# The Noonday Argument: Fine-Graining, Indexicals, and the Nature of
Copernican Reasoning

Brian C. Lacki<EMAIL_ADDRESS>Breakthrough Listen, Astronomy
Department, University of California, Berkeley CA, USA

###### Abstract

Typicality arguments attempt to use the Copernican Principle to draw
conclusions about the cosmos and presently unknown conscious beings within it.
The most notorious is the Doomsday Argument, which purports to constrain
humanity’s future from its current lifespan alone. These arguments rest on a
likelihood calculation that penalizes models in proportion to the number of
distinguishable observers. I argue that such reasoning leads to solipsism, the
belief that one is the only being in the world, and is therefore unacceptable.
Using variants of the “Sleeping Beauty” thought experiment as a guide, I
present a framework for evaluating observations in a large cosmos: Fine
Graining with Auxiliary Indexicals (FGAI). FGAI requires the construction of
specific models of physical outcomes and observations. Valid typicality
arguments then emerge from the combinatorial properties of third-person
physical microhypotheses. Indexical (observer-relative) facts do not directly
constrain physical theories. Instead they serve to weight different
provisional evaluations of credence. These weights define a probabilistic
reference class of locations. As indexical knowledge changes, the weights
shift. I show that the self-applied Doomsday Argument fails in FGAI, even
though it can work for an external observer. I also discuss how FGAI could
handle observations in large universes with Boltzmann brains.

##  I Background

What can we learn from the simple fact that we exist where and when we do? The
answer may bear on many profound questions, including the nature and size of
the cosmos, the existence and types of extraterrestrial intelligence, and the
future of humanity.

Attempts to reason about the cosmos from the fact of our existence have been
called anthropic reasoning [e.g., 1, 2]. The anthropic principle argues that
our existence is expected even if the events that lead up to it are rare [3].
In its weakest form, it simply asserts that humanity’s existence is
probabilistically likely in a large enough universe [4]. The strongest
versions of the anthropic principle have an opposite premise at heart [2]: our
existence is not a fluke, but somehow necessary in a logical sense [5].
Anthropic reasoning has been extended to include Copernican principles, which
emphasize the typicality of our environment. Weak forms point out our
evolution was not a special rupture in the laws of physics, but one possible
outcome that can be repeated if given enough “trials” in sufficiently many
cosmic environments like ours. Strong forms state that conscious beings
(“observers”) like ourselves are common.

Anthropic reasoning frequently tries to synthesize the anthropic and
Copernican principles: we should regard our circumstances as typical of
observers like ourselves. Bostrom [2] has formalized this notion as the Self-
Sampling Assumption (SSA). The group of observers considered similar enough to
us for Copernican reasoning to be valid is our “reference class”. It may be as
wide as all possible sentient beings or as narrow as people exactly identical
to your current self. There is no consensus on a single reference class, or
indeed whether we might use a multitude [e.g., 6, 7, 2], although the more
extreme Copernican formulations apply the universal reference class of all
observers. Typicality arguments are often justified in normal experiments to
derive conclusions when unusual outcomes are expected given enough “trials”.
In fact, some kind of typicality assumption seems necessary to reason about
large cosmologies, where thanks to the anthropic principle there will exist
observers like ourselves with certainty even in Universes distinct from ours –
otherwise observations have no power to constrain the nature of the world [8,
9]. Typicality is commonly invoked in discussions of cosmology as the
“principle of mediocrity” [10].

The seemingly reasonable Copernican statement of the SSA has led to
controversy, as it can be applied to constrain the cosmological contexts of
as-of-yet unobservable intelligences in the Universe. Few applications of the
Copernican principle are more contentious than the Doomsday Argument. In its
most popular form as presented by Gott [11], we are most likely “typical”
humans and therefore are unlikely to be near humanity’s beginning or end [see
also 12]. The Bayesian Doomsday Argument, most strongly defended by Leslie
[1], has a more robust basis: our birthrank (the number of humans before us)
is treated as a uniform random variable drawn from the set of all birthranks
of the final human population. A larger human population has more “outcomes”,
resulting in a smaller likelihood of “drawing” your specific birthrank and a
Bayesian shift favoring a short future for humanity [see also 13, 2]. A
generalized non-ranked variant has been brought to bear to evaluate the
existence of beings unlike ourselves in some way, as in the Cosmic Doomsday
argument [14].

The Doomsday Argument and similar typicality arguments would have profound
implications for many fields. For example, in the Search for Extraterrestrial
Intelligences [SETI 15] the prevalence of technological societies is
critically dependent on the lifespan of societies like our own [e.g., 16, 17,
18]. Doomsday would be an extraordinarily powerful argument against prevalent
interstellar travel, much less more exotic possibilities of astronomical-scale
“megastructures” [19, 20]. The instantaneous population of a galaxy-spanning
society could be $\sim 10^{20}$ [11, 14] while predictions of intergalactic
travel and astro-engineering suggest populations greater than $10^{50}$ [21,
22]. Yet the Doomsday Argument applies huge Bayes Factors against the
viability of these possibilities or indeed any long future, essentially
closing off the entire field of SETI as difficult to futile [11]. Similar
Bayesian shifts might drastically cut across theories in other fields, like
landscape hypotheses.

This sheer power cannot be stressed enough. These likelihood ratios for broad
classes of theories ($>10^{40}$ in some cases!) imply more than mere
improbability, they are far more powerful than those resulting from normal
scientific observation. If we have any realistic uncertainty in such futures,
even the slightest possibility of data being hoaxed or mistaken (say
$10^{-9}$) results in epistemic closure. If we discovered a galaxy-spanning
society, or if we made calculations implying that the majority of observers in
the standard cosmology live in realms where the physical constants are
different from ours, then the evidence would force us to conclude that
scientists are engaged in a diabolical worldwide conspiracy to fake these
data. The SSA even can lead to paradoxes where we gain eerie “retrocausal”
influence over the probabilities of _past_ events by prolonging humanity’s
lifespan [23].

Given the unrealistic confidence of the Doomsday Argument’s assertions, it is
not surprising that there have been many attempts to cut down its power and
either tame or refute typicality [e.g., 24, 25, 26, 27, 28, 6, 2, 29, 30].
These include disputing that our self-observation can be compared to a
uniformly drawn random sampling [24, 26, 30], arguing for the use of much
narrower reference classes [6, 2, 31], or rejecting the use of a single
Bayesian credence distribution [32, 29]. The most common attack is the Self-
Indication Assumption (SIA): if we really are drawn randomly from the set of
possible observers, then any given individual is more likely to exist in a
world with a larger population. The SIA demands that we adopt a prior in which
the credence placed on a hypothesis is directly proportional to the number of
observers in it, which is then cut down by the SSA using our self-observation
[24, 25, 28, see also Neal 6 for further discussion]. Yet this prior too
posits absurd levels of confidence, not at all like how we actually reason: we
do not actually start out assuming a posthuman galactic future is $\gtrsim
10^{20}$ times more likely than one where we go extinct in 2100, and then
update to equal credence upon learning we live in 2021. Moreover, the SIA
leads to the Presumptuous Philosopher problem: the SIA prior leads us to favor
cosmologies where the Universe is large, possibly to a ridiculous degree (say,
$\sim 10^{100}$) without ever making any observations [33, 34]. But the
Doomsday Argument is a Presumptuous Philosopher problem in spirit too, just in
the opposite direction – one develops extreme certainty about far-off
locations without ever observing them.

In this paper, I critically examine and deconstruct the Copernican typicality
assumptions used in the Doomsday Argument and present a framework for
understanding them, Fine Graining with Auxiliary Indexicals (FGAI). I start
with an analysis of the self-applied Doomsday Argument and how it can lead to
solipsism (Section II). Using the “Sleeping Beauty” thought experiment, I
illustrate the disanalogy between these arguments and similar-seeming valid
arguments (Section III). I then lay the groundwork for FGAI by showing how
fine-graining cosmological theories can account for many of the uses for
typicality without making reference to indexical propositions (Section IV).
Section V reintroduces indexical credences as weights for physical credence
distributions. Finally, I discuss the nature of the Copernican Principle and
summarize why the self-applied Doomsday Argument fails in FGAI (Section VI).

Throughout this paper, I use the terms “Small” and “Large” to broadly group
theories about unknown observers [as in 1]. In Small theories, the majority of
actually existing observers are similar to us and make similar observations of
their environment, while Large theories propose additional, numerically
dominant populations very dissimilar to us. When specifically talking about
the lifespan of a population, a “Short” history is one that is Small while a
“Long” history is one that is Large (e.g., $N_{\rm total}\gg 10^{11}$ for
human history).

##  II The Doomsday Argument and its terrible conclusion

The Doomsday Argument is arguably the most far-reaching and contentious of the
arguments from typicality. It generates enormous Bayes factors against its
Large models, despite relatively plausible (though still very uncertain)
routes to Large futures. By comparison, we have no specific theory or forecast
implying that inorganic lifeforms [c.f., 6] or observers living under very
different physical constants elsewhere in a landscape dominate the observer
population by a factor of $\gg 10^{10}$. Cases where we might test typicality
of astrophysical environments, like the habitability of planets around the
numerically dominant red dwarfs [35], generate relatively tame Bayes factors
of $\sim 10\text{--}100$ that seem plausible (a relatively extreme value being
$\sim 10^{4}$ for habitable planets in elliptical galaxies from [36, 37]). For
this reason, it is worth considering Doomsday as a stringent test of the
“Copernican Princple”.

Let us consider how the Doomsday Argument works and the SSA’s role in it.

###  II.1 Noonday, a parable

A student asked their teacher, how many are yet to be born?

The teacher contemplated, and said: “If the number yet to be born equaled the
number already born, then we would be in the center of history. Now, remember
the principle of Copernicus: as we are not in the center of the Universe, we
must not be in such a special time. We must evaluate the $p$-value of a
possible final population as the fraction of people who would be closer to the
center of history.

“Given that 109 billion humans have been born so far,” continued the teacher,
citing Kaneda and Haub [38], “The number of humans yet to be born is not
between 98 and 119 billion with 95% confidence. There may be countless
trillions yet to be born, or none at all, but if you truly believe in
Copernicus’s wisdom, you must be sure that the number remaining is not 109
billion.”

And so all who spoke of the future from then on minded the teacher’s Noonday
Argument.

###  II.2 Noonday, $x$-day , and the frequentist Doomsday Argument

In its popular frequentist form, the Doomsday Argument asks Wouldn’t it be
strange if we happened to live at the very beginning or end of history? It
applies whether considering the temporal lifespan of humanity, the number of
people ever born, or any other measure $z_{\rm present}$ of how much history
has passed so far compared to humanity’s total history measure $z_{\rm
total}$. Let $F\equiv z_{\rm present}/z_{\rm total}$ be the fraction of
history that has passed, a value between $0$ and $1$. If we regard $F$ as a
uniform random variable, the probability of drawing $F$ closer to $0$ or $1$
than a value $f$ is $\overline{p}=\min(2f,1-2f)$. We construct a confidence
interval with probability $p\equiv 1-\overline{p}$ by bounding our $F$ to be
between $\overline{p}/2$ and $1-\overline{p}/2$: $z_{\rm
present}/(1-\overline{p}/2)\leq z_{\rm total}\leq z_{\rm
present}/(\overline{p}/2)$ with probability $p$ [11].

This form does not specifically invoke the SSA: $F$ can parametrize measures
like lifespan that have nothing to do with “observers” and require no
reference class. In fact, it is motivated by an analogy to externally-applied
Doomsday arguments, in which the lifespan of a phenomenon is inferred by
making an age measurement of a randomly selected member of a population [11].
Unlike the Bayesian Doomsday Argument, it penalizes models where $F=1$
(Section II.3).

The parable of the Noonday Argument demonstrates the weakness of the
frequentist Doomsday Argument: it is not the only confidence interval we can
draw. The Noonday Argument gives another, one arguably even more motivated by
the Copernican notion of us not being in the “center” [c.f., 27]. An infinite
or zero future is maximally compatible with the Noonday Argument. But there is
no reason to stop with Noonday either. Wouldn’t it be strange if our $f$
happened to be a simple fraction like $1/3$, or some other mathematically
significant quantity like $1/e$? Or if we pick a random number from $[0,1]$
with the aid of, say, radioactive decay, surely that cannot predict our future
by being unusually close to our $f$?

Doomsday and Noonday are just two members of a broad class of _$x$ -day
Arguments_. Given any $x$ in the range $[0,1]$, the $x$-day Argument is the
observation that it is extremely unlikely that a randomly drawn $F$ will just
happen to lie very near $x$. To handle edge cases, we regard the interval
$[0,1]$ as cyclic, just like the frequentist Doomsday Arugment. The $x$-day
confidence interval with confidence level $p=1-\overline{p}$ is $[z_{\rm
present}/(x+(1-\overline{p}/2)),z_{\rm present}/(x+\overline{p}/2)]$ if
$x\leq\overline{p}/2$, $[z_{\rm present},z_{\rm
present}/(x+\overline{p}/2)]\cup[z_{\rm present}/(x-\overline{p}/2),\infty)$
if $\overline{p}/2<x<1-\overline{p}/2$, or $[z_{\rm
present}/(x-\overline{p}/2),z_{\rm present}/(x+(1-\overline{p}/2))]$ if
$1-\overline{p}/2\leq x$. The Doomsday Argument is simply the $0$-day Argument
and the $1$-day Argument, whereas the Noonday Argument is the ½-day Argument.

By construction, all frequentist $x$-day Arguments are equally valid
frequentist statements if $z_{\rm present}$ truly is a random variate with a
uniform distribution. Any notion that being near the beginning, the end, or
the center is “strange” is just a subjective impression, one that has no
bearing on these constructions. The confidence intervals are disjoint but
their union covers the entire range of possibilities, with an equal density
for any $p$ covering any $z_{\rm present}$ value [27]. In fact, one can
generalize the $x$-day argument to all kinds of confidence intervals – there
is no reason we could not construct one for any measurement by “excluding” a
small range of values closest to a measurement.

