ using the likelihood function
L ∗ (µ0 ) := Pr(T ≥ tnow | T ≥ S , µ = µ0 ).
That is, they treat as background evidence the fact that humans at somepoint become observers (T ≥ S ) and ask what, against that background,the additional evidence T ≥ tnow tells us about µ. While they consider
a variety of distributions for S , we can get a sense of their results byconsidering their ‘Model 4’, according to which observerhood occurs
at a fixed time tobs . Thus
L ∗ (µ0 ) = Pr(T ≥ tnow | T ≥ tobs , µ = µ0 ) = e −µ0 (tnow −tobs ) .
Let’s consider the extreme case in which we only recently passed thethreshold tobs for observerhood, so tobs = tnow = 200 kyr. Then L ∗ (µ0 ) =1 independent of µ0 . This conclusion is supposed to represent mathematically the thought that if observerhood requires at least 200 kyr of20

gestation, then observing such a past is uninformative about the rate ofextinction risk.
There are two problems with this analysis, as far as I can see. First,
L ∗ tells us to what extent the observation T ≥ tnow supports various
values of µ, given background evidence T ≥ S . However, T ≥ S is
itself significant evidence, and what we want to know is what all thisevidence tells us about µ. Second, we in fact know something stronger
than T ≥ tnow and T ≥ S ; we know T ≥ tnow ≥ S .20
For these two reasons, the likelihood function that accounts for all
our evidence in the model is not L ∗ but
L ∗∗ (µ0 ) := Pr(T ≥ tnow ≥ S | µ = µ0 ).
Using the definition of conditional probability, we can factor this as
L ∗∗ (µ0 ) = Pr(tnow ≥ S | T ≥ tnow , µ = µ0 ) × Pr(T ≥ tnow | µ = µ0 )= FS (tnow ) × L (µ0 ).
Recall that FS (tnow ) does not depend on µ0 , so L ∗∗ is just L times aconstant. Therefore the estimates for µ that we get using L ∗∗ will be
exactly the same as the ones we got using L , before we tried to takeobservation selection effects into account. For example, it is still thecase that our evidence supports a low value µ ≈ 10−8 a million times
more strongly than it supports a high value of µ ≈ 7 × 10−5 . There are
no observation selection effects, as far as this model goes.

Tegmark and Bostrom
The story about Tegmark and Bostrom (2005a) is similar in its essentials,
and I will treat it more briefly, with less conceptual discussion.21 Theseauthors want to estimate the rate µ of events like vacuum decay, measured in events per gigayear, so that the typical timescale τ := 1/µ has
units of gigayears. Such an event would annihilate any planets nearby
and prevent any new ones from forming. The authors attempt to estimate µ (or equivalently τ) in a way that is immune to the effects of theanthropic shadow.
20 For simplicity, I’m assuming that we can rely on 200 kyr as a precise estimate ofhumanity’s current age.
21 I refer in this section to the extended version of their paper, Tegmark and Bostrom(2005b).

21

Let Tv be the time of the first vacuum decay after the big bang.
Vacuum decay is assumed to happen at a constant rate 1/τ, so that
Pr(Tv ≥ t | τ = τ0 ) = e −t /τ0 . Now, naively, we could take tnow :=13.7 Gyr as an estimate of Earth’s current age, and generate estimatesfor τ using the likelihood function
L (τ0 ) := Pr(Tv ≥ tnow | τ = τ0 ) = e −tnow /τ0 .
However, in an effort to circumvent the anthropic shadow, the authors complicate things by introducing two further events: a ‘formation’ event, occuring at time Tf , which results in the formation of Earth
if there has not yet been vacuum decay (Tf ≥ Tv ); and an ‘observerhood’ event, occuring at time Tf + To , which results in the evolutionof observers on Earth, if there has still not yet been vacuum decay(Tv ≥ Tf + To ). The authors use a particular model of planetary formation to give a prior probability distribution for Tf ; the distribution
for To turns out not to matter, assuming that Tv , Tf , and To are independent.
Henceforth take tf = 9.1 Gyr as an estimate of Earth’s actual formation time. To get an estimate for τ, Tegmark and Bostrom consider the
likelihood function
L ∗ (τ0 ) := Pr(Tf ≥ tf | Tv ≥ Tf + To , τ = τ0 ).
L ∗ (τ0 ) is the chance that Earth would form at least as late as it actually did, given that observers managed to evolve. If τ is low, then thischance is low (more likely, Earth evolved early, allowing more time for
evolution to do its work). Thus, the fact that Earth formed so late is (inthis context) evidence that τ is high. Tegmark and Bostrom find thesmallest τ0 such that L ∗ (τ0 ) ≤ 0.05. This provides a lower estimate forτ, namely, τ ≥ 2.5 Gyr.22
Although the method here is not quite parallel to the one used by
Snyder-Beattie et al., it raises essentially the same two issues.
First, the above analysis effectively treats the evolution of observers(Tv ≥ Tf +To ) as background evidence, and against this background asks
what the late formation of the Earth tells us about the value of τ. But
we should also consider what the background evidence tells us. In fact,
22As they put it in standard frequentist terminology, ‘we can rule out the hypothesisthat τ < 2.5 Gyr at 95% confidence’ (p. 3). They give estimates at other levels ofconfidence as well, and some other considerations about the robustness of their model.

22second, we should ask what is supported by the entirety of our evidence
about the parameters of the model. We have ballpark figures for both
Tf and To , and, more importantly, we know that Earth is tf = 13.7 Gyrold, so Tv ≥ 13.7 Gyr ≥ Tf + To . Thus the likelihood function that
captures all our evidence in this model is
L ∗∗ (τ0 ) = Pr(Tv ≥ 13.7 Gyr, Tf ≈ 9.1 Gyr, To ≈ 4.5 Gyr | τ = τ0 ).
On the standing assumption that Tf and To are independent of Tv , wefind that L ∗∗ is the same as L up to a constant scale-factor. We could,
then, just as well have used L to generate estimates for τ. For example,
the condition L (τ0 ) ≤ 0.05 gives a lower estimate of τ ≥ 4.5 Gyr.

6 Conclusion
My analysis of the anthropic shadow highlights three main points.
First (as discussed throughout, but especially in section 3), the basic
worry about the anthropic shadow is misguided. Jack and Jill differ
with respect to whether their existence depends on the absence of afatal supervolcano eruption, but their evidence does not differ in any
way relevant to the rate of eruptions.
Second (as discussed mainly in section 3), a full analysis of the situation depends on theoretical choices about the circumstances underwhich one should align one’s credences with the known chances. My
own view is that Jacob and Jill are both in the right circumstances to doso. Even if the reader is not convinced of this view, the general point isinteresting because these theoretical choices are not explicitly discussed
in the small literature on the anthropic shadow, even as they are contentious within the wider literature on ‘anthropics’.
Third, analyses that do claim to find an effect of the anthropic
shadow are misleading because of a form of base-rate neglect. For example, Ćirković et al. do not take into account the fact that a longtime without omnicidal events is good evidence that the rate of potentially omnicidal events is low. And Snyder-Beattie et al. do not take
into account the fact that a long gestation period for humanity is goodevidence that the extinction rate for humanity is low. Of course, thecogency of this critique depends on the first point made above. Jill the
Martian can obviously appeal to these long ‘safe’ periods as evidence
about extinction rates—and what goes for Jill should go for us as well.
Thus is the anthropic shadow finally dispelled.
23

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Elga, Adam (2004). ‘Defeating Dr. Evil with self-locating belief ’, Philosophy and Phenomenological Research 69(2):383–396.
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