Content uploaded by Vesa Palonen
Author content
All content in this area was uploaded by Vesa Palonen on May 13, 2013
Content may be subject to copyright.
1
Bayesian considerations on the multiverse explanation
of cosmic fine-tuning
V. Palonen
1
Department of Physics, P.O. Box 43, FI-00014 University of Helsinki, Finland
(Dated: September 3rd, 2009)
Short title: Bayesian considerations on the multiverse
The fundamental laws and constants of our universe seem to be finely tuned for life. The
various multiverse hypotheses are popular explanations for the fine tuning. This paper
reviews the four main suggestions on inference in the presence of possible multiple
universes and observer selection effects. Basic identities from probability theory and
previously unnoticed conditional dependencies of the propositions involved are used to
decide among the alternatives.
In the case of cosmic fine-tuning, information about the observation is not independent of
the hypothesis. It follows that the observation should be used as data when comparing
hypotheses. Hence, approaches that use the observation only as background information
are incorrect. It is also shown that in some cases the self-sampling assumption by
Bostrom leads to probabilities greater than one, leaving the approach inconsistent. The
“some universe” (SU) approach is found wanting. Several reasons are given on why the
“this universe” (TU) approach seems to be correct. Lastly, the converse selection effect by
White is clarified by showing formally that the converse condition leads to SU and its
absence to TU. The overall result is that, because multiverse hypotheses do not predict the
fine-tuning for this universe any better than a single universe hypothesis, the multiverse
hypotheses fail as explanations for cosmic fine-tuning. Conversely, the fine-tuning data
does not support the multiverse hypotheses.
Keywords: Multiple universes; Multiverse; Bayes; Belief networks; Probability; Observer
selection effects; Anthropic selection, Anthropic principle, This universe
PACs:
02.50.Tt, 98.80.Qc
1
email: vesa.palonen@helsinki.fi
2
Introduction
The fine-tuning of our universe’s fundamental laws and constants [1] [2] [3] [4] [5] [6]
requires an explanation [2] [7] [8]. For example, it seems that the strength of gravity has
to be fine-tuned to an accuracy of 1/10
36
[4] for life to exist. Depending on the researcher,
the number of recognized fine-tuning constraints posed by life varies from some tens to a
hundred.
A popular way to explain the happy coincidences are the varied multiverse hypotheses.
Namely, if there are sufficiently many universes, it will not be surprising to find at least
one universe with fine-tuned constants. Because we cannot observe our own non-
existence, the observer selection effect (OSE) is thought to ‘filter out’ all the universes
where observers can not exist. Indeed, many have taken the combination of multiverse
theories with the observer selection effect as sufficient to explain fine-tuning [2] [9] [10].
Several inferences in physics have been made using anthropic constraints. Given that
observers (we) exist, one can use this information and constrain the estimates on physical
parameters or make predictions about them. This represent inference within a physical
hypotheses and both the hypothesis and observability are used as background
information, basically as constraints. This kind of inference should not be confused with
the goal of the present paper which is inference concerning the overarching hypotheses
themselves. As will be seen, when comparing hypotheses, the information about
observability enters the comparison as data, not merely as a constraint.
A good reminder of problems with the use of observer selection effects is given by Leslie
[2]: A prisoner being executed by a firing squad finds that all of the marksmen have
missed and against all odds he is alive. The surviving prisoner now notes that one can
only observe one’s own survival and uses the observer selection effect to infer that his
survival was just as likely to be due to chance than due to design. There is nothing to
explain, claims the prisoner. In fact, any ridiculous hypothesis with a nonzero probability
for survival is an equally good explanation. But this seems intuitively false.
The above paradox is even more visible when we modify the example a bit. Let us
consider a case where the prisoner is blindfolded and cannot hear anything. The prisoner
only knows that he has just survived an execution by a firing squad. Furthermore, the
prisoner knows that there are two squads around, one with two marksmen, the other with
a thousand marksmen. Which firing squad is more probable to have carried out the
execution? If the prisoner uses the selection effect as advocated e.g. by E. Sober [10], he
will infer that both squads are equally probable. Again, this kind of inference seems
intuitively false. It seems that at least in some cases hypotheses which give a low
probability for the observation (survival) should also be given a low probability in the
hypothesis comparison.