The Noonday Argument demonstrates that not every plausible-sounding Copernican
argument is useful, even when technically correct. A proper Doomsday-style
Argument therefore requires something more than the probability of being near
a “special” time. When dealing with continuously varying parameters and a
continuous likelihood function, _any_ specific parameter value can be excluded
simply by changing the bounds of integration. Confidence intervals merely
summarize the effects of likelihood – which is small for Large models, as
reflected by the relative compactness of the 0/1-day intervals.

###  II.3 Bayesian Doomsday and Bayesian Noonday

Bayesian statistics is a model of how our levels of belief, or credences, are
treated as probabilities conditionalized by observations. We have a set of
some models we wish to constrain, and we start with some prior credence
distribution over them. The choice of prior is subjective, but a useful prior
when considering a single discrete parameter $\lambda$ of unknown scale is the
flat log prior: $P^{\rm prior}(\lambda)\propto 1/\lambda$. This credence
distribution is then modified in proportion to the likelihood ${\cal
L}(\bm{\lambda_{i}}|D)$ of the observed data $D$ in each model
$\bm{\lambda_{i}}$, which is simply the probability $P(D|\bm{\lambda_{i}})$ in
that model that we observe $y$. Bayes’ theorem then gives us a posterior
credence distribution, with an updated credence for the model parameters
$\bm{\lambda_{i}}$:

$P^{\rm pos}(\bm{\lambda_{i}}|D)=\displaystyle\frac{P^{\rm
prior}(\bm{\lambda_{i}}){\cal
L}(\bm{\lambda}|D)}{\sum_{\bm{\lambda_{j}}\in\Lambda}P^{\rm
prior}(\bm{\lambda_{j}}){\cal L}(\bm{\lambda_{j}}|D)},$ (1)

where $\Lambda$ is the set of possible $\bm{\lambda}$.111Bayes’ theorem can be
adapted to continuous parameters by replacing $P^{\rm prior}$ and $P^{\rm
pos}$ with probability distributions and the sum in the denominator with an
integral over all possible parameter values.

Bayesian probability provides a more robust basis for the Doomsday Argument,
and an understanding of how it supposedly works. In the Bayesian Doomsday
argument, the parameter we seek to constrain is the final total population of
humanity and its inheritors, $N_{\rm total}$, using birthranks as the
observable. The key assumption is to apply the SSA with all of humanity and
its inheritors as our reference class. If we view ourselves as randomly
selected, our birth rank $N_{\rm past}$ is drawn from a uniform distribution
over $[0,N_{\rm total}-1]$, with a likelihood of $0$ if $N_{\rm total}<N_{\rm
past}$ and $1/N_{\rm total}$ if $N_{\rm total}\geq N_{\rm past}$. Starting
from the uninformative flat prior $P^{\rm prior}(N_{\rm total})\propto
1/N_{\rm total}$, applying Bayes’ theorem then results in the following
posterior:

$P^{\rm pos}(N_{\rm total})\propto\left\\{\begin{array}[]{ll}0&(N_{\rm
total}<N_{\rm past})\\\ 1/N_{\rm total}^{2}&(N_{\rm total}\geq N_{\rm
past}).\end{array}\right..$ (2)

The posterior in equation 2 is strongly biased against Large models, with
$P^{\rm pos}(N_{\rm total}\geq N)\approx N_{\rm past}/N$.

Birthrank is indeed the most natural basis for the Doomsday Argument, and thus
the Argument depends on observers (or observer-moments). Birthrank is
necessarily distributed uniformly. In contrast, age measurements are likely to
be biased – in an exponentially growing population, for example, most
measurements are made by people living near the end. This emphasizes the role
of the SSA.

The parable’s trick of excluding $N_{\rm total}$ values near $2N_{\rm past}$
is irrelevant here. Bayesian statistics does define credible intervals
containing a fraction $p$ of posterior credence and we could draw Noonday-like
intervals that include Large models but exclude a narrow range of Small
models. But this is simply sleight-of-hand, using well-chosen integration
bounds to hide the fundamental issue that $P^{\rm pos}(N_{\rm total}\gg N_{\rm
past})\ll 1$.

But could there be a deeper $x$-day Argument beyond simply choosing different
credible intervals? Perhaps not in our world, but we can construct a thought
experiment where there is one. Define an $x$-ranking as $N_{x}=N_{\rm
past}-xN_{\rm total}$; allowed values are in the range of $[-xN_{\rm
total},(1-x)N_{\rm total}]$. If through some quirk of physics we only knew our
$N_{x}$ 222Neal [6] briefly discusses a case related to $x=1$, with a
deathrank motivated by a hypothetical asteroid impact., we would treat $N_{x}$
as a uniform random variable, calculate likelihoods proportional to $1/N_{\rm
total}$ and derive a posterior of $1/N_{\rm total}^{2}$ for allowed values of
$N_{\rm total}$ ($N_{\rm total}\geq-
N_{x}/x~{}\textrm{and}~{}N_{x}<0~{}\textrm{or}~{}N_{\rm total}\geq
N_{x}/(1-x)~{}\textrm{and}~{}N_{x}>0$). Thus all Bayesian $x$-day Arguments
rule out Large worlds, including the Bayesian Noonday Argument, even though
the parable’s Noonday Argument implies we should be perfectly fine with an
infinite history. This is true even if $x$ is not “special” at all: most of
the $x$-day Arguments are perfectly consistent with a location in the
beginning, the middle, or the end of history – anything is better than having
$f\approx x$, even if the maximum likelihood is for a “special” location. For
example, the Bayesian Doomsday Argument, which uses our $0$-ranking, predicts
a maximum likelihood for us being at the very end of history. The oddity is
also clear if we knew a cyclic Noonday rank $N_{1/2}^{\prime}$ that wraps from
the last to the first human: the most likely $N_{\rm total}$ value would be
$N_{1/2}^{\prime}+1$, a seemingly anti-Copernican conclusion favoring us being
adjacent to the center of history.

The Bayesian $x$-day Arguments provide insight into the heart of the Bayesian
Doomsday Argument, which has nothing to do with the beginning, middle, or end
of history being “special”. What powers all of these arguments is the SSA
applied with a broad reference class: the $1/N_{\rm total}$ factor in the
likelihood. According to these SSA-based arguments, the reason Large worlds
are unlikely is because measuring _any_ particular value of $N_{x}$ whatsoever
is more unlikely as $N_{\rm total}$ increases. This is a generic property when
one actually is randomly drawing from a population; it does not even depend on
numerical rankings at all. Ultimately, the result follows from the Bayesian
Occam’s Razor effect, in which Bayesian probability punishes hypotheses with
many possible outcomes [39].

###  II.4 The Presumptuous Solipsist

The SSA is doing all of the work in the $x$-day Arguments, but nothing
restricts its applications solely to birth rankings or $N_{x}$ in general. In
fact, the unrestricted SSA argues against Large models of all kinds. It can be
used to derive “constraints” on all kinds of intelligences. One example is the
Cosmic Doomsday argument against the existence of interstellar societies,
solely by virtue of their large populations regardless of any birthrank [14].
If thoroughly applied, however, it would lead to shockingly strong evidence on
a variety of matters, from cosmology to astrobiology to psychology [as noted
by 6, 40].

The existence of non-human observers is widely debated. Several theories of
physics predict a universe with infinite extent (in space, time, or
otherwise), including the open and flat universes of conventional cosmology,
eternal inflation, cyclic cosmology, and the many-world interpretation of
quantum mechanics.333In fact, Leslie [41] does use the Doomsday argument to
reject the many-world interpretation on temporal grounds, since that
interpretation predicts that the number of observers grows exponentially. In
each of these models, every possible observer is almost surely realized, with
a nearly endless panoply of Earths at least slightly different from ours. But
according to the unrestricted SSA, the probability that a randomly selected
observer would find themselves in _this_ version of Earth might as well be
zero in a sufficiently big universe. The unrestricted SSA also rules out the
existence that most of the Universe’s sentient population is not humanlike in
some way, like having a non-carbon based biology or living in the sea. In fact
it obliterates the case for SETI even without the Doomsday Argument, for if
many aliens existed in the Universe, what would the probability be of being
born a human instead any of the panoply of extraterrestrial species that are
out there?444Hartle and Srednicki [40] argue the unrestricted SSA is absurd
because we could rule out aliens on Jupiter this way. But this example could
be counted as a success: Jupiter really is uninhabited despite the many
astronomers of past centuries who believed other planets were similar to Earth
on typicality grounds [42]. The real problem is that the unrestricted SSA
rules against intelligent life everywhere else, even the nearest $10^{100}$
Hubble volumes. Animal consciousness is a better example as we have reason to
suspect many animals are indeed conscious in some way [e.g., 43]. It also
makes short work of the question of the question of animal consciousness.
Isn’t it strange that, of all the creatures on the Earth, you happen to be
human? The SSA would be a strong argument against the typical mammal or bird
being a conscious observer, to say nothing of the trillions of other
animals.555Neal [6] makes a similar point with non-animal consciousness, using
a thought experiment where we have a theory about bacterial self-awareness
with low but not insignificant prior probability.

Unrestrained application of the SSA leads to a far more radical, and ominous
conclusion, however. Why not apply the SSA _to other humans_? We can do this
even if we restrict our reference class to humanity and remain agnostic about
multiverses, aliens, and animal consciousness.666Bostrom [2] argues that an
anthropic prediction is robust if it is the same for a wide range of reference
classes. By this criterion, solipsism seems to be one of the most robust
conclusions possible, holding whether the reference class is all possible
observers, observers within our universe, organic lifeforms, vertebrates,
humans, humans living on the Earth, humans living at this very moment, and by
the SSSA, humans who are exactly you at different points in your life. It
fails only if your reference class is yourself in this instant, or something
nearly as restrictive, in which case the evidential power of typicality
arguments generally disappears. Solipsism, the idea that one is the only
conscious being in existence and everything else is an illusion, is an age-old
speculation.777If “observers” must be conscious, we could evaluate the more
conservative hypothesis that other humans exist but have no qualia, merely
behaving as if conscious [44]. We do not even have to reject physicalism for
this, merely postulate that qualia result from some quirk of your individual
brain. Obviously, most people do not favor solipsism _a priori_ , but it
cannot actually be disproved, only ignored as untenable.

What happens when we apply the SSA to our sliver of solipsistic doubts? Let
$\epsilon$ be the credence you assign to all solipsistic ideas that are
constrained by SSA. To be sure, we usually do not grant much credence to
solipsism, so perhaps $\sim 10^{-9}$ is a reasonable value. But if one is not
unduly prejudiced against solipsism, $\epsilon$ cannot be zero. If one woke up
to find one’s perceptions of the Universe melting away, leaving one as an
isolated Boltzmann brain in a tiny collapsing cosmos, for example, surely one
would want to at least be able to entertain that idea (see Section V.10). The
remaining credence, $1-\epsilon$ will be assigned to the idea that other
people are real. What are the odds that you are _you_ , according to the SSA?
The principle assigns a likelihood of $\leq 1/10^{11}$ for a realist
worldview; only the normalization factor in Bayes’ theorem preserves its
viability. But the SSA indicates that, according to solipsism, the likelihood
that an observer has your data instead of the “data” of one of the
hallucinatory “observers” you are imagining is _1_. From the point of view of
solipsism, the premise that other people actually exist is a Large model.
Applying Bayes’ theorem, the credence you should now assign to it is $P_{\rm
solip}\geq\epsilon/[1/N_{\rm past}+\epsilon(1-1/N_{\rm past})]$:

$\displaystyle P_{\rm solip}$
$\displaystyle\gtrsim\left\\{\begin{array}[]{ll}1-1/(\epsilon N_{\rm
past})&(\epsilon N_{\rm past}\gg 1)\\\ \epsilon N_{\rm past}&(\epsilon N_{\rm
past}\lesssim 1)\end{array}\right..$ (5)

If your prior willingness to entertain solipsism was above $\sim 10^{-11}$,
the SSA magnifies it into a virtual certainty.888We might try to rescue
realism by including every perceived person in our reference class, so even in
solipsism there are $10^{11}$ “humans”. Yet this is almost as bizarre as
solipsism itself. We would include fictional characters or hallucinations in
the real world and might gain enormous retrocausal powers by simply intending
to imagine populated galaxies.

Actually, the situation is much worse: Bostrom [2] further proposes a Strong
Self-Sampling Assumption (SSSA), wherein individual conscious experiences are
the fundamental unit of observations. What are the odds that you happen to
make this observation at _this_ point of your life instead of any other?
Suppose it takes 100 seconds to fully make the observation that one exists in
this moment, presumably a generous estimate. Given a mean human life
expectancy over history of ten years [38], the SSSA increases $N_{\rm past}$
by a factor of $\sim 3$ million. Your credence in extreme solipsism, where
only your current observer-moment exists, should be amplified by a factor
$\sim 3\times 10^{17}$, and more if one believes there is good evidence for an
interstellar future, animal consciousness, alien intelligences, or the
existence of a multiverse. Unless one is unduly prejudiced against it –
invoking the absurdly small prior probabilities that are the problem in the
SIA – the principle behind the Doomsday Argument impels one into believing
nobody else in existence is conscious, not even your past and future selves.
The SSSA does allow there to be multiple copies of you, exactly identical to
yourself, since you have no way of telling which of these selves you are, but
that is hardly any consolation.

One might object that “me, now” is a superficial class, that everyone in the
real world is just as unique and unrepresentative, so there is no surprise
that you are _you_ instead of everything else. This is invalid according to
unrestricted Copernican reasoning. You are forced to grapple with the
fundamental problem that your solipsistic model has only one possible
“outcome” but realist models have many because there really do exist many
different people – in the same way that drawing a royal flush from a stack of
cards on the first try is very strong evidence it is rigged even if every
possible draw is equally unlikely with a randomly shuffled deck. Nor can one
argue a distinction in that all currently living people are equally typical as
distinct individuals, whereas a Large future would have us living in an
atypically “special” time in a Large future. The $x$-day Arguments show that
there exists an $N_{x}$ that is small for each individual in any population.
We have no reason to limit Doomsday to birthrank instead of any measure of
similarity between people – to do so is ignoring data.

