ISSN (Print) 0473-453X

Discussion Paper No. 1111

ISSN (Online) 2435-0982

THE PERCEPTION
OF CLIMATE SENSITIVITY:REVEALING PRIORS FROM POSTERIORS

Masako Ikefuji
Jan R. Magnus

December 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The perception of climate sensitivity:
Revealing priors from posteriors
Masako Ikefuji
Faculty of Humanities and Social Sciences, University of Tsukuba
and ISER, Osaka University

Jan R. Magnus
Department of Econometrics and Data Science,
Vrije Universiteit Amsterdam and Tinbergen InstituteDecember 4, 2020
Abstract: A Bayesian typically uses data and a prior to produce a posterior.
In practice, the data and the posterior are often observed but not the prior.
We shall follow the opposite route, using data and the posterior information
to reveal the prior. We then apply this theory to (equilibrium) climate sensitivity as reported by the Intergovernmental Panel on Climate Change in an
attempt to get some insight into their prior beliefs. It appears that the IPCC
scientists have agreed a priori on a value for the climate equilibrium between
3 and 4 degrees Celsius, while judging the occurrence of a real disaster much
more likely than the previous report predicts.
JEL Classification: C11, Q54.
Keywords: Revealed prior; climate sensitivity; IPCC.
Corresponding author: Masako Ikefuji, Faculty of Humanities and Social Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577,
Japan.
Email addresses:
ikefuji.masako.gn@u.tsukuba.ac.jp (Ikefuji),
jan@janmagnus.nl (Magnus).

1

1

Introduction

Everybody has priors, Bayesians and non-Bayesians alike. The priors may
be vague and difficult to make explicit, but they are there and they may be
important. The purpose of this paper is to show that we can make priors
explicit from our knowledge of the data and the posterior, and to apply thistheory to the perception of climate sensitivity.
Imagine a group of people (the ‘committee’) with a collective prior, perhaps based on knowledge and experience, perhaps on political beliefs, perhaps on short-term profit. The committee meets privately and we have no
information about their discussions. But we do have scientific data (official‘objective’ statistics and scientific results) and we do have access to their
published predictions or policy recommendations, which they present to the
public. In other words, we have the data and the posterior, but not the priorwhich the committee does not reveal and possibly may not even be able toformulate or quantify. Can we recover the prior from the data and the posterior? Yes, this is indeed possible and we shall study the properties of therecovered prior in some detail.
We apply this theory to the estimation of the equilibrium climate sensitivity (ECS), which is an important diagnostic in climate modeling. The data
come from various studies as listed and reported by the Intergovernmental
Panel on Climate Change (IPCC). The IPCC then formulates its own conclusions (the posterior). What we are interested in is to recover the priorbeliefs of the IPCC.
The idea of reversing Bayesian thought and — rather than obtain a posterior from data and prior — recover the prior from data and posterior, does
not seem to have received much attention. The current paper attempts tofill this gap. Of course, the list of possible applications is endless. A politicalparty uses scientific data and publishes reports. From these two sources we
can recover their priors. Do these conform to the party program? Scientists
use data and write papers. The results in these papers may well be influenced
by prior beliefs or non-scientific reasons. Can this influence be quantified?
Such questions can, in principle, be studied by the theory developed in this2

paper.
In Section 2 we introduce the ECS and summarize the findings of the
IPCC: the posterior. In Section 3 we analyze how to recover the prior fromthe data and the posterior within the framework of the normal distribution.
In Section 4 we consider the case when there is only one parameter of interest, and extend the framework to monotonic reparametrizations of thenormal distribution, like the lognormal distribution. Next, in Section 5, we
discuss the data which result from fifteen studies used by the IPCC. Given
the posterior and the data, we then recover and discuss the prior mean andstandard deviation (Section 6). Section 7 concludes. An Appendix discusses
the role of monotonic reparametrizations in general, and the lognormal distribution in particular.