This paper reviews and evaluates four popular suggestions on how to use the observer
selection effect in the case of fine-tuning. These are the “assume the observation (AO)”
approach above by Sober, the “self-sampling assumption (SSA)” approach by Bostrom
[11], the “some universe (SU)” approach by e.g. Manson and Thrush [12], and the “this
universe (TU)” approach by e.g. White [13] and Dowe [14]. Among these, mainly the
SSA has been claimed to be a general theory for selection effects. AO and SSA are
discussed first and reasons are given for why they are likely to be incorrect. The
3
discussion then proceeds to SU and TU and reasons are given for why cosmic fine-tuning
seems to be a TU case.
The main argument in this paper concerns the proper use of the probabilities. The precise
shape of the probability distributions or the numerical values are not central to the
argument. The argument remains essentially the same for all reasonable distributions.
Ways of estimating some of the underlying distributions exist [15], but uniform
probabilities are used here for clarity. Although some shortcuts would be available, the
equations are mostly written down in detail because in my opinion in the end this is the
clearest way of carrying out the inference.
Principles of Bayesian hypothesis comparison
In this section the basic equations for Bayesian hypothesis testing are reviewed. Readers
familiar with Bayesian methods may want to skip this section.
Probabilities will be used in the epistemic sense, denoting plausibilities first and
frequencies of occurrence only as a result [16] [17]. It is to be understood that all
probabilities are conditional on some background information
I
.
I
will mostly be
shown explicitly in the equations as this may help the reader to see the difference
between its use and the use of other propositions in the calculation.
Bayes’ theorem can be derived from the product rule of probability, which states that the
probability for
A
and
B
can be written as
( ) ( | ) ( )
p A B p A B p B
∧ =
, (1)
where by
A B
∧
we mean a logical conjunction (
A
and
B
are true) and
( | )
p A B
denotes a conditional probability of
A
being true given that B is true. Alternatively one
can also write
( ) ( | ) ( )
p A B p B A p A
∧ =
. (2)
Equating the right sides of eqs. (1) and (2) one obtains the Bayes’ theorem
( ) ( | )
( | )
( )
p B p A B
p B A
p A
=
. (3)
Eq. (3) is a general tool for probabilistic inversion; If the probability of
A
given
B
,
( | )
p A B
, is known, we can calculate the probability of
B
given
A
,
( | )
p B A
. However,
this presupposes that we can estimate
( )
p B
and
( )
p A
, the prior probabilities of
B
and
A
.
Let us assume that
, 1,...,
i
C i n
=
are a complete set of propositions. The law of total
probability states
( ) ( )
i
i
p A p A C
= ∧
∑
. (4)
4
Hence, if we have the joint probability density for
A
and
i
C
, we can sum over all the
possible
i
C
to get the probability for
A
. This process of eliminating propositions by
summing is called marginalization.
Let
n
H
be some hypothesis and
D
some measured data. Bayes’ theorem (eq. (3)) gives
the probability of the hypothesis
n
H
given the data
D
( | ) ( | )
( | )
( | )
n n
n
p H I p D H I
p H D I
p D I
∧
∧ =
, (5)
where
( | )
n
p D H I
∧
is the probability of the data for the hypothesis
n
H
(often called the
likelihood) and
( | )
n
p H I
is the prior probability of
n
H
. The term
( | )
p D I
is often called
a marginal probability and can be viewed as just a normalization constant common to all
hypotheses.
Using eq. (5), we can compare the probabilities of several hypotheses by using the ratio
of their probabilities given the data
( | ) ( | ) ( | )
( | ) ( | ) ( | )
n n n
m m m
p H D I p H I p D H I
p H D I p H I p D H I
∧ ∧
=
∧ ∧
, (6)
where on the right side the first term is the ratio of the prior probabilities of the
hypotheses. The second term is the ratio of probabilities for the measured data D under
each hypothesis. It is therefore a ratio of each model’s prediction of the data D.
In order to clarify the use of the above equation, Figure 1. depicts the predictions of two
hypotheses for the variable
E
. It is important to note that probability distributions are
normalized so that the total probability, which is a sum or an integral over all possible
values, equals one. In the figure, hypothesis
1
H
gives a rather broad prediction and due
to normalization, the prediction is low overall. Hypothesis
2
H
gives a rather strict
prediction for
E
near to the value 2 and the distribution is high in this area. Now, if we
measure
2
E
=
,
1
( 2 | )
p E H I
= ∧
has a moderate value but
2
( 2 | & )
p E H I
=
is large.