Later sections will provide a way out of solipsism even if one accepts the SSA
(Section V.2). This is to fine-grain the solipsism hypothesis by making
distinctions about what the sole observer hallucinates, or which of all humans
is the “real” one. The distinctions between specific observers are thus
important and cannot be ignored. In the context of the Doomsday Argument, we
can ask what is the likelihood that the sole solipsist observer imagines
themself to have a birthrank of $10^{11}$ instead of $1$ or $10^{50}$. Indeed,
Section IV argues that fine-graining is an important part of understanding the
role of typicality. As we will see, using fine-grained models can impose
constraints that prevent its use in Doomsday Arguments.

###  II.5 The anti-Copernican conclusion of the maximal Copernican Principle

The Copernican Principle is inherently unstable when adopted uncritically.
Even a small perturbation to one’s initial prior, a sliver of doubt about
there _really_ being 109 billion people born so far, is magnified to the point
where it can completely dominate one’s views about the existence of other
beings. This in turn leads to epistemic instability, as everything one has
learned from the external world is thrown into doubt. Although the SSA started
out as a way of formalizing the Copernican Principle, it has led to what may
be considered an anti-Copernican conclusion in spirit. Weaker Copernican
principles suggest that you consider yourself one of many possible minds, not
considering yourself favored, just as the Earth is one of many planets and not
the pivot of the Universe. But this strong formulation suggests that you are
the _only_ kind of mind, unique in all of existence. Instead of a panoply of
intelligences, we get at most an endless procession of copies of you and no
one else. Like the anthropic principle, the most extreme versions of the
Copernican principle presumptuously tell you that _you_ are fundamental.

##  III Deconstructing typicality

###  III.1 Why is typicality invoked at all?

Typicality, the principle behind the SSA, has been invoked to explain how we
can conclude anything at all in a large Universe. Since the Universe appears
to be infinite, all possible observations with nonzero probability will almost
surely be made by some observers somewhere in the cosmos by the anthropic
principle. That is, the mere fact that there exists an observer who makes some
observation has likelihood $1$ in every cosmology where it is possible.
Furthermore, a wide range of observations is possible in any given world
model. Quantum theories grant small but nonzero probabilities for measurements
that diverge wildly from the expected value: that a photometer will detect no
photons if it is pointed at the Sun, for example, or that every uranium
nucleus in the Earth will spontaneously decay within the next ten seconds.
Most extreme are Boltzmann brains: any possible observer (for a set of
physical laws) with any possible memory can be generated wherever there is a
thermal bath. Thus we expect there exist observers who make any possible
observation in infinite cosmologies that can sustain cognition. Without an
additional principle to evaluate likelihoods, no evidence can ever favor one
theory over another and science is impossible [8, 9, 32, 2].

The common solution has been to include indexical information in our
distributions. Indexicals are statements relating your first-person experience
to the outside world. They are not meaningful for a third-person observer
standing outside the world and perceiving its entire physical content and
structure. The SSA, and the SIA in reply, attempt to harness indexicals to
learn things about the world: they convert the first-person statement into a
probabilistic objective statement about the world, by treating you as a
“typical” observer. Frequently, the physical distinctions between observers
are left unspecified, as if they are intrinsically identical and only their
environments are different.

When we make an observation, we do at least learn that we are an observer with
data $D$, which is indexical information. The idea of these arguments, then,
is to construct a single joint distribution for physical theories about the
third-person nature of the cosmos and indexical theories about which observer
we are. Often this is implicit – rather than having specified theories about
which observer we are, indexical information is evaluated with an indexical
likelihood, the fraction of observers in our reference class with data $D$.
While all possible observations are consistent with a theory in an infinite
Universe, most observations will be clustered around a typical value that
indicates the true cosmology [2]. Thus the indexical likelihoods result in us
favoring models that predict most observers have data similar to ours over
those where our observations are a fluke.

Typicality is usually fine for most actual cosmological observations, but it
yields problematic conclusions when attempting to choose between theories with
different population sizes. These problems will motivate the development of
Fine-Graining with Attached Indexicals (FGAI) approach to typicality over the
next three sections.

###  III.2 Sleeping Beauty as three different thought experiments

The core of the Doomsday Argument can be modeled with a thought experiment
known as the Sleeping Beauty problem [45]. In this thought experiment, all
participating observers are treated as indistinguishable. Imagine that you are
participating in experiment in which you wake up in a room for some number of
days. Each day your memory of any previous days has been wiped, so you have no
sense of which day it is. Now, suppose you knew that the experiment was either
Short, lasting for just Monday, or Long, lasting for Monday and Tuesday. In
the original formulation of the thought experiment (SB-O), the experimenters
flip a coin, running the Short version if it came up Heads and the Long
version if it came up Tails [45]. You wake up in the room, not knowing how the
coin landed, ignorant of whether you are in a Short run or a Long run. What
probability should you assign to the possibility that the coin landed heads
and the run is Short, $1/2$ or $1/3$?999To emphasize the potential for
unrealistic Bayes factors, consider the case where you are immortal and the
Long version lasts for a trillion days. If you use the SIA, could you ever be
convinced the experimenters are truthful if they come in and tell you the coin
landed on Heads? An intuitive answer might be that you have absolutely no
basis to choose between Short and Long because the coin is fair, and that you
should have an uninformative prior assigning weight $1/2$ to each
possibility.101010This paper uses thought experiments where there is a simple
binary choice between Small and Large models, for which $1/2$ is the
uninformative prior probability. If there are many choices for $N_{\rm total}$
spanning a wide range of values, a flat log prior in $N_{\rm total}$ is more
appropriate. However, the SIA implies that the probability should be $1/3$,
because each possible day in each outcome is equally plausible as your current
day. More awakenings happen in the Long run, so you’d be more likely to be
among them on this day.

If you then learn that it is Monday, the SSA would increase your credence in
the Short run. For a prior probability of $1/3$, you would come to believe
there is $1/2$ probability you are in the Short run. Applied to the
uninformative prior, your credence would instead be $2/3$. The thought
experiment is directly analogous to the Doomsday Argument; one either starts
with a bias against Short runs (with the SIA) or ends with a bias against Long
runs (without the SIA).

The conventional Sleeping Beauty thought experiment has been used to examine
the SSA and competing principles [6], but we have to be careful about drawing
analogies to our own predicament. We especially must be clear about what
probability actually represents, whether it is solely Bayesian and subjective
or frequentist and objective. The following versions of the
experiment111111Garisto [30] independently notes the distinction between an
“inclusiverse” like SB-A where all possibilities exist and an “exclusiverse”
like SB-B where only some exist, arguing there is a weighting factor in SB-A
that makes $1/3$ the correct initial credence. As per Garisto, $P^{\rm
prior}({\rm S})=1/2$ in SB-B because of exclusive selection over high-level
theories. In Weighted FGAI (Section V.3), however, there is no observer
selection effect when learning “today is Monday”, because there is no single
likelihood to interpret that observation but instead multiple provisional
likelihoods with shifting weights. clarify this distinction (see Figure 1):

  * (SB-A) You know that the experiments proceed with a Short run followed by a Long run of the experiment, and you are participating in both. Each day, you wake up with an identical psychological state. Today you wake up in the room. With what probability is today one the day you awaken during the Short run?

  * (SB-B) The experimenters have decided, through some unknown deterministic process, to run only either the Short or the Long version of the experiment. You have absolutely no idea which one they have decided upon. Each day, you wake up with an identical psychological state. Today you wake up in the room. What credence should you assign to the belief that the experimenters have chosen to do the Short run?

  * (SB-C) The experimenters have decided, through some unknown deterministic process, to run only either the Short or a variant of the Long version of the experiment. In the modified Long experiment, they run the experiment on both Monday and Tuesday, but only wake you on one of the two, chosen by another unknown deterministic process. Today you wake up in the room. What credence should you assign to the belief that the experimenters have chosen to do the Short run? If you learn it is Monday, what credence should you then calculate for the Short run?

When we treat indexical facts on the same footing as physical propositions, we
are tempted to make analogies between SB-A and SB-B or between SB-C and SB-B.
In each case, only one in three of the possible outcomes is in the Short
experiment, and only one of the two possible Long experiment awakenings occur
on a Monday. Yet these analogies are deeply flawed. Both SB-A and SB-C have
obvious uninformative priors yielding the same result with or without the SIA,
but they point to different resolutions of the Sleeping Beauty problem: $1/3$
for SB-A and $1/2$ for SB-C.

Figure 1: Illustration of how different methods of treating indexicals affect
Bayesian credences about Short (S) and Long (L) runs in SB-A, SB-B, and SB-C,
before and after learning it is Monday (M) instead of Tuesday (T). The
physical theory has already been determined in SB-A, while the observer
position in each possible physical theory is determined in SB-C. Ruled out
hypotheses are colored in black and do not count towards the normalization.
The SSA, with or without the SIA, leads to presumptuous conclusions in SB-B.
In naive FGAI, the indexical and physical distributions are not mixed.
Instead, there is an overarching physical distribution, and each model has an
associated indexical probability distribution (indicated by the arrows).

SB-A and SB-B posit an _indexical_ uncertainty mapping your current first-
person experience to one of the identical days. SB-A has this as the only
uncertainty, with all relevant objective facts of the world known with
complete certainty. Thus, the combined SIA and SSA amount to simply counting
days and grouping them; they will give you the right answers most of the time.
Because only an indexical is at stake, _there can be no Presumptuous
Philosopher problem_ in SB-A – you are _already_ absolutely certain of the
“cosmology”. But although it has been reduced to triviality in SB-A, there is
actually a second set of credences for the third-person physical facts of this
world: our 100% credence in this cosmology, with both a Short and Long run.

In both SB-B and SB-C, you are in a state of total uncertainty about the
objective state of the world itself. A Bayesian probability distribution about
whether you are in the Short or Long versions has no relation to any objective
fact about the matter. The objective frequentist probability that you are in
the Short run with SB-B is neither $1/2$ nor $1/3$ – it is either $0$ or $1$.
Instead the Bayesian prior is solely an internal one, used by you to weigh the
relative merits of different theories of the cosmology of the experiment.
Thus, it makes no sense to start out implicitly biased against the Short run,
so the probability $1/2$ is more appropriate for a Bayesian distribution. For
all you know, the experimentalists are anti-Long run extremists. Minor
evidence for that – like finding an anti-Long screed crumpled in the trash can
written by a junior associate experimenter – should result in the prior
slightly favoring the Short run, which is not possible if we adopt a $1/3$
prior credence in the Short theory.

Attempts to make analogies between situations like SB-A and those like SB-B
lead to ambiguous conclusions. Some thought experiments attempt to motivate
the Doomsday Argument by making analogies with SB-A-like scenarios – clearly
we should believe we are in the more “typical” Long population in that case
(e.g., Leslie 1’s “emerald” thought experiment). Yet, arguably this instead
justifies the use of the SIA – in SB-A, one is more likely to exist in the
Large context of the Long experiment, so we should be predisposed to favor a
Large model. But in SB-A-like thought experiments, the physical world model is
completely specified and there is no doubt about whether the Large population
exists. This same distinction breaks the analogy between these thought
experiments and the Doomsday Argument. SB-B posits that you are not just
trying to figure out where you are in the world, but the nature of the world
in the first place.121212Some attempt to resolve the experiment by considering
how the participant should bet against the experimenter, using expectation
values. But the frequentist expectation value is meaningless in SB-B because
the choice is deterministic. Rerunning the experiment will result in the same
outcome.

We can also contrast SB-B with SB-C, which is more like a typical experiment
in that the Large theory has more outcomes but the same number of observers.
In SB-C, there are actually two Long hypotheses that split the original $1/2$
credence: one where you awaken on Monday only and one where you awaken on
Tuesday only. The big difference between SB-B and SB-C is that there are two
possible physical outcomes in SB-C’s Long variant, but only one in SB-B’s –
the observations made by all observers in SB-B are determined by whether Short
or Long is chosen. Only the limited indexical information available to SB-B’s
participant makes it seem like there are two outcomes. This indexical
dependence is not present in SB-C, just as there is no physical distinction in
SB-A. In SB-C, learning “it is Monday” should result in a changed credence,
because learning the physical fact that “a participant awakened on Monday” is
not a determined outcome in the SB-C Long theory. This is not a Presumptuous
argument because it involves a scientific observation that directly follows
from the theory, not a philosophical assumption about typicality. But in SB-B,
finding out it’s Monday is consistent with both a Short and Long run, and the
likelihood of either given that simple physical fact on its own is $1$,
suggesting we should not simply just favor the Short run.

SB-C does not accurately model $x$-day or any of the solipsistic arguments
either, where the number of observers itself is at stake. This is illustrated
by considering a thought experiment where we try to determine if fish are
conscious, supposing that they outnumber humans by a hundred to one. Our
theories clearly should be of the form i.) all eight billion humans are
conscious but zero fish are, or ii.) all eight billion humans plus the eight
hundred billion fish are conscious, possibly with intermediate theories where
only some fish are conscious. But the equivalent of SB-C would require the
fish-consciousness hypothesis to be ii.) only eighty million humans plus 7.92
billion fish are conscious, an absurdity. It would be absurd even if we didn’t
know the world’s population: there is no physical mechanism by which the
existence of conscious fish somehow prevents conscious humans from
existing.131313We can assume for the purpose of this argument that physical
constraints on resources and the total carrying capacity of the world are too
weak to bear on the question of sentient fish. Presumably we already know fish
exist and outnumber humanity, even if we do not know their population, and are
just wondering about their cognitive capabilities. The same disanalogy with
SB-C applies to arguments about exotic extraterrestrial intelligences,
interstellar futures, and multiverses.

###  III.3 Separating indexical and physical facts

Simple Bayesian updating does give reasonable results in SB-A and SB-C, where
only indexical or only physical propositions are being considered. This
suggests that any theory of typicality should reduce to Bayesian updating
under these circumstances. But Presumptuous Philosopher-like problems arise in
SB-B whether we adopt the SSA alone or the SSA with the SIA. This suggests
that it is the mixing of physical and indexical facts into one joint
distribution that leads to trouble.

The reason we cannot mix indexical and physical propositions could be because
they are fundamentally different. Indexicals are like statements about
coordinate systems. We are free to set the origin wherever we want as long as
we are consistent, but that freedom does not fundamentally change the way the
Universe objectively works. It follows that _purely indexical facts cannot
directly constrain purely physical world models_. The paradoxes are a result
of trying to force indexical data into working like physical data.141414It
might even be possible to create similar paradoxes by creating joint
probabilities with other types of facts, like moral or aesthetic facts. A key
difference, though, is that indexical uncertainty leads to uncertainty in how
to even evaluate physical likelihoods.

Since Bayesian updating does work in SB-A and SB-C, we can suppose there are
in fact two types of credence distributions, physical and indexical. In FGAI
there is an overarching physical distribution describing our credences in
physical world models. Attached to each physical hypothesis, there is also an
indexical distribution (Figure 1). Each indexical distribution is updated in
response to indexical information [c.f., 32]. The physical distribution is
insulated from changes in the indexical distribution. Within the context of a
particular world-model, one may apply typicality assumptions like the SSA/SIA
to the indexical assumption. The separation between physical and indexical
distributions protects world models from the extremely small likelihoods of
both the SSA and the SIA.

In the Sleeping Beauty thought experiment, when the experimenter announces it
is the first day of the experiment, you learn both an physical fact (“a
participant wakes up on Monday”) and an indexical fact (“I’m the me waking on
Monday”). The physical fact is consistent with both a Short and Long run in
all variants of the thought experiment, so the physical prior is unchanged. In
SB-B, the indexical fact leaves the Short hypothesis indexical distribution
unchanged, but it changes the indexical distribution for the Long theory. FGAI
confines the effects of indexical information. In SB-B, the physical prior
remains unchanged, and remains at $1/2$ even after learning it is Monday. So
the prior probability of a Short run in Sleeping Beauty can be either $1/2$ or
$1/3$ [c.f., 46]: it is $1/2$ in the physical model distribution used in SB-B
and SB-C and $1/3$ in the indexical distribution used in SB-A.

###  III.4 The frequentist limit and microhypotheses

Nonetheless, the frequentist $1/3$ probability does in a way slip back in if
the experiment is modified to more closely match frequentist assumptions.
Consider:

  * (SB-Bn) You know for certain that the experiment is being run for $n$ times where $n\gg 1$. Whether a given run is Short or Long is determined through some deterministic but pseudo-random process, such that any possible sequence of Shorts and Longs is equally credible from your point of view. Each day, you wake up with an identical psychological state. Today you wake up in the room. What credence should you assign to the belief that today is happening during a Short run?151515Bostrom [46] proposes a similar thought experiment, the “Three Thousand Weeks”, concluding that $1/3$ is the correct probability to use.

Now there is a large number of competing physical hypotheses, $2^{n}$ in
total, one for each possible sequence of Shorts and Longs. As before, each of
those hypotheses has an attached indexical distribution (Figure 2). From a
practical point of view, however, in most of these hypotheses about half of
the runs are Short and half are Long. Although the physical hypotheses may
differ in detail, the statistical properties of the great majority are
similar. Hence, one becomes nearly certain of a roughly even proportion
between Short and Long runs simply through combinatorics. Furthermore, this
statistical similarity translates to the indexical distributions: in most
indexical distributions, $\sim 1/3$ of the awakenings happen in the Short run.
As $n$ tends to infinity, the rare outliers with unbalanced Short/Long runs
become insignificant, and the physical distribution becomes trivial when
viewed from a coarse-grained perspective; only the indexical distributions
remain unconstrained. The probability you would most care about is then $1/3$.
161616Interestingly, this suggests that if the Universe is infinite and the
decision between Short and Long is indeterministic (e.g., chosen by
radioactive decay), then you should assume the probability you are in a Short
run is $1/3$. This is because an infinite universe actually implements the
frequentist limit [see also 30]. You essentially know the statistical
properties of the physical “cosmology” with $100\%$ certainty. On the other
hand, if the experimenters are using a deterministic strategy, or the universe
is small enough, there is only one trial with two outcomes, just as in SB-B,
implying your credence in Short should be $1/2$. This does lead to the odd
situation where our credences depend on whether or not a multiverse exists
[30].

Figure 2: If the Sleeping Beauty experiment is run many times, with each
possible sequence of Long and Short _a priori_ equally likely, a vast number
of microhypotheses about the sequence of Short and Long is generated. Shown
here are the 64 microhypotheses when there are $n=6$ runs. Attached to every
single one of these fine models is an indexical distribution.

SB-Bn, where most of the individual sequences are statistically similar, calls
to mind statistical mechanics, where a vast number of physical microstates are
grouped into a small number of distinguishable macrostates. By analogy, I call
each possible detailed world model a _microhypothesis_ , which are then
grouped into _macrotheories_ defined by statistical properties.

##  IV Replacing typicality

###  IV.1 The fine-grained approach to typicality

How are we to make inferences in a large universe, then, without directly
mixing indexicals into the credence distribution? I propose that most of the
work performed by typicality can instead be performed by _fine-graining_.
Fine-graining of physical theories is the first principle of FGAI.

The common practice is to treat observers in a fairly large reference class as
interchangeable when discussing their observations, but this is merely a
convenience. In fact, we can make fine distinctions between observers –
between Earth and an inhabited planet in Hubble volume # 239,921, for example,
or between me and you, or even between you in 2021 and you in 2020.
Macrotheories often cannot predict exactly which specific observer measures a
particular datum. Thus, every theory is resolved into myriads of
_microhypotheses_ , each of which does make these predictions. Because the
distinctions between these observers are _physical_ , statements about a
specific observer making a particular measurement are evaluated as purely
physical propositions, without invoking indexicals. Some microhypotheses will
be consistent with the data, others will not be. The resulting credences in
the macrotheories are entirely determined by summing the posterior weight over
all microhypotheses, in many cases through simple counting arguments.
Typicality then follows from the likelihood values of the microhypotheses – as
it indeed does in conventional probability, where specific “special” events
like getting a royal flush from a randomly shuffled deck are no more rare than
specific mundane outcomes [47]. Thus, in most cases, there is no need to
invoke any separate Copernican principle, because it is a demonstrable
consequence of our theories.

The other main precept of FGAI is that purely indexical facts do not directly
constrain third-person propositions about the physical world, rather modifying
indexical distributions attached to each world model. Observations must be
treated as physical third-person events when constraining the physical
distribution. Statements like “I picked ball 3 out of the urn” or “I am the
109 billionth human” must be recast into third-person statements like “Brian
Lacki picked ball 3 out of the urn” or “Mary the Probability Scientist is the
109 billionth human”. Each microhypothesis requires an _observation model_ , a
list of possible observations that each observer may make. Observation models
necessarily impose physical constraints on which observations can be made by
whom, forbidding impossible observations like “Hypatia of Alexandria observed
that her peer was Cyborg 550-319447 of Gliese 710” from being considered as
possible outcomes.

The fine-graining is most straightforward when every microhypothesis predicts
that all physically indistinguishable experiments lead to the same outcome.
This follows when we expect conditionalization to entirely restrict possible
observations. For example, the Milky Way could be the product of an
indeterministic quantum fluctuation, but copies of our Earth with its data
(e.g., photographs of the Milky Way from inside) do not appear in elliptical
galaxies except through inconceivably contrived series of coincidences. More
difficult are purely indeterministic cases, when any specific observer can
observe any outcome, which is true for most quantum experiments. I will argue
that even then we can form microhypotheses by assuming the existence of an
appropriate coordinate system (Section IV.5).

A more serious difficulty is what to do when different plausible indexical
hypotheses would lead to different likelihood evaluations for microhypotheses.
That is, we may not know enough about where we are to determine whether a
microhypothesis predicts an observation or not. In this section, I will adopt
the perspective that I call Naive FGAI: we adopt the maximum possible
likelihood over all observers we could be, as in the treatment of SB-B in
section III.3, because we have no basis for accepting any further conclusion.

###  IV.2 Naive FGAI

FGAI constructs the physical probability distributions using the Hierarchical
Bayes framework, dividing theories into finer hypotheses about internal
parameters, possibly with intermediate levels. Suppose we have $M$
macrotheories $\tau_{1}$, $\tau_{2}$, … $\tau_{M}$. Each macrotheory
$\tau_{k}$ has $m_{k}$ microhypotheses $\mu_{k,1}$, $\mu_{k,2}$, …,
$\mu_{k,m_{k}}$. Each microhypothesis inherits some portion of its parent
macrotheory’s credence or weight. Sometimes, when the microhypotheses
correspond to exact configurations resulting from a known probabilistic (e.g.,
flips of an unfair coin), then the prior credence in each $P^{\rm
prior}_{k,j}$ can be calculated by scaling the macrotheory’s total prior
probability accordingly. In other cases, we have no reason to favor one
microhypothesis over another, and by the Principle of Indifference, we assign
each microhypothesis in a macrotheory equal prior probability: $P^{\rm
prior}_{k,j}=P^{\rm prior}_{k}/m_{k}$. Some macrotheories are instead
naturally split into mesohypotheses describing intermediate-level parameters,
which in turn are fine-grained further into microhypotheses. Mesohypotheses
are natural when different values of these intermediate-level parameters
result in differing numbers of outcomes – like if a first coin flip determines
the number of further coin flips whose results are reported. Finally, in each
$\mu_{k,j}$, we might be found at any of a number of locations. More properly,
as observers we follow _trajectories_ through time, following a sequence of
observations at particular locations, as we change in response to new data
[c.f., 46]. The set $O_{k,j}(D)$ is the set of possible observer-trajectories
we could be following allowed by the microhypothesis $\mu_{k,j}$ and the data
$D$. The set $O_{k,j}$ describes our reference class if $\mu_{k,j}$ is true.

In Naive FGAI, prior credences $P^{\rm prior}_{k,j}$ in $\mu_{k,j}$ are
updated by data $D$ according to:

$P^{\rm pos}_{k,j}=\frac{\displaystyle P^{\rm prior}_{k,j}\times\max_{i\in
O_{k,j}(D)}{\cal L}(\mu_{k,j}|o@i\to
D)}{\displaystyle\sum_{x=1}^{M}\sum_{y=1}^{m_{x}}P^{\rm
prior}_{x,y}\times\max_{z\in O_{x,y}(D)}{\cal L}(\mu_{x,y}|o@z\to D)},$ (6)

where ${\cal L}(\mu_{k,j}|o@i\to D)$ is the likelihood of $\mu_{k,j}$ if the
observer ($o$) located at position $i$ in $O_{k,j}(D)$ observes data $D$. Of
course if the likelihoods are equal for all observers in the reference class
$O_{k,j}(D)$, equation 6 reduces to Bayes’ formula. The posterior credences in
the macrotheories can be found simply as:

$P^{\rm pos}_{k}=\sum_{j=1}^{m_{k}}P^{\rm pos}_{k,j}.$ (7)

Naive FGAI is sufficient to account for many cases where typicality is
invoked. Paradoxes arise when the number of observers itself is in question,
as in Doomsday, requiring a more sophisticated treatment (Section V).

###  IV.3 A simple urn experiment

In some cases, microhypotheses and observations models are nearly trivial.
Consider the following urn problem: you are drawing a ball from an urn placed
before you that contains a well-mixed collection of balls, numbered
sequentially starting from $1$. You know the urn contains either one ball
(theory A) or ten (theory B), and start with equal credence in each theory.
How does drawing a ball and observing its number constrain these theories?
Both theory A and theory B have microhypotheses of the form “The urn contains
$N$ balls and ball $j$ is drawn at the time of experiment”. In theory A, there
is only one microhypothesis, which inherits the full 50% of Theory A’s prior
weight. Theory B has 10 microhypotheses, one for each possible draw and each
of equal credence, so its microhypotheses start with 5% credence each. Each
microhypotheses about drawing ball $j$ has an observation model containing the
proposition that you observe exactly ball $j$ – this observation is a physical
event, since you are a physical being.

Then the likelihoods of an observed draw is either $1$ (if the ball drawn is
that predicted in the microhypothesis) or $0$ (if the ball is not the
predicted one). If we draw ball $3$, for example, the credence in all
hypotheses except Theory B’s “Ball 3 is drawn” microhypothesis is zero. Then
the remaining microhypothesis has 100% credence, and Theory B has 100%
credence as well. If instead ball $1$ is drawn, Theory A’s sole
microhypothesis and one of Theory B’s microhypotheses survive unscathed, while
the other nine microhypotheses of Theory B are completely suppressed. That is,
100% of Theory A’s credence survives, while only 10% of Theory B’s credence
remains; therefore, post-observation, the credence in Theory A is $10/11$ and
the credence in Theory B is $1/11$. This, of course, matches the usual
expectation for the thought experiment.