From eq. (6) we get
1
2
( | 2 ) 0.35
1
( | 2 ) 1.6
p H E I
p H E I
= ∧
=
= ∧
and hence hypothesis
2
H
is more
probable given the data. However, if we measure
3
E
=
, which does not hit the high peak
of
2
H
,
1
2
( | 3 )
0.35
1
( | 3 ) 0.001
p H E I
p H E I
= ∧
=
= ∧
and hence hypothesis
1
H
is more probable. It is
seen that because of the normalization, hypotheses are penalized for making broad
predictions. They give a moderate probability for many possible values but cannot predict
anything well.
5
Figure 1. Two hypotheses H1 and H2 have a prediction concerning the value of E. Due to
normalization of the probability density, a hypothesis with a broad prediction does not predict
anything very well.
In addition to the above equations, some tools for handling the conditional independence
of statements, originally discovered in the field of Bayesian belief networks, will be used.
They will be introduced below.
Nomenclature
The following symbols will be used frequently, the reader may wish to skim casually
through them for now and refer back to the definitions in detail later if necessary:
n
H
The proposition that hypothesis
n
H
is true.
M
Multiverse hypothesis in particular
Ω
The set conceivable values for the vector of physical constants
E
A vector of the coordinate variables of corresponding to the physical constants
and parameters.
D
Our present data (concerning a realization of E),
D
↔
”E = the present physical
constants and parameters.”
Later in the paper we will need to distinguish between “this universe is fine-tuned” and
“some universe is fine-tuned”. For this reason the following two are introduced:
t
D
This universe is fine-tuned. The index
t
represents the ‘true index’ of our
universe.
'
D
Some universe is fine-tuned
And we will also be using the following:
,
E n
Ω
A set of all possible values for E, given hypothesis
n
H
,
O n
Ω
A set of all values of E that can be observed given hypothesis
n
H
6
O
Ω
A set of a all values of E that can be observed, an union of all
,
O n
Ω
,
,
O O n
n
Ω ↔ Ω
∪
O A proposition “The realization of E can be observed”,
,
( )
n O n
n
O H E↔ ∧ ∈Ω
∪
S(
Ω
) The size (cardinality) of
Ω
For clarity, we will call a universe inside a multiverse a subverse. Our universe, which is
possibly one subverse, will be called ourverse.
Assume the observation (AO)
E. Sober [10] and Ikeda and Jefferys [18] have argued that because one cannot observe
one’s own non-existence, the observability O should be taken as background information
for hypothesis comparison. If this is the case, using the Bayes’ theorem to obtain the
probability of hypothesis
n
H
given the fine-tuning data
D
, one would get
,
,
( | ) ( | ) ( | ) / ( | )
( | )
( | )
1
,
( )
n n n
n
n O n
O n
p H D O I p H I p D H O I p D I
p D H O I
p D H E I
S
∧ ∧ = ∧ ∧
∝ ∧ ∧
= ∧ ∈Ω ∧
≈
Ω
(7)
where we have neglected
( | )
n
p H I
, the prior probability for the hypothesis, and
( | )
p D I
, the marginal probability in the second line. The prior and marginal probabilities are
roughly the same for all hypothesis and will roughly cancel out when
n
H
is compared to
other hypotheses. The remaining term
( | )
n
p D H O I
∧ ∧
is the prediction of the
hypothesis for the data
D
. The important thing to note in the above result is that
O
is
used as a condition for the probabilities in the same way as the background information
I
is. This filters out all unobservable events and is equivalent to assuming that all
hypotheses will only produce observable events. The prediction of fine-tuning for each
hypothesis is therefore improved to certainty in the AO approach. This is a bit like asking
for the probability of a six in a dice throw knowing that you will get a six. Hence,
assuming AO and taking a uniform prediction for the cosmic constants in the last line of
eq. (7), the probability of a hypothesis in the comparison is only dependent on the size of
the observation-space.
Note that usually in Bayesian hypothesis comparison hypotheses are penalized for
making broad predictions because the prediction is a normalized probability distribution.
Yet, on AO, the non-observable part of the prediction is filtered out and the hypotheses
are not penalized for predicting however broadly in the event-space.