But what if instead you and 99 other attendees at a cosmology conference were
drawing from the urn with replacement and you were prevented from telling each
other your results? If we regard all one hundred participants as exactly
identical observers, the only distinct microhypotheses seem to be the
frequency distribution of each ball $j$ being drawn. Then naive FGAI without
modification would prevent any significant update to Theory B’s credence if
you draw ball $1$ because all you know is that you are a participant-observer
and there exists an observer-participant who draws ball $1$, which is nearly
certain to be true (but see Section IV.5). In this case, inference would seem
to require something like the SSA, where the likelihood of a “typical”
participant drawing ball $1$ is $10\%$. Yet this is not necessary in practice
because if this experiment were carried out at an actual cosmology conference,
the participants _would_ be distinguishable. We then can fine-grain Theory B
further by listing each attendee by name and specifying which ball they draw
for each microhypothesis, and forming an observation model where names are
matched to drawn balls. For example, we could order the attendees by
alphabetical order and each microhypothesis would be a 100-vector of integers
from $1$ to $10$.171717Even if all the observers start as identical, they can
be distinguished by assigning identifiers, perhaps based on their distinct
locations at some instant.

With fine-graining, there is only $1^{100}=1$ microhypothesis in Theory A –
all participants draw ball $1$ – but $10^{100}$ microhypotheses in Theory B.
Furthermore, in only $10^{99}$ of those microhypotheses do you specifically
draw ball $1$ and observe ball $1$. Thus only 10% of Theory B’s
microhypotheses survive your observation that you drew ball $1$. The credence
in Theory B is again $1/11$, but is derived without appealing to typicality.
Instead, typicality _follows_ from the combinatorics. In fact, the original
one-participant version of this thought experiment can be regarded as a
coarse-graining of this 100-participant version, after marginalizing over the
unknown observations of the other participants.

Appendix A presents a simple worked example with explicit microhypotheses.

###  IV.4 Implicit microhypotheses: A thought experiment about life on Proxima
b

In other cases, the microhypotheses can be treated as abstract, implicit
entities in a theory. A theory may predict an outcome has some probability,
but provide no further insight into which situations actually lead to the
outcome. This happens frequently when we are actually trying to constrain the
value of a parameter in some overarching theory that describes the workings of
unknown physics. Historical examples include basic cosmological parameters
like the Hubble constant. Not only do we not know their values, we have no
adequate theories to explicitly predict them. Yet these parameters are subject
to cosmic variance; some observers in a big enough Universe should deduce
unusual values far from their expectation values. Naive FGAI can be adapted
for such theories by positing there are _implicit_ microhypotheses that we
cannot specify yet. The probability that we observe an outcome is then treated
as if it is indicating the fraction of microhypotheses where that outcome
occurs.

Suppose we have two models about the origin of life, L-A and L-B, both equally
plausible. L-A predicts that all habitable planets around red dwarfs have
life. L-B predicts that the probability that a habitable zone planet around a
red dwarf has life is $10^{-100}$. Despite this, we will suppose that the
conditions required for life on the nearest potentially habitable exoplanet
Proxima b are _not_ independent of our existence on Earth, and that any copy
of us in a large Universe will observe the same result. The butterfly effect
could impose this conditionalization – small perturbations induced by or
correlated with (un)favorable conditions on Proxima b may have triggered some
improbable event on Earth necessary for our evolution. We wish to constrain
L-A and L-B by observing the nearest habitable exoplanet, Proxima b, and we
discover that Proxima b does have life on it.

If we apply Copernican reasoning, we might say that in L-B, only one in
$10^{-100}$ inhabited G dwarf planets would observe life around the nearest
red dwarf, and that as typical observers, we should assign a likelihood of
$10^{-100}$ to L-B. Thus, L-B is essentially ruled out.

L-B does not directly specify which properties of a red dwarf are necessary
for life on its planets; it merely implies that the life is the result of some
unknown but improbable confluence of properties. Nonetheless, we can interpret
L-B as grouping red dwarfs into $10^{100}$ equivalence classes, based on
stellar and planetary characteristics. Proxima Centauri would be a member of
only one of these. L-B then would assert that only one equivalence class of
$10^{100}$ bears life. Thus, L-B actually implicitly represents $10^{100}$
microhypotheses, each one a virtual statement about which equivalence class is
the one that hosts life (Figure 3). In contrast, L-A has only one
microhypothesis since the equivalence class contains all red dwarfs. We then
proceed with the calculation _as if_ these microhypotheses were known.

Figure 3: Extremely simplified representation of how FGAI treats theories like
L-B. Probabilities of observing a particular outcome in a macrotheory may be
presumed to result from some unknown fine-scale division of parameter space
into regions delineating different equivalence classes. Then microhypotheses
would be constructed by considering all possible outcomes for all regions. For
L-B, we observe the red dwarf nearest to Earth (Proxima $\oplus$) and see
whether it is inhabited (flower symbol). The observation of an inhabited
Proxima $\oplus$ reduces the number of allowed microhypotheses. a.: observers
on other planets distinguishable from Earth would observe red dwarfs with
different properties (Proxima X, Y, and Z) and probe different classes. b.:
many identical Earths observe distinct Proxima $\oplus$ (red stars). These are
treated by assuming there is some indexing that allows microhypotheses to be
constructed, and their likelihoods calculated by symmetry.

If we observe life on Proxima b, then the sole microhypothesis of L-A survives
unscathed, but implicitly only one microhypothesis of L-B of the $10^{100}$
would survive. Thus, after the observation, L-A has a posterior credence of
$100\%/(1+10^{-100})$, while L-B has a posterior credence of
$10^{-100}/(1+10^{-100})$. As we would hope, FGAI predicts that we would be
virtually certain that L-A is correct, which is the result we would expect if
we assumed we observed a “typical” red dwarf.

What of the other inhabited G dwarf planets in an infinite Universe? The
nearest red dwarfs to these will have different characteristics and most of
them will belong to different equivalence classes (Figure 3). In principle, we
could construct microhypotheses that specify what each type of these observers
will around their nearest red dwarf, and implicitly we assume they exist. If
we failed to develop the capability to determine whether Proxima b has life
but aliens from 18 Scorpii broadcast to us that their nearest red dwarf has
life, the result would be the same. Because we do not know what the
equivalence classes are, we cannot be sure about which types of observers
share equivalence classes. We might instead _define_ equivalence class #1 to
have life, and then consider each microhypothesis to be a mapping between
distinct types of observer and equivalence class number.

###  IV.5 Fine-graining and implicit coordinate systems

The strictest interpretation of the separation of physical and indexical facts
is that we cannot constrain physical models if the observed outcome happens to
_any_ observer physically indistinguishable from us. This is untenable, at
least in a large enough universe – quantum mechanics predicts that all non-
zero probability outcomes will happen to our “copies” in a large universe. But
this would mean no measurement of a quantum mechanical parameter can be
constraining. Surely if we do not observe any radiodecays in a gram of
material over a century, we should be able to conclude that its half-life is
more than a nanosecond, even though a falsely stable sample will be observed
by _some_ copy of us out there in the infinite universe.181818Leslie [1] takes
the position that indeterminism blunts SSA-like arguments, but only if the
result has not been decided yet, because it is obvious the probability of an
indeterministic future event like a dice throw cannot be affected by who we
are. I believe this attempt to soften Doomdsay fails, because we are
considering the probability _conditionalized on you being you_. The anthropic
principle is based entirely on such conditionalizing.

Strict indeterminism is not necessary for this to be a problem, either. In the
last section, we might have supposed that the existence of life on Proxima b
depends on its exact physical microstate five billion years ago, and that
these microstates are scattered in phase space. Yet, Proxima b is more massive
than the Earth and has a vaster number of microstates – by the pigeonhole
principle, in an infinite Universe, most Earths exactly identical to ours
would neighbor a Proxima b that had a different microstate and could observe a
different outcome about whether it has life (Figure 3).

But the separation of indexicals and physical theories need not be so strict.
Indexicals are regarded in FGAI as propositions about coordinate systems. In
physical theories we can and do use coordinate systems, sometimes arbitrary
ones, as long as we do not ascribe undue objective significance. We can treat
these situations by imposing an implicit indexing, as long as we do not
ascribe undue physical significance to it. Thus, in an infinite Universe, we
label “our” Earth as Earth 1. The next closest Earth is Earth 2, the third
closest Earth is Earth 3, and so on. We might in fact only implicitly use a
coordinate system, designating our Earth as Earth 1 without knowing details of
all the rest. Our observations then are translated into third-person
propositions about observations of Earth 1. The definition of the coordinate
system imposes an indexical distribution where we must be on Earth 1. Of
course, this particular labeling is arbitrary, but that hardly matters because
we would reach the same conclusions if we permuted the labels. If we came into
contact with some other Earth that told us that “our” Earth was Earth 3,296 in
their coordinate system, that would not change our credence in a theory.

In a Large world, then, the microhypotheses consists of an array listing the
outcome observed by each of these implicitly indexed observers. Only those
microhypotheses where Earth (or observer) 1 has a matching observation
survive. If the observed outcome contradicts the outcome assigned to Earth 1
by the microhypothesis, we cannot then decide we might actually be on Earth
492,155 in that microhypothesis because Earth 492,155 does observe that
outcome. The coordinate system’s definition has already imposed the indexical
distribution on us. Using an implicit coordinate system lets us apply a fine-
graining treatment, where typicality is a consequence of combinatorics.

There are limits to the uses of implicit coordinate systems, however, if we
wish to avoid the usual Doomsday argument and its descent into solipsism. In
SB-B, could we not assign an implicit coordinate system with today at index 1,
and the possible other day of the experiment at index 2? A simplistic
interpretation would then carve the Long run theory into two microhypotheses:
observer 1 is on Monday and observer 2 is on Tuesday, and observer 1 is on
Tuesday and observer 2 is on Monday. Upon learning it is Monday, this fine-
graining would lead us to favor the Short run theory.

There is a very important difference between SB-B and these examples. First,
the number of observers is the same when deciding between L-A and L-B or in
other typical experiments. Furthermore, in this section’s thought experiments,
our physical theory did not in any degree predict which observers get a
particular outcome. Thus the likelihood distribution over the outcomes is
exactly identical for all observers. This is why we are able to assign
likelihoods even though the mapping between our implicit coordinate system and
some external coordinate system is unknown. In SB-B, though, our physical
theory does predict which observer gets a particular outcome. The likelihood
of observing “it is Monday” is not identical for each day. Instead, “it is
Monday” needs to be interpreted as an indexical fact. We can indeed create an
implicit coordinate system with today at index 1 and the other day at index 2.
But we _cannot calculate likelihoods in this coordinate system_ – the
referents of the indices in the theory are unknown, and until we can connect
the implicit coordinates with the coordinate system used by the Long theory,
we cannot update credences either. All we can say is that if observer 1 is
located on Monday, they observe “it is Monday” with certainty; if observer 1
is located on Tuesday, they observe “it is Tuesday” with certainty. In these
kinds of situations, a more sophisticated theory is needed.

###  IV.6 Naive FGAI and $x$-day arguments

####  IV.6.1 Classes of $x$-day models in FGAI

One advantage of underpinning typicality arguments with fine-grained
hypotheses is that doing so forces one to use a well-specified model that
makes the assumptions explicit. In an $x$-day Argument, each microhypothesis
corresponds to a possible permutation of people born as well as a complete set
of possible observations by each observer (whether a human or an effectively
independent external observer like an alien). This entails an observation
model, a set of constraints on who can observe whom. In particular, realistic
theories of observation are _causal_ – an observer cannot “observe” people
living in the future – and _local_ – an observer cannot “observe” another
person without some physical mechanism linking them.

In this section, I will discuss four general classes of fine-grained $x$-day
schemas, resulting in microhypotheses with different combinatorial properties.
These fine-grainings are illustrated by a simplified model where we consider
only two theories [c.f., 1]: a Short/Small theory with final human population
$N_{S}$ and a Long/Large theory with final human population $N_{L}$, where
$N_{L}>N_{S}$. Each human at $x$-rank $r$ is drawn from a set of possible
humans $H$, and measures their $x$-rank to be $r$, specifying the observation
model. The possible humans in $H$ correspond to different genetic makeups,
microbiomes, life histories, memories encoded in the brain, and so on. The
details of how humans at rank $r$ are selected from $H$ form the basis for
each of the different model classes (Figure 4).

Figure 4: Different fine-grainings of the $x$-day Argument in FGAI have
different assumptions about the interchangeability of humans. The four basic
schema shown here correspond to IO, a Sleeping Beauty-like case where all
observers are treated as indistinguishable; UO, where there are many possible
humans, each of whom may be assigned to arbitrary ranks; USO, where possible
humans are grouped into subsets and each rank may drawn from only one subset,
although the order of the subsets is unknown; and KSO, where each rank draws
from a fixed subset of possible humans, although which observer from each
subset is selected is arbitrary. The clouds represent an unknown or random
selection from a set. The shown configurations are chosen so that observer $A$
(blue circle) exists with rank $1$, but this is a contingent selection in the
models.

The four schemas differ by their restrictions on the possible permutations:

  * Indistinguishable Observers (IO) – Every possible human is treated as identical, with $H=\\{A\\}$. There is only one possible microhypothesis for each theory because the permutations are indistinguishable. Any information about $x$-rank is treated as purely indexical. IO is essentially the same kind of scenario as SB-B.

  * Unconstrained Observers (UO) – Humans are drawn from a very large set $H$ (with $|H|={\cal N}$) of distinguishable observers, and any $h\in H$ can be born at any rank $r$. In UO, people with different names, identities, and memories could be treated as distinct members of $H$, but these details would no correlation with historical era.

  * Unconstrained Subset Ordering (USO) – The set of possible humans $H$ is partitioned into ${\cal R}$ ($\geq N_{L}$) mutually disjoint subsets $H_{r}$, with one for each $r$. Humans with rank $r$ can only be drawn from $H_{r}$. This reflects the fact that individuals are the result of a vast constellation of historical circumstances that should never be repeated again. In USO, we can specify the contents of these subsets, but we do not know which subset is assigned to each $r$. In an infinite Universe, every “copy” of you will have the same $r$ as you.