7
Weisberg [19] has noted that our information in the case of observer selection effects is
actually of the form ”If we observe, we will observe
O
” not ”we will observe
O
”, as AO
assumes. So, we really have only the conditional
" "
D O
→
as background information.
Weisberg shows that this information does not raise the prediction for fine-tuning.
Already the above argument seems enough to discredit AO but we will nevertheless press
on to facilitate further understanding on the topic by using some of the mathematics
developed for inference with Bayesian belief networks [20] [21]. In general, the
probability for the hypothesis given the relevant information will depend on the joint
probability, as can be seen from the definition of conditional probability
( | ) ( ) / ( ).
n n
p H D O I p D O H I p D O I
∧ ∧ = ∧ ∧ ∧ ∧ ∧
(8)
Now, the joint probability will in general factor as
( ) ( ) ( | ) ( | ) ( | )
n n n n
p D O H I p I p H I p O H I p D O H I
∧ ∧ ∧ = ∧ ∧ ∧
. (9)
Only in the case that the observation
O
and the hypothesis
n
H
are independent,
|
n
O H
⊥ ∅
(or
|
n
O H I
⊥
) [20] [21], does one get
( ) ( ) ( | ) ( | ) ( | )
( ) ( | ) ( | ) ( | )
( | ) ( | ),
n n n n
n n
n n
p D O H I p I p H I p O H I p D O H I
p I p H I p O I p D O H I
p H I p D O H I
∧ ∧ ∧ = ∧ ∧ ∧
= ∧ ∧
∝ ∧ ∧
(10)
which, when used in eq. (8), is the AO method. In the last line we show only the terms
dependent on the hypothesis and hence relevant for hypothesis comparison. Hence, a
necessary and sufficient condition for Sober’s method is the independence of the
observation from the hypothesis. Figure 1a shows the necessary probabilistic dependency
structure as a graph [22] [23].
In the case of cosmic fine-tuning, the probability of observation is not independent of the
hypothesis but instead the observation is a probabilistic product of the hypothesis, and
one has to use eq. (9). The case corresponds to the graph in Figure 1b. Hence, in the case
of cosmic fine-tuning, probability theory requires one to use also the prediction for the
observation,
( | )
n
p O H I
∧
, in the hypothesis comparison. It’s use is also required by the
strong condition to use all available information. As a result, hypotheses purporting to
explain fine-tuning must be penalized for having a low prediction for observability. So,
when one has survived a very dangerous accident, the capability to observe ones own
survival cannot be taken as background information, thereby assuming that the survival
was certain, but as part of the data. Hypotheses must be favored according to how high a
probability they give for the survival and the corresponding observation. The conditional
statement
" "
D O
→
by Weisberg can legitimately be taken as background information
but this does not raise the prediction for fine-tuning.
8
H
O
D
n
H
n
O
D
a)
b)
Figure 2. a) When the observation
O
is independent of the hypothesis
n
H
conditional on the data
D
, the observation cannot be used as data in the hypothesis comparison, resulting in a result
equivalent to the assume the observation (AO) approach. b) In the case of cosmic fine-tuning the
observation is a product of the hypothesis and hence
O
has to be used as data in the hypothesis
comparison. It follows that the AO approach is incorrect for cosmic fine-tuning.
The self-sampling assumption (SSA)
Interesting work has been done by N. Bostrom [11] on developing a theory of how to
take selection effects into account, along with a good explanation of the issues involved.
Bostrom’s suggestion for inference in cases involving selection effects is the self-
sampling assumption (SSA), which states that one should reason as if one were a random
sample from the set of all observers in one’s own reference class. The exact definition of
‘one’s own reference class’ remains open, leaving room for movement under criticism
[24]. Bostrom has developed a modified version of SSA, called the observation equation
or SSSA-R (from Strong SSA Revised), which mainly corrects for SSA’s bias to select
hypotheses with big amounts of observers. SSA was developed inductively from
examples and from attempted solutions to some encountered paradoxes. This makes the
method somewhat ad-hoc. It is based on intuitive solutions, which may or may not be
truly correct solutions, to thought-experiments, which may or may not be analogous to
fine-tuning. I my opinion SSSA-R and variants try to give directly something that should
be the result of rigorous probabilistic inference.
Leslie has argued that SSA leads to observer-relative chances [25]. Bostrom has admitted
that it does [26], but does not think that this poses a problem. Yet it does seem
problematic that, even if two observers exchange information so that they have the same
information except that the one has ‘I am A, he is B’ and the other has ‘I am B, he is A’,
they will not arrive at the same probabilities with SSA. If they start gambling, both
cannot win. Moreover, in principle the observers could ‘rise above’ their circumstances
and, like we, calculate the probability that the other would use and hence both of them
will obtain two contradictory probabilities for the case. A further question is, what
probability should we use as outsiders, and why should the observers involved not use
that probability also? To me it does not seem rational that the probability will change just
because of the identity of the person doing the inference. Indeed, this is contrary to the
starting assumptions of some derivations of probability theory [16] which may lead to
contradictions in SSA since probability theory was used in the derivation of SSA and
because SSA purports to give out probabilities.
9
We will now show that, in addition to the above problems, SSSA-R can lead to
probabilities bigger than one. We will use Bostrom’s God’s Coin Toss (GC) example in
reference [11] and modify it a bit: A coin is thrown. If the coin lands heads (H), a world
with an amount
h
β
of black bearded and
h
α
red bearded observers is created. If the coin
lands tails (T), a world with
t
β
black bearded and zero red bearded observers is created.
Let
( ) ( ) 1/ 2
p H p T
= =
and let
D
β
represent the evidence available to the observer who
has discovered that he has a black beard. Now, with Bostrom’s own usage of SSSA-R
(the original approach where all observers are in the same reference class), we get
1
1 1 1
( | )p H D
h h t h h
β
α β β α β
−
= +
+ +
. (11)
and
1
1 1 1
( | )p T D
h h t t
β
α β β β
−
= +
+
. (12)
But a problem becomes apparent when we calculate
( ) ( ) ( ) ( ) ( | ) ( )
p T p T D p T D p T D p T D p D
β β β β β
= ∧ + ∧ ¬ = ∧ =
, (13)
and hence
( )
( )
( | )
p T
p D
p T D
β
β
=
. (14)
Now because
( ) 1/ 2
p T
=
and
( | )
p T D
β
can be arbitrarily small, SSSA-R can lead to
arbitrarily large values for
( )
p D
β
. For example, taking
2
h h
β α
= =
and
10
t
β
=
, we get
1/ 2 7
( )
2 / 7 4
p D
β
= =
. But by definition a probability cannot be bigger than one. A similar
result can be obtained for SSA. Hence, SSA and variants are inconsistent with probability
theory.
This universe (TU) or some universe (SU)?
We will now turn our attention on whether one should really use
t
D
⇔
“this universe is
fine-tuned” (TU) or
'
D
⇔
=“some universe is fine-tuned” (SU) as data. Of course, this
choice has profound implications on the inference concerning fine-tuning because in a
sufficiently large multiverse it is virtually certain that some subverse will be fine-tuned.
10
White has argued for the use of TU as data in the fine-tuning case [13]. White points out
that some-propositions are weaker and contain less information than corresponding this-
propositions. Hence, White argues that because the some-proposition
'
D
contains less
information than the this-proposition
t
D
, which we have available,
t
D
should not be
replaced with
'
D
because doing so would violate the strong desiderata to use all
available information. So, when a this-proposition is available as data, the use of the
corresponding some-proposition is warranted if and only if it leads to the same
conclusion as using the this-proposition. We will first try to answer the main criticisms of
TU and then show why cosmic fine-tuning is indeed a TU case.
Manson and Thrush argue against the use of TU on the basis that those arguing for TU do
not seem to use “this planet” (TP) as data [12]. This criticism is dependent on two
assumptions:
(1) We should not use TP as data.
(2) Subverses and planets are equivalent as concerns the inference.
Consider a simple case where we did not have any observations of the other planets. Let
us also say we had two hypotheses about solar system formation. The good hypothesis
1
H
would predict the observed properties of the earth and the sun with good accuracy and the
bad hypothesis
2
H
would give a near-zero prediction for the data. Science normally
proceeds by selecting the hypothesis which better explains the phenomena, in this case
1
H
. Yet, if we were to use “some planet is such and such” (SP) as data, even the bad
hypothesis would predict that the data will be observed somewhere with a probability of
one when there is an infine amount of planets. The two hypotheses would remain equal.
It therefore seems that the past, present, and future success of astrophysics depends on
our steering clear of SP.