  * Known Subset Ordering (KSO) – As in USO, the set of possible humans is partitioned into disjoint subsets $H_{r}$, with humans at rank $r$ drawn only from $H_{r}$. Unlike USO, we already know beforehand the ordering of these subsets – which one is $H_{1}$, $H_{2}$, and so on. All that remains to be discovered is the final human population $N$ and which humans, in fact, are selected out of each $H_{r}$. KSO models emphasize how we already know, by virtue of our historical knowledge, our place in history before applying an $x$-day Argument.191919The Doomsday Argument yields a “correct” result for most people in history, leading Leslie [1] (and with more cavaets, Bostrom 2) to conclude that we should use it as well, as should have early humans. This logic fails in the KSO model. Of course, the Doomsday Argument works for most people by construction, but in KSO, it always fails for early humans in their Large future theories. We are not interested in whether Doomsday “works” for most people in history, but whether it applies to us _specifically_.

None of these models exactly corresponds to how we would approach the Doomsday
Argument, since we do not actually know the set $H$ or exactly how it is
partitioned – these details are implicit for us. But by making explicit
models, they illustrate the problems it can run into. Of these, KSO arguably
is most analogous to our situation when we apply the Doomsday Argument, since
humans who know they live in the year 2021 cannot be born in the Paleolithic
or an interstellar future (to the extent our memories are reliable).202020The
Simulation Argument proposes that long-lived societies create vast numbers of
emulations of their ancestors, suggesting we are likely emulations too [48].
Although the literature treats computer simulations specifically, I believe it
applies to _any_ technology that can falsely convince someone they are living
in our epoch (e.g., virtual reality, brains in vats, lucid dreaming, etc.).
The fundamental point is our growing ability to manipulate perception. I adopt
the realist viewpoint that our birthrank measurements are reliable except when
specifically considering solipsistic hypotheses. Our Short and Long theories
actually posit that everyone who has lived exist at their birthranks, and then
propose an additional $N_{\rm future}$ future people after us, with the
likelihood of us existing at birthrank $10^{11}$ being $1$ by assumption. This
is not just a tautology because who you are is shaped by your place in
history; realistic Small and Large models both predict that you, with all your
personality, memories, and beliefs, could only appear this point in history
[as in 26].

Although previous discussion in this paper has made it seem like there are
“purely indexical” measurements, in a strict sense this cannot be true. All
measurements are physical events. When someone learns their position, they are
actually interacting with a physical environment that is location-dependent
and changing as a result. Therefore, a truly physical theory cannot regard
observers as identical if they have different indexical knowledge. Instead,
such observers can only be found in specific locations. For example,
hypotheses about individual $A$ of Figure 4 really should be fine-grained into
hypotheses about $A_{1}$, who can only exist at rank $1$, and $A_{2}$, who can
only exist at rank $2$, and so on. In this sense, USO and KSO are more
physically grounded in how they interpret indexical observations than UO or
IO.

####  IV.6.2 Self-applied $x$-day Arguments in Naive FGAI

An $x$-day Argument to constrain final population may be self-applied by a
member of the population being constrained (as in the Bayesian Doomsday
Argument), or applied by an external observer who happens upon the population
at some time (analogous to some of Gott 11’s examples). Self-applied $x$-day
Arguments are by far more problematic. Naive FGAI provides a partial – though
incomplete – accounting of how self-applied $x$-day Arguments like Bayesian
Doomsday fare.

I will start under a “single-world” assumption that there is only one copy of
humanity in the cosmos (or that every copy of humanity yields the same exact
permutations of extant humans). Then it is relatively simple to calculate the
number of microhypotheses in IO, UO, USO, and KSO with combinatorial
arguments, given the number of humans in $H$ (and each $H_{r}$). In Naive
FGAI, the likelihoods for each macrotheory about $N$ is then given by the
fraction of microhypotheses consistent with observations. Table 1 presents the
results of such calculations, where the observer in question is labeled $A$
and is located at $N_{x}=1$ without loss of generality.

Table 1: Microhypotheses counts for self-applied $x$-day Arguments in single world models  | IO | UO | USO | KSO
---|---|---|---|---
|  | No replacement | With replacement | All $|H_{r}|$ equal |
Total permutations | $1$ | $\displaystyle\frac{{\cal N}!}{({\cal N}-N)!}$ | ${\cal N}^{N}$ | $\displaystyle{\cal N}_{r}^{N}\frac{{\cal R}!}{({\cal R}-N)!}$ | $\displaystyle{\cal N}_{r}^{N}$
Permutations where $A$ selected | $1$ | $\displaystyle N\frac{({\cal N}-1)!}{({\cal N}-N)!}$ | ${\cal N}^{N}-({\cal N}-1)^{N}$ | $\displaystyle N{\cal N}_{r}^{N-1}\frac{({\cal R}-1)!}{({\cal R}-N)!}$ | $\displaystyle{\cal N}_{r}^{N-1}$
Permutations with $A$ at rank $1$ | $1$ | $\displaystyle\frac{({\cal N}-1)!}{({\cal N}-N)!}$ | ${\cal N}^{N-1}$ | $\displaystyle{\cal N}_{r}^{N-1}\frac{({\cal R}-1)!}{({\cal R}-N)!}$ | $\displaystyle{\cal N}_{r}^{N-1}$
Fraction where $A$ exists | $1$ | $N/{\cal N}$ | $1-(1-1/{\cal N})^{N}$ | $\displaystyle\frac{N}{{\cal N}_{r}{\cal R}}$ | $1/{\cal N}_{r}$
Fraction with $A$ at rank $1$ | $1$ | $1/{\cal N}$ | $1/{\cal N}$ | $1/({\cal R}{\cal N}_{r})$ | $1/{\cal N}_{r}$

In the single world model, the four schemas all arrive at the ultimate
conclusion that the surviving microhypothesis fraction if $A$ measures their
rank to be $1$ is independent of $N$. In Naive FGAI, this means the $x$-day
Argument is nullified in all of them. Yet they take different routes to get
there. In IO, $A$ is the only possible individual, and there only ever is one
microhypothesis per theory, which can never be ruled out – $A$ merely learns
indexical information. In UO and USO, there is an SIA-like effect when $A$
observes themself to exist: there are more locations for $A$ to be born in the
Large theory, so more microhypotheses are consistent with that observation.
Unlike the SIA, this indication effect is not in the prior itself but as a
result of the self-observation.212121$A$ having more locations to be born is a
physical statement about the number of actual physically occurring trials,
with each outcome being a physically distinct history. It does not require
“souls waiting to be born” [c.f., 1, 2]. Then $A$ measuring their rank of $1$
culls proportionately more microhypotheses in the Large theory, canceling this
effect.222222Actually, USO can be much more complicated if the different
$H_{r}$ have different sizes. Then the fraction of surviving microhypotheses
(and posterior probabilities) is $N$-dependent if each possible permutation of
humans is given equal initial credence within each macrotheory (e.g., if
$N=1$, $H_{a}=\\{A_{a}\\}$, and $H_{b}=\\{A_{b},B_{b}\\}$, then the
microhypotheses of $A_{a}$ only, $A_{b}$ only, and $B_{b}$ only are given
equal credence). A more natural weighting is to first divide each macrotheory
into “mesohypotheses” about the ordering of the $H_{r}$, and each permutation
microhypothesis inherits an equal fraction of its mesohypothesis’s credence
(e.g., $A_{a}$ only would get twice as much initial credence as $A_{b}$ only).
See Appendix B for a worked example. Finally, in KSO, where $A\in H_{1}$ _a
priori_ , $A$’s self-observation automatically implies that $A$ is at rank $1$
no matter the theory.

We can also consider “many worlds” models in which there are many copies of
humanity out the larger cosmos, each with different rank permutations of human
individuals. Now there would be a distribution of lifetimes for all the
humanities, but for simplicity we can consider models where historical forces
(e.g., nuclear war or environmental constraints) always either end humanity in
the near-future or fail to prevent its explosion into a Large interstellar
future. In IO, birthrank measurement continues to be a purely indexical
measurement. In the other models, the probability any given individual $A$
exists somewhere in a vast cosmos converges to $1$. In order to make any
inferences in Naive FGAI, we would need to use the implicit coordinates
described previously, with each $A$ labeling their world as world I and
considering microhypotheses about who is at rank $1$ in world I. In KSO and
USO this allows a kind of coarse-graining of the microhypotheses yielding
similar results to the “single world” model.

UO yields the most ambiguous results in “many worlds”. There will exist worlds
where $A$ measures themselves to have any possible rank. What $A$ infers
depends on what kind of indexing scheme they use. If they index only over
worlds, then $A$ is more likely to exist in world I’s history in a Large $N$
model, balanced by being less likely to exist at rank $1$. But previous
discussion of these implicit coordinate systems suggest $A$ should index over
their copies: not worlds in general, but only worlds containing observers
identical to themself. Then, because $A$ is guaranteed to exist in a vast
enough cosmos, there _is_ a Doomsday Argument. Any given $A$ is more likely to
observe themself at rank $1$ if there are fewer possible ranks they could be
born at, as in the Short histories. Note, however, in practice this requires
anachronistic birth orders with any individual being born with equal frequency
in every epoch in history. There would have to actually be as many worlds
where you specifically, with all your memories, would be the first human as
ones where you live in 2021 – a physically improbable outcome and thus
incompatible with the spirit of FGAI. This anachronism is prevented in USO and
KSO, which have no Doomsday effect even in a vast cosmos.

####  IV.6.3 Externally-applied $x$-day arguments in Naive FGAI

Externally-applied $x$-day Arguments are structurally similar to “urn” thought
experiments, while seeming to also justify the self-applied Doomsday Argument
by analogy. Under some circumstances, one can infer something about the
lifespan distribution of an external phenomenon by drawing a sample of it at
random (as in the most popularized examples of Gott 11). But the key
difference with self-applied Doomsday arguments is the existence of an
additional “observer” that is the one carrying out the experiment, who is
completely outside the population being constrained. This adds another
physical degree of freedom, which changes the microhypothesis counts in Naive
FGAI to allow for a Doomsday Argument.

Suppose an observer $X$ happens upon humanity, with individuals drawn from $H$
according to IO, UO, USO, or KSO. All of the possible sequences of human
individuals listed by the microhypotheses still are valid, and would have the
same relative prior probabilities. Yet there is a final observable quantity,
which is actually what $X$ measures: the relative time $t$ between humanity’s
start and $X$ happening upon us. Even if the evolution of $X$ is deterministic
enough to demand they arrive at a specific time, $X$ does not have enough
information beforehand to select a particular $t$; presumably such factors are
microscopic and chaotic. Therefore, every single one of the microhypotheses of
the previous section must now be divided into $T$ even finer microhypotheses,
one for each possible $t$ interval $X$ can arrive in.

Whether a Doomsday Argument applies depends on the selection process, in
particular how $X$ found humanity. If $X$ could plausibly have arrived before
or after humanity’s existence, then $T$ is both much larger than humanity’s
lifespan ${\cal T}_{H}$ (or its equivalent) and also independent of ${\cal
T}_{H}$. This selection method is quite common: if one looks for examples of a
rare phenomenon at a particular location, then most likely one will be
searching in the vast epochs before or after it passes. It is in fact less of
a coincidence for $X$ to arrive at the right time to observe humanity if $N$
is large, favoring a Large history – the fraction of surviving microhypotheses
scales as $\sim{\cal T}_{H}/T$.232323This is well known in SETI, where the
abundance of currently observable extraterrestrial intelligences scales
directly with their mean lifespan in the Drake equation. Now suppose $X$
measures the age of humanity. The fraction of microhypotheses consistent with
$X$ arriving at this particular epoch of human history ($\sim 1/T$) is the
same in Small and Large models. Thus there is no Doomsday Argument as such;
$X$ cannot constrain humanity’s lifespan.242424For this reason, if we
specifically observe Proxima b and discover a very young technological
society, we are not justified in concluding they typically live for a short
period before intelligence goes extinct on a planet forever, because of the
sheer improbability of making that discovery in _any_ theory. We could
legitimately suspect that every planet gives rise to a long procession of many
such short-lived societies, however.

But what if there is no possibility of $X$ failing to make an observation of
humanity? One could suppose that rather than searching a particular location,
$X$ seeks out the nearest technological society, scouring the infinite cosmos
if they have to until they find one. (We will also assume all such societies
have exactly the same lifespan for the sake of argument.) This would be more
like picking someone on the street to ask them their age. The only possible
values of $t$ are values within humanity’s lifespan – each Large history
theory has a larger number of possible outcomes. The microhypothesis counts in
Table 1 are multiplied by $N$ to account for the extra degree of freedom in
$t$. Only $\sim 1/{\cal T}_{H}$ of them are consistent with any particular
measurement of humanity’s current age. In FGAI, $X$ now favors Small
histories, and the externally-applied Bayesian Doomsday argument is in
effect.252525And what if $X$ then announces their finding to Earth – should we
agree that humanity’s lifespan is short? Actually, we do not have enough
information to tell. It would be strong evidence if $X$ is the only entity
carrying out a survey like this, or if there is one or more target society per
survey. If there are many such surveys/observers, however, then we expect an
observer like $X$ to show up frequently, even in the early epochs of a Large
history. In fact, $X$ being the first such observer implies our history is
Long – otherwise, the multitudes of observers like $X$ destined to choose our
Earth would have had to squeeze into our short historical epoch before we went
extinct.

These are not the only possible models for the time of $X$’s arrival. Actual
observations may deliberately be carried out at a specific time during its
history, limiting the microhypotheses about $t$ to a narrow range independent
of ${\cal T}_{H}$ [as in the wedding example of 49]. Furthermore, the age of
the Universe sets an upper limit on the relative $|t|$. We cannot rule out
astrophysical objects having lifespans of trillions of years – as we suspect
most do [50]. Finally, the observational outcomes will be biased if the
population varies with time. If we survey an exponentially growing population
of human artifacts, for example, Gott [11]’s Doomsday Argument necessarily
underpredicts the lifespan.