Second, there is a difference in inference concerning planets and universes: we can
observe other planets but not other subverses. So, for planetary formation, our data
consists also of the other observed planets and hypotheses are compared based on all of
the planets. In the case of the multiverse, we do not have data on the other universes and
therefore our inference will be based only on this universe. As will be shown below, this
difference is important when considering whether one should use this- or some-type
propositions. Hence, subverses and planets are not analogous for the present discussion.
Bostrom’s analysis of fine-tuning also uses SU. Bostrom argues against TU [11] first by
pointing out that White’s approach is committed to
( | ' ) ( | ')
i
p M D D p M D
∧ <
for every
subverse
i t
≠
. The above means that the probability for a multiverse is smaller when TU
is used as data than when SU is used as data. So far so good, but next Bostrom
misinterprets the inequality to mean that any finely tuned universe other than ourverse
would make
M
more likely, which he then sees as a problem for TU. But what makes
( | ')
p M D
bigger is not the state of any particular universe but instead the relaxation of
the information that we have about the particular universe. A small modification of
White’s coffee table example [13] shows that this is a natural inequality resulting from
the difference between ‘some’ and ‘this’: The information that someone at the coffee
table was drunk last night increases the probability for there being several persons at the
11
table. Yet the use the information that in fact the person who was drunk last night was
John cancels the increase of probability for there being several people at the table.
Second, Bostrom tries to prove wrong White’s assertion that a multiverse does not predict
fine-tuning for this universe any better than a single universe by pointing out that there
may be correlations among subverses. Bostrom seems to shift the burden of proof to
White, claiming that unless it is proved that a multiverse cannot gain a better prediction
for fine-tuning due to correlations, a multiverse is a good explanation. But possible
correlations to other subverses cannot improve a multiverse’s prediction for fine-tuning
because we do not possess relevant information about the other possible subverses.
Neither do we have sufficient knowledge about the correlation function. Using epistemic
probabilities we can marginalize over our ignorance and as the sum is taken over the
possible states of the other subverses, the prediction about ourverse will of course iron
out irrespective of the correlations. And marginalization over the possible correlation
functions would have the same effect. Hence, correlations among the subverses do not
help the multiverse.
Third, Bostrom goes on to claim that White is committed to denying the above-
mentioned inequality
( | ' ) ( | ')
i
p M D D p M D
∧ <
because, in Bostrom’s view,
'
D
can
not be relevant to
M
unless information about one subverse
x
can be probabilistically
relevant to another subverse
y
. But this is clearly wrong. Even if the events of a
hypothesis are independent, compound information about the events can be very relevant.
It is hard to think of a case where this would not hold. It seems that while White’s view is
not committed to paradoxes, Bostrom’s criticism is, for, were Bostrom’s criticism of TU
true, we would all be forced to use “some” in almost every inference, and as a result
would be committing the gambler’s fallacy perpetually. It seems that none of the
criticisms of TU carry much weight.
In addition to reasons given by White, other points can be made in support of TU:
• We do seem to live in this universe. Our uncertainty about ourverse concerns the
‘true index’ or ‘true label’ of ourverse, not the ourverse itself.
• The use of TU is needed if one wants to make any inference in a large multiverse.
This is because, in a sufficiently large multiverse, almost everything happens
somewhere. It is curious that multiverse hypotheses have been used rather
exclusively as explanations for fine-tuning. Using SU would lead to the
predictions of almost all theories being very nearly unity for almost all
conceivable data. This would end scientific inference.
• A TU analysis will not change drastically should we discover the true index or
true ‘name’ of ourverse. Yet this information seems to be purely indexical since
for the present purposes we already know that the index is something (call it
t
).
• A TU analysis seems to minimize the expectation value of observers being wrong.
To summarize, there are good arguments for using TU, no good reasons against it, and
good reasons against SU. This seems sufficient to establish the use of TU for cosmic fine-
tuning. Yet, one can further advance the point by considering more carefully the available
information.
12
TU and the converse conditional
Presently we know that we live in this universe but we do not know the true index or the
‘name’ of ourverse. Is this a problem for TU? Note that if we do not know the true index
of the ourverse, it follows that we do not know the true index of anything within it. For
example, for any given dice throw in the ourverse there may be corresponding dice
throws in other subverses. If the correct inference would be to use ‘some event in the
multiverse’ in these cases, no inference could be done. Clearly, probability theory works
mostly fine when we use “this”, so we can expect that our not knowing the true index of
ourverse will not be a problem for TU. Let us now look at how the unknown true index
can result in “this” or “some”, depending on the case at hand.