##  V Reconstructing typicality

###  V.1 The indication paradox in Naive FGAI

Naive FGAI, supplemented by the use of implicit microhypotheses and implicit
coordinate systems, is sufficient to handle many practical cases where
typicality is invoked. It also suggests at least one route for addressing the
Doomsday Argument, KSO. As noted, though, it sometimes favors Large theories
where a specific individual is more likely to exist. Now, sometimes this can
be a reasonable assumption. Suppose SB-B was being carried out not with one
participant, but two already extant observers, Alice and Bob, who remember
their pre-experiment lives. Further suppose one was chosen at random to be
wakened on each day, and both wake up at the end. If Alice wakes in the
experiment, she plausibly can conclude the Long version of the experiment is
more likely because she then would be more likely to be awakened. But would it
really make sense if a new participant is instead created each day with no
memories and never waking up again?262626Garisto [30] emphasizes this
distinction between situations where one outcome is “Picked” and those where
each outcome is simply a different location one can “Be”.

Allowing for this SIA-like effect can lead to bizarre consequences. Consider
this compounded version of SB-B:

  * (SB-B′) You wake up in the room. The experimenters tell you they are running SB-B, but each day they will also flip a fair coin each day the experiment is run and tell you the result. But a team member gains your trust and tells you they are lying! The other experimenters have either adopted Strategy A or Strategy B. In Strategy A, they tell you the coin lands Heads on Monday, and if you awaken on Tuesday, they will tell you it landed Tails. In Strategy B, the supposed outcomes are opposite Strategy A’s. Suspecting your new friend’s duplicity, the other experiments exclude them from the decision of which Strategy to go with. Now the experimenters inform you the coin landed Heads. How does that affect your credence in a Short or Long run?

It is obvious the coin flip “result” should not affect our beliefs in a Short
or Long experiment, because of the symmetry between the coin flip outcomes.
Yet in Naive FGAI, _either_ coin flip “result” leads us to favoring a Long
experiment (Figure 5)! The reason is that in the Long experiment, you will
experience each outcome at least once in either Strategy’s microhypothesis,
but only one of the two Strategies in the Short theory results in a Heads
“observation”. Thus only one Short microhypothesis survives compared to both
of Long’s.

The indication paradox is that _any_ datum about the false coin flip favors
the Long run. It is analogous to asserting that since your existence is the
result of innumerable unlikely factors, the Universe is surely very large
given that you as a specific individual do in fact exist.

###  V.2 Revising FGAI to confront the paradox

This is the fundamental issue: how should we calculate likelihoods when we
don’t even know who or where we are? When discussing Naive FGAI, there was a
symmetry between the observers being considered in that all of them genuinely
had equal probabilities distributions for observed outcomes. But in cases like
SB-B and SB-B′, otherwise identical observers at different locations
necessarily observe different outcomes. There is no symmetry to exploit. Naive
FGAI simply used the maximum likelihood among all possible observers, but this
is not the only possible solution. The results of applying other formulations
of FGAI to SB-B′ are illustrated in Figure 5.

Figure 5: The indication paradox of SB-B′, and how it is treated with
different modifications of FGAI. Each box, representing the credence in a
microhypothesis, lists the observed outcomes, ordered by index in the used
coordinate system. An underlined, bold outcome is treated as the one the
observer measures for the purposes of calculating likelihoods. In the Weighted
FGAI treatment, the normalized weights applied to each microhypotheses are
given to the left of each provisional physical credence.

_Implicit FGAI_ – The solution closest to Naive FGAI is labeled Implicit FGAI,
because we reject the formulation of SB-B′ and adopt an implicit coordinate
system referring to “today” and “the other day”. We then fine-grain according
to mixings of the pure Strategies A and B: we instead consider Strategies A′
and B′. In Strategy A′, the experimenters tell us the coin lands Heads today
and Tails on the “other day” if we awaken on “the other day”; in Strategy B′,
the outcomes of A′ are flipped. In this situation, a Heads “observation” today
rules out both B′ microhypotheses, preventing an indication effect. A further
observation that “today is Monday” is treated as purely indexical, with no
relevance for our physical credences.

We can use the symmetry to perform a “coordinate transformation” between
strategies referring to external labels and strategies referring to our
labels. In this case, the symmetry is restored in the participant-centric
coordinates, but this is not generally possible – if our friend told us at the
start the experimenters had decided on Strategy A, there would be only one
allowed microhypothesis in Short, but two in Long, leading to a Doomsday-like
effect. Without this symmetry, implicit FGAI does not let us translate
predictions from one indexical system to another, severely limiting its use.
Labels like “Monday” or “Tuesday” are meaningless until one can connect them
to the implicit coordinate system, propositions like “Strategy A is adopted”
or “Strategy B is adopted” lack the referents to connect them to observations.
Because of this inability, Implicit FGAI is inadequate.

_SSA+FGAI: Doomsday without solipsism?_ – A more radical solution is to
reintroduce the SSA: instead of applying the maximum possible likelihood for
an outcome among all observers we might be, we applying the mean likelihood
among these possible observers. Equation 6 is modified to:

$P^{\rm pos}_{k,j}=\frac{\displaystyle P^{\rm prior}_{k,j}\times\sum_{i\in
O_{k,j}(D)}\frac{{\cal L}(\mu_{k,j}|o@i\to
D)}{|O_{j,m}(D)|}}{\displaystyle\sum_{x=1}^{M}\sum_{y=1}^{m_{k}}P^{\rm
prior}_{x,y}\times\sum_{z\in O_{x,y}(D)}\frac{{\cal L}(\mu_{x,y}|o@z\to
D)}{|O_{x,y}(D)|}}.$ (8)

Then, in SB-B′, the result of the “coin flip” does not affect our credences in
Short and Long. Both microhypotheses of Long would survive the observation,
but they would be penalized by a $1/2$ factor because only one in two
participants is given a Heads outcome. But this penalty continues to apply
even when it is no longer needed: when we now apply the observation that
“today is Monday” (and thus Strategy A was adopted) to the resulting
distribution, Short is now favored two to one.

Although the SSA+FGAI provides a route for the Doomsday Argument in FGAI, this
may not be as bad as it seems. The fine-graining “tames” the unrestricted
SSA’s solipsistic tendencies in many cases, when there is no “ranking”: note
that in SB-B′, we disfavor the Long hypothesis _only_ after we learn it is the
first day of the experiment. The frustration of solipsism is illustrated by a
minimal model in which we consider possible extant observers $A$ and $B$, with
a Small world theory that only one of them exists and a Large world theory
where both exist. In the Small world model, the two microhypotheses are only
$A$ existing and only $B$ existing. But in the Large world model, because
there is no ordering, there is only one microhypothesis: both $A$ and $B$
exist. Thus, $A$’s self-observation allows one of two Small microhypothesis
with its full weight and the sole Large microhypothesis with just half weight
to survive, leaving Small and Large with equal credence. No further data is
possible, and thus Small and Large remain in equal standing.

This is the crucial difference between the Doomsday Argument and the other
solipsistic arguments that SSA+FGAI exploits. When considering $x$-day
Arguments, it is impossible for observers with high $N_{x}$ to exist without
observers with low $N_{x}$ also existing. It would be like if the Small world
model _only_ had the microhypothesis that $A$ exists. When considering other
solipsistic thought experiments, we can include hypotheses that only observers
_unlike_ us exist in our Small theory prior – microhypotheses where _only_
fish are conscious instead of humans, or _only_ Jupiter is inhabited instead
of Earth.272727SSA+FGAI would neutralize some formulation of the “Cosmic
Doomsday” Argument – although it would “refute” _our_ becoming an interstellar
society, it is perfectly consistent with elliptical galaxy inhabitants, dark
matter lifeforms, or aliens who evolve a trillion years from now creating one.
Indeed, we must include them to avoid solipsism, although these possibilities
usually are not included when “Copernican” arguments are made.282828A similar
phenomenon occurs even in Weighted FGAI if we consider two models with the
same number of observers but with a different number of possible outcomes, as
in SB-C. This is the Bayesian Occam’s Razor at work. In some cases, using the
Naive FGAI prior leads to an SIA-like effect favoring Large theories, however
(see Section V.7). Even Doomsday itself may be neutralized in the KSO model if
we apply the SSA to the narrow reference class of observers identical to
ourselves: every microhypothesis where we exist has us living in 2021.

Despite being “tamer” than unrestricted SSA, SSA+FGAI is unsatisfactory both
in that it allows (potentially unrealistically powerful) Doomsday Arguments at
all and because it never “forgives” the penalties it imposes. Recall, the
$1/2$ factor applied to Long’s likelihood in SB-B′ arose from the fact that
you could plausibly either be awake then on Monday or Tuesday. Because of your
indexical uncertainty, you had no way to assign one value to the likelihood,
and the SSA’s purpose was to provide it in the face of this ambiguity. That
circumstance no longer applies if you then learn “today is Monday” and the
indexical uncertainty is eliminated, but there is no way to undo the SSA.
Essentially, this results in SSA+FGAI double-counting the evidence against
Large theories: penalizing all the microhypotheses because we do not know who
we are, and then penalizing some of the microhypotheses because we _do_ know
who we are. In contrast, if you had learned “today is Monday” first, then you
would have had enough indexical information to calculate a single likelihood
for each microhypothesis. SSA+FGAI is unsatisfactory because it either reaches
different conclusions based on which order data is learned, or it demands the
widest possible reference class always be used no matter what we learn.

_Rebooted SSA+FGAI_ – “Rebooted SSA+FGAI” is a modification of SSA+FGAI to
eliminate the order-dependence of learning data. It is like SSA+FGAI, except
that the credence distribution is reset to the original prior whenever new
indexical information is learned. Then, all physical and indexical data is re-
applied simultaneously. The likelihood of microhypotheses are evaluated
according to the SSA applied only among observers with all your current
indexical data. Essentially, with each reboot, the reference class shrinks.
The rebooting undoes the penalty applied by the SSA.

###  V.3 Weighted FGAI

The most advanced of the FGAI frameworks is Weighted FGAI. Like Rebooted
SSA+FGAI, Weighted FGAI effectively allows re-evaluation of likelihoods as new
indexical information is learned. It also provides a more formal framework for
keeping track of indexical information.

Weighted FGAI’s premise is that there is no single physical distribution for
the microhypotheses. Instead, every microhypothesis has many physical
credences, one for each possible observer location. These are the _provisional
physical credences_ , each evaluated as if we were certain that we were at its
associated location. The set of provisional physical credences is then bound
by a set of _indexical weights_ , which is an indexical distribution evaluated
according to the indexical knowledge we have. At any given time, we evaluate
our credences in the various microhypotheses by averaging over the provisional
physical distributions using the indexical weights.292929To use a loose
analogy, the combination of indexical weights and provisional physical
distributions can be compared to density matrices in quantum mechanics.
Density matrices are generally needed to describe the mixed state of a part of
a system. Metaphorically speaking, Weighted FGAI is needed when an observer
sees only a part of the consequences of a microhypothesis and no single pure
distribution can describe credences.

In Weighted FGAI, we consider the set $O$, collecting every possible observer-
trajectory among all the microhypotheses in each macrotheory. Within each
macrotheory $\tau_{k}$, each possible observer-trajectory $i$ we might be
located on has an indexical weight $w_{k,i}$. In real experiments, we may know
enough to eliminate some possible observer-trajectories, so these are excluded
from $O$ (e.g., we restrict $O$ to humans). Note that all microhypotheses in
each macrotheory share these weights, but they are not shared between
macrotheories (see Section V.5). Only some subset $O_{k,j}$ of $O$ are
possible observer-trajectories in a given microhypothesis $\mu_{k,j}$, and
each $\mu_{k,j}$ has its own normalization for the weights: ${\cal W}^{\rm
pos}_{k,j}\equiv\sum_{i\in O_{k,j}}w^{\rm pos}_{k,i}$. In the absence of any
prior data or other constraints on uninformative priors, the weights are
initialized with an uninformative distribution $w^{\rm prior}_{k,i}=1$. This
is the effective role of the SSA in Weighted FGAI – it is the principle for
choosing initial indexical weights.

The provisional physical credence $p_{k,j,i}$ is the credence in $\mu_{k,j}$
calculated by an observer along observer-trajectory $i\in O_{k}(D)$. They are
not normalized, since that would require the indexical weights that are
calculated for other microhypotheses, and are not true probability
distributions, although they are closely related to actual physical credence
distributions. Unless otherwise required based on prior knowledge or to ensure
an uninformative credence distributions (Section V.7), initial prior credences
at all observer-locations are assigned as in Naive FGAI: $p^{\rm
prior}_{k,j,i}=P^{\rm prior}_{k,j}$.

We update the provisional physical distributions when observations when an
observation give us data $D$. For each microhypothesis, the observation model
combined with the assumption that we are the observer along trajectory $i$
yields a single likelihood for that data. The provisional physical
distributions are thus “updated” using these likelihoods, neglecting
normalization: $p^{\rm pos}_{k,j,i}=p^{\rm prior}_{k,j,i}\times{\cal
L}(\mu_{k,j}|o@i\to D)$. As in Naive FGAI, the provisional physical
distributions are insulated from indexical information.

The data may also yield indexical information – if we learn “today is Monday”,
“I am a human on Earth”, or “my birthrank is 108 billion”, for example. The
indexical weightings is updated independently according to the probability we
have data $D$ if we are the observer along trajectory $i$, averaged over the
macrotheory’s microhypotheses:

$w^{\rm pos}_{k,i}=w^{\rm prior}_{k,i}\frac{\sum_{j:i\in O_{k,j}}p^{\rm
prior}_{k,j,i}P(o@i\to D|\mu_{k,j})}{\sum_{j:i\in O_{k,j}}p^{\rm
prior}_{k,j,i}}$ (9)

In situations like SB-A when we already are certain of the physical nature of
the world, $p^{\rm prior}_{k,j,i}$ is nonzero only for the known world model
$\mu_{k,j}$. In that case, $w^{\rm pos}_{k,i}$ is an unnormalized indexical
credence and equation 9 reduces to normal Bayesian updating. The Bayesian
limit more generally occurs if $P(o@i\to D|\mu_{k,j})$ is the same for all
remaining microhypotheses. Indexical data can be neutral with respect to the
different locations. In other cases, we may learn exactly which observer we
are (such as “My birthrank is exactly 108 billion” in a Doomsday Argument),
resulting in a nonzero weight for a single observer-trajectory because the
likelihoods for all others are zero.