As before, let
t
D
be the proposition that this universe is finely tuned for life and let also
t
be a variable for the true index of ourverse. Let the number of subverses be
N
. It is
important to note that what is uncertain here is the indexing or naming, not the subverse
itself. That is, the observed fine-tuning will not move to another subverse because of a
change in our knowledge, just the indexing we are using may move. Now, because we do
not know the value of
t
, we will express our ignorance by marginalizing (summing) over
it:
1 1
1 1
1
( ( ) | ) ( ( ) | )
( | ) ( | ) ( | )
1
( | ) ( | ).
N N
t t
i i
N N
t t t
i i
N
t t
i
p t i D H p t i D H
p t i D H p D H p t i D H
p D H p D H
N
= =
= =
=
= ∧ = = ∧
= = ∧ = = ∧
= =
∑ ∑
∑
∪ ∪
(15)
The above result is equivalent to TU. Note that our knowledge about
t
D
does not affect
the probability about the true index of ourverse. For all we know, ourverse’s index could
be any index and hence the principle of indifference suggests a uniform prior for
( | )
t
p t i D H
= ∧
. It follows that because cosmic fine-tuning is a TU case, a multiverse
does not predict fine-tuning any better than a single universe and conversely fine-tuning
does not support a multiverse.
White points out that what makes the difference between “this” and “some” seems to be
what White calls the converse observational selection effect [13]. This refers to whether
any event happening entails our observing it, e.g. whether, using the notation above,
i
D t i
→ =
is true. We will denote this conditional by
' '
c i
S D t i
↔ → =
. Of course, in the
case of cosmic fine-tuning, we will not observe every universe that is fine-tuned, instead
we observe ourverse only. Hence
c
S
does not hold for cosmic fine-tuning. However, in
cases where the converse conditional
c
S
is true and we observe one
t
D
, we will typically
get
13
1
1
1
1
1
( | )
( | )
( | ) ( | ) ( | )
( | ) ( | ) ( | ),
( | ) ( | )
N
t j c
i j i
N
i j c
i
j i
N
i i c j
i
j i
N
i j
i
j i
N
t
p t i D D H S
p t i D D H S
p D H p t i D H S p D H
p D H p t i t i H p D H
N p D H p D H
α
= ≠
=
≠
=
≠
=
≠
−
= ∧ ∧ ¬ ∧
= = ∧ ∧ ¬ ∧
= = ∧ ∧ ¬
= = = ∧ ¬
= ¬
∑
∑
∏
∑
∏
∪ ∩
∩
(16)
which is a “some”-type result. It can also be shown that in the more general case where
we observe
m
events and given that we would observe all events for which
D
α
is true
(converse condition), the likelihood will be of the form
!
( | ) ( | )
( )! !
m N m
N
p D H p D H
N m m
α α
−
¬
−
, (17)
which clearly is a “some”-type result. Hence, the above clarifies why the converse
conditional leads to “some” and why, when the converse conditional does not hold, as is
the case with cosmic fine-tuning, the correct approach is “this” analysis. As an example,
a fisherman fishing with a fishnet and capable of seeing all the fish couth should use a
“some”-type analysis like the one in eq. (17). Yet a fish, if it cannot see other fish, should
use a “this” analysis like the one in eq. (15). The thing which makes the difference is
information the observer has, not the identity.
Conclusions
The four most viable approaches for inference in a possible multiverse and in the
presence of an observer selection effect were reviewed.
Concerning the ‘assume the observation’ (AO) approach advocated by Sober, Ikeda, and
Jefferys, it was shown that this kind of an observer selection effect is justified if and only
if the observation is conditionally independent of the hypothesis. In the case of cosmic
fine-tuning the observation would be a child of the hypothesis and the two are not
independent. It follows that one should use the observation as data and not as a
background condition. Hence, the AO approach for cosmic fine-tuning is incorrect.
The self-sampling assumption approach by Bostrom was shown to be inconsistent with
probability theory.
Several reasons were then given for favoring the ‘this universe’ (TU) approach and main
criticisms against TU were answered. A formal argument for TU was given based on our
present knowledge. The main result is that even under a multiverse we should use the
proposition “this universe is fine-tuned” as data, even if we do not know the ‘true index’
14
of our universe. It follows that because multiverse hypotheses do not predict fine-tuning
for this particular universe any better than a single universe hypothesis, multiverse
hypotheses are not adequate explanations for fine-tuning. Conversely, our data on cosmic
fine-tuning does not lend support to the multiverse hypotheses. For physics in general,
irrespective of whether there really is a multiverse or not, the common-sense result of the
above discussion is that we should prefer those theories which best predict (for this or
any universe) the phenomena we observe in our universe.
Bibliography
[1]
J. Barrow and F. Tipler, The Anthropic Cosmological Principle, Oxford
University Press, 1988.
[2]
J. Leslie, Universes, Routledge, 1990.
[3]
M. Denton, Nature’s Destiny: How the Laws of Biology Reveal Purpose in the
Universe, Free Press, 2002.
[4]
R. Collins, "Evidence for fine-tuning", in God and Design, Routledge, 2003, pp.
178-199.
[5]
G. Gonzalez and J. Richards, The Privileged Planet, Regnery Publishing, 2004.
[6]
M. Rees, Just Six Numbers, Basic Books, 2001.
[7]
R. Swinburne, The Existence of God, 2nd ed., Oxford University Press, 2004.
[8]
R. Collins, "The Teleological Argument", in The Blackwell Compation to
Natural Theology, 2009.
[9]
S. Weinberg, "Living in the Multiverse", in Universe or Multiverse?, Cambridge
University Press, 2007, arXiv:hep-th/0511037.
[10]
E. Sober, "The Design Argument", in The Blackwell Guide to the Philosophy of
Religion, Blackwell Publishing, 2004.
[11]
N. Bostrom, Observation Selections Effects and Probability, Doctoral
dissertation, London School of Economics. Available at anthropic-
principle.com, 2000.
[12]
N. A. Manson and M. J. Thrush, "Fine-tuning, Multiple Universes, an
d the “This
Universe” Objection", Pacific Philosophical Quarterly 84, pp. 67-83, 2003.
[13]
R. White, "Fine-Tuning and Multiple Universes", Nous 34, p. 260–76, 2000.
[14]
P. Dowe, "Response to Holder: Multiple Universe Explanations are not
Explanations”, Science and Christian Belief 11, pp. 67-68, 1999.
[15]
J. Koperski, "Should We Care about Fine-Tuning?”, British Journal for the
Philosophy of Science 56(2), pp. 303-319, 2005.
[16]
E. T. Jaynes, Probability Theory: The Logic of Science, Cambridge University
Press, 2003.
[17]
D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms,
Cambridge University Press, available at
http://www.inference.phy.cam.ac.uk/mackay/itila/book.html, 2002.
15
[18]
M. Ikeda and W. H. Jefferys, "The Anthropic Principle Does Not Support
Supernaturalism", in The Improbability of God, Prometheus Press, 2006, pp.
150-166.
[19]
J. Weisberg, "Firing Squads and Fine Tuning: Sober on the Design Argument",
British Journal for the Philosophy of Science 56(4), 2005.
[20]
J. Pearl, "Bayesian Networks", UCLA Cognitive Systems Laboratory, Technical
Report (R-246), http://ftp.cs.ucla.edu/pub/stat_ser/R246.pdf.
[21]
A. P. Dawid, "Influence Diagrams for Causal Modelling and Inference", Intern.
Statist. Rev. 70, pp. 161–189,
http://www.ucl.ac.uk/Stats/research/Resrprts/abs01.html#221, 2002.
[22]
J. Whittaker, Graphical Models in Applied Multivariate Statistics, John Wiley &
Sons, 1990.
[23]
R. E. Neapolitan, Probabilistic Reasoning in Expert Systems, John Wiley &
Sons, 1990.
[24]
K. D. Olum, "Conflict Between Anthropic Reasoning and Observation",
arXiv:gr-qc/0303070v2, 2003.
[25]
J. Leslie, "Observer-relative Chances and the Doomsday Argument", Inquiry 40,
pp. 427-436, 1997.
[26]
N. Bostrom, "Observer-relative Chances in Anthropic Reasoning?”, Erkenntnis
52, pp. 93-108, 2000.