Our posterior credence in microhypothesis $\mu_{k,j}$ is then generated from
the provisional physical distributions by applying the indexical weights:

$\displaystyle P^{\rm pos}_{k,j}$
$\displaystyle=\frac{\displaystyle\frac{1}{{\cal W}^{\rm pos}_{k,j}}\sum_{i\in
O_{k,j}}w^{\rm pos}_{k,i}p^{\rm prior}_{k,j,i}{\cal L}(\mu_{k,j}|o@i\to
D)}{\displaystyle\sum_{x=1}^{M}\sum_{y=1}^{m_{x}}\frac{1}{{\cal W}^{\rm
pos}_{x,y}}\sum_{z\in O_{x,y}}w^{\rm pos}_{x,z}p^{\rm prior}_{x,y,z}{\cal
L}(\mu_{k,j}|o@i\to D)}$ (10) $\displaystyle=\frac{\displaystyle\frac{1}{{\cal
W}^{\rm pos}_{k,j}}\sum_{i\in O_{k,j}}w^{\rm pos}_{k,i}p^{\rm
pos}_{k,j,i}}{\displaystyle\sum_{x=1}^{M}\sum_{y=1}^{m_{x}}\frac{1}{{\cal
W}^{\rm pos}_{x,y}}\sum_{z\in O_{x,y}}w^{\rm pos}_{x,z}p^{\rm pos}_{x,y,z}}.$

The posterior credence $P^{\rm pos}_{k}$ of a macrotheory is once again found
by summing the credences of its microhypotheses. The use of the weights means
that Weighted FGAI is not a strictly Bayesian theory. We cannot calculate the
“actual” physical credence until we can assign a single, unambiguous
likelihood, and we cannot do this in FGAI. Instead the posterior credence is a
construction built from possible physical credences using the indexical
weights.

Equation 10 can be viewed as Weighted FGAI’s rendition of the Observer
Equation proposed by Bostrom [2]. It has several useful limits. First, if the
likelihood ${\cal L}(\mu_{k,j}|o@i\to D)$ is the same for all remaining
possible observer trajectories $i$ ($W^{\rm pos}_{k,i}>0$) in each
microhypothesis, then this application reduces to Bayes’ formula applied on
the indexical weight-averaged prior probabilities. In particular, in
situations where indexical information is entirely irrelevant, $p^{\rm
prior}_{k,j,i}=P^{\rm prior}_{k,j}$ and all indexical dependence vanishes from
equation 10. Second, suppose the data $D$ picks out a subset
$O^{\prime}\subset O$ possible observer-trajectories, but is equally
consistent with all of them. Then, $O^{\prime}$ is our location reference
class, and equation 9 applied to equal prior weights yields an equal prior
weight for each element in $O^{\prime}$. This amounts to an assertion of the
SSA applied to the narrowed reference class $O^{\prime}$. Equation 10 then
implements SSA+FGAI (equation 8). Of course, if we know our exact observer-
trajectory, then that trajectory’s indexical weight is $1$, and we are left
with Bayes’ formula with likelihoods calculated using our known location.
Finally, if we are already certain that $\mu_{k,j}$ is correct, then the prior
provisional credences are nonzero only for this microhypothesis, and equation
10 gives $P^{\rm pos}_{k,j}=1$.

As required in FGAI, the indexical weights are entirely separated from the
(provisional) physical distributions, albeit with a more active role than in
Naive FGAI. The weights shift adaptively in response to indexical data, as we
can more accurately assess likelihoods over microhypotheses. The novel feature
of Weighted FGAI is the shifting indexical weights’ ability to “revive”
theories that were provisionally disfavored by the SSA. This lets us implement
the “rebooting”, where we evaluate all the likelihoods again with fresh
indexical knowledge. If we had only a single joint probability distribution,
evidence can only eat away the prior probability in a theory because
likelihoods are always $\leq 1$. The only way a theory survives with a single
credence distribution is if the other theories’ prior credence is also
consumed at a comparable rate. In Weighted FGAI, the provisional probability
invested in disfavored observer trajectories is not lost but re-assigned to
the microhypotheses consistent with likely observer trajectories.

This formulation of Weighted FGAI is based on the assumption that there are
only a finite number of locations or trajectories, parameterized by a finite
set of indexical weights. Infinite volumes are a common feature of modern
cosmologies, however. One way to address this problem would be to replace the
discrete distributions with densities in a way that recovers conclusions found
for an arbitrarily large finite number of locations. Another is to note that
the number of expected configurations of a Hubble volume is expected to be
finite [e.g., 51]. The ultimate motivation for including the weights is that
likelihood calculations depend on observer-trajectory and evolution. If two
locations are located in two indistinguishable Hubble volumes, their
likelihood calculations should also be indistinguishable. Because the weights
would no longer refer to a single location with a unique evolution, trouble
could arise if all observations are be explained away by “updating” to
whichever unlikely sequence of Hubble volume configurations just happens to
fit the data. We might prevent this by allowing indexical updates solely on
information available in the past light cone when observations begin. This
would let us update indexical information to account for our birthrank or our
location in a spiral galaxy, for example, without letting us pretend that we
just happen to be in the Hubble volume where all observed positrons have
remained stable despite having a half-life of a femtosecond.

###  V.4 What are the indexical weights?

The indexical weights probabilistically define our current reference class,
which evolves as we learn new information. These weights are calculated using
likelihoods averaged over all microhypotheses. But who is doing the
observation along that trajectory depends on the microhypothesis, just as who
is born at birthrank $1$ in the Doomsday Argument varies. Thus the indexical
weights describe a reference class of locations or contexts (or more properly,
trajectories), not observers. The indexical weights tell us _where_ we are,
not _who_ we are.

Equation 9 includes a dependence on the prior physical credence $p^{\rm
prior}$. When normalized, the indexical weights themselves form a joint
physical-indexical probability distribution, even though weighted FGAI rejects
such distributions for the physical credence. Unlike previous joint
distributions, the indexical weights implement an inverse SSA. Instead of
evaluating physical hypotheses under the assumption we are a typical observer,
it evaluates indexical hypotheses under the assumption that a typical physical
hypothesis within a macrotheory is true.

Indexical weights can also be viewed as a second-order probability
distribution over a series of trivial indexical distributions. Srednicki and
Hartle [32] first proposed that we could test the correctness of various
indexical distributions called xerographical distributions by comparing their
predictions in physical theories with our evidence. A key difference with
Srednicki and Hartle [32], though, is that only an orthonormal set of
xerographic distributions, each implying absolute certainty in one’s
trajectory, is used in Weighted FGAI. 303030Friederich [31] noted that a
xerographic distribution with nonzero credence for multiple possible observers
is nonetheless consistent with learning our identity with certainty. That is
because the likelihood of receiving that information is nonzero for one of the
observers, in turn yielding a nonzero likelihood for the xerographic
distribution as a whole. Thus, a xerographic distribution expressing
uncertainty survives learning perfect indexical information. Weighted FGAI
avoids this issue because indexical locations themselves are weighted.

Unlike the attached indexical distributions of Section III, the indexical
weights are averaged over all possible microhypotheses. The indexical weights
can be non-zero for locations completely inconsistent with the data – indeed,
this is how Weighted FGAI is able to constrain these microhypotheses at all.
If we are just interested in the indexical distribution for a single
microhypothesis, it can be calculated as:

$\xi_{k,j,i}=\frac{w^{\rm prior}_{k,i}p^{\rm prior}_{k,j,i}P(o@i\to
D|\mu_{k,j})}{\sum_{z\in O_{k,j}}w^{\rm prior}_{k,z}p^{\rm
prior}_{k,j,z}P(o@z\to D|\mu_{k,j})}.$ (11)

One unresolved issue is what counts as a separate trajectory. If the
experimenters in SB-B′ assigned you to room I if they chose Strategy A but
room II if they chose Strategy B, or if they put you in room I on the Heads
day and room II in the Tails day, different hypotheses about strategies
naively have their own indexical weights. This leads to an SIA-like effect,
even if the rooms are completely indistinguishable from your perspective (see
also Section V.7). A deeper understanding of locations and trajectories in
Weighted FGAI is needed to resolve the issue, however.

###  V.5 How should the indexical weights be linked?

According to equation 10, the weights are shared among all microhypotheses and
all macrotheories, but is this necessary?

Figure 6: Illustration of how using microhypothesis-level weighting in SB-B′
results in the indication paradox where any data about the “coin flip”
automatically favors the Long theory.

Perhaps the most natural solution would be to give each microhypothesis its
own set of weights, updated according to equation 11. This scheme is invalid:
the averaging of the weights regulates how much each weight can “adapt” to new
data. Microhypothesis-level weights are too sensitive – they find whichever
observer happens to have your data $D$ and declare that you are that observer.
In a large enough universe, there is likely to be some observer at some
location with your data $D$, and the result is credence evaluation according
to this optimal location, resulting in a likelihood of $1$. It would be like
flipping a coin a thousand times, always getting heads, and then concluding
that the coin is fair but you just happen to be the fluke observer who will
happen to observe a thousand heads in a vast cosmos (c.f. Section IV.5).
Figure 6 illustrates what happens when microhypothesis-level weights are used
to resolve SB-B′: the observer automatically favors the Long theory upon
learning the “result” is Heads because a Heads observation is more likely to
exist somewhere within the Long theory.

Another natural solution would be to share indexical weights between
macrotheories. Indexical weights would be updated as in equation 9 by
averaging over all possible microhypotheses in every macrotheory where the
observer-trajectory exists. When the probability of an observer along
trajectory $i$ observing data $D$ is independent of macrotheory, macrotheory-
level weighting gives the same results as universal weights. Universal weights
are problematic because the averaging depends on macrotheory credences
themselves. Suppose in SB-B, we woke up and then learned the experimenters had
chosen to run the Long variant without learning what day it was. With
universal weights, we would update the Monday weight to $w_{M}=1/2$, resulting
in us favoring it being Tuesday.

How weights are shared determines how “aggressive” Weighted FGAI is in making
conclusions. Imagine there is an experiment with two indistinguishable
participants in two rooms, both of which contain a coin. The participants
watch a coin flip and observe its results but are not allowed to inspect the
coins. In variant A of the experiment, both coins are fair, but in version B,
room I’s coin always lands on Heads. Both variants are given equal prior
credence. The microhypotheses list the possible outcomes of each coin flip in
each variant. Not knowing which room they are in ($W^{\rm
prior}_{\textrm{I}}=W^{\rm prior}_{\textrm{II}}=1/2$), one participant
observes their coin lands Heads. How should the participant update the
indexical weights?

If both variants A and B have their own weights, then the participant may
update their variant B weights to $W_{B,\textrm{I}}=2/3$ and
$W_{B,\textrm{II}}=1/3$, while the variant A weights are unaffected. Then the
participant evaluates $P^{\rm pos}(B)=5/8$. But if there is just one set of
weights for both A and B, they are instead updated them to
$w_{\textrm{I}}=1/2\times(2/8+2/4)=3/8$ and
$w_{\textrm{II}}=1/2\times(2/8+1/4)=2/8$. The resulting evaluated credence in
variant B is $P^{\rm pos}(B)=(3/5\times 2/4+2/5\times 1/4)/[(3/5\times
2/8+2/5\times 2/8)+(3/5\times 2/4+2/5\times 1/4)]=8/13$. Linking the weights
on the macrotheory-scale results in a faster convergence to the belief that
one is in room I and variant B is being run.313131For comparison, the credence
in variant B is only $4/7$ if we use microhypothesis-level weighting, $3/5$ if
we use uninformative weights, and as high as $2/3$ if we are certain we are
located in room I.

This remains another open issue in Weighted FGAI, without an entirely clear
justification for sharing weights on the macrotheory-level aside from the fact
that it gives us the “right” answers.

###  V.6 Weighted FGAI and Sleeping Beauty

Figure 7: Treatment of the fundamental variations of the Sleeping Beauty
problem in Weighted FGAI. In SB-A, only the indexical weights shift, while in
SB-C, the physical credence distributions entirely drive the conclusion. In
SB-B, the indexical information shifts the indexical weights assigned to each
provisional distribution, ensuring that neither the Short nor Long
distributions are favored.

Weighted FGAI’s treatment of the Sleeping Beauty examples is illustrated in
Figure 7. Because both the physical provisional distributions and indexical
weights are updated according to Bayes’ equation (as required by Section
III.3), we can recover the naive Bayesian predictions of SB-A and SB-C.

In the SB-B variant, there are only the Short and the Long theories, with one
trivial microhypothesis each, and each having $1/2$ prior credence. The Short
theory has a normalized indexical weighting of $1$ for Monday. The Long theory
has a Monday provisional probability and a Tuesday provisional probability,
both still equal to $1/2$. The Long theory also has an indexical weighting of
$1/2$ for Monday and $1/2$ for Tuesday. Now we learn “today is Monday”. The
likelihood that this observation is made is $1$ if we are the Monday observer
so the Monday provisional probabilities have a likelihood of $1$. The
likelihood this observation is made is $0$ if we are the Tuesday observer, so
the Tuesday provisional probability in the Long theory is eliminated. However,
the Long theory gives us 100% confidence in our being the Monday observer if
today is Monday, and its indexical weights shift to $1$ for Monday and $0$ for
Tuesday. As a result, the probability lost in the Tuesday provisional
distribution is irrelevant. Our credence in the Long theory is $(1\times
1/2+0\times 0)/[(1\times 1/2)+(1\times 1/2+0\times 0)]=1/2$, just as before.