2

Climate sensitivity, the reported posterior

When the radiation balance of the Earth is perturbed, the temperature willchange. By how much is measured by the equilibrium climate sensitivity(ECS): the long-term temperature rise that is expected to result from adoubling of the atmospheric CO2 concentration, usually relative to the preindustrial level (around 1750). It is a prediction of the new global mean nearsurface air temperature once the CO2 concentration has stopped increasing
and most of the feedbacks have had time to have their full effect. The ECS
is an important diagnostic in climate modelling, but it cannot be measured
directly and forms a large source of uncertainty. CO2 levels rose from 280
parts per million (ppm) in the eighteenth century (IPCC, 2013, p. 100) toabout 416 ppm by 2020, an increase of almost 50%. In the same period, theEarth’s temperature rose by a little over one degree Celsius. The ECS will
be our parameter of interest, and we shall denote it by β.
In estimating β we rely exclusively on the Intergovernmental Panel on
Climate Change (IPCC) reports. So far, five so-called Assessment Reports
have appeared, the first in 1990, the fifth in 2013. In these reports we findestimates (and precisions) of studies on the ECS (our data) and the IPCC’s
own estimates (our posterior). From this information we will attempt to3

recover the IPCC’s priors.
In the fifth report, more precisely the Working Group I contribution(IPCC, 2013), hereafter IPCC5, the authors state that ‘no best estimate
for equilibrium climate sensitivity can now be given because of a lack ofagreement on values across assessed lines of evidence and studies’ (IPCC5,
p. 16, footnote). But later in the same report they do provide estimates, as
follows:‘. . . ECS is likely in the range 1.5℃ to 4.5℃ with high confidence. ECS is positive, extremely unlikely less than 1℃ (highconfidence), and very unlikely greater than 6℃ (medium confidence)’.

(IPCC5, pp. 83–84)

The IPCC also provides a precise interpretation of terms like ‘extremely
unlikely’ and ‘medium confidence’ (IPCC5, p. 36), which differs slightly fromthe interpretation in the previous Assessment Report (IPCC, 2007, p. 22) byexplicitly taking into account the degree of ‘agreement’ in the team about theevidence provided by each study. Given this interpretation, IPCC5 concludes
thatPr(1.5 < ECS < 4.5) = 0.67,
Pr(ECS < 1.0) < 0.05, andPr(ECS > 6.0) < 0.10.
In addition (pp. 75 and 817), they summarize information of experiments by
the Coupled Model Intercomparison Project Phase 5 (CMIP5) who reporta range 2.1–4.7 for the ECS, without however stating the likelihood of thisrange.
Assuming the ECS β to be lognormally distributed, so that log β ∼N(b2 , σ22 ), we seek combinations (b2 , σ2 ) such that the posterior probabilities
closely match the probabilities in the IPCC report. This leads to Table 1,

where we have calculated the required possibilities for various credible valuesof b2 and σ2 . There is no unique lognormal distribution that fits our data,
but b2 = 1.07 and σ2 = 0.53 (in bold) seems a reasonable approximation and4

Table 1: Posteriors on climate sensitivity, Fifth Assessment Report
Lognormal approx.
b2
σ2

Probabilities for ECS (%)
1.5–4.5 < 1.0 > 6.0 2.1–4.7

0.95
1.00
1.05
1.06
1.07
1.08
1.09
1.10
1.15

66.5
68.0
69.2
69.1
68.9
69.5
69.3
69.9
70.1

0.57
0.55
0.53
0.53
0.53
0.52
0.52
0.51
0.49

4.8
3.5
2.4
2.3
2.2
1.9
1.8
1.6
0.9

7.0
7.5
8.1
8.4
8.7
8.6
8.9
8.7
9.5

49.5
52.1
54.6
54.7
54.8
55.8
55.9
56.9
58.9

is also in line with Hwang, Reynès, and Tol (2013, Figure 4) where b2 = 1.071
and σ2 = 0.527.

0.2
0.1
0.0

Probability density

0.3

0.4

Figure 1: Posterior lognormal distribution with b2 = 1.07 and σ2 = 0.53

0

2

4

6

ECS

5

8

10

The selected posterior distribution, plotted in Figure 1, satisfies
Pr(1.5 < β < 4.5) = 68.9% (about 67%),
Pr(0 < β < 1.0) = 2.2% (less than 5%), andPr(β > 6.0) = 8.7% (less than 10%),in accordance with the IPCC report’s conclusions. In addition, the interquartile range is Pr(2.04 < β < 4.17) = 50% and there is a 1% probability ofβ > 10.0. The skewness of the distribution is well illustrated by the fact that2

mode and median of β are quite different: The mode is eb2 −σ2 = 2.20, while
the median is eb2 = 2.92.

3

From posterior to prior under normality

In order to highlight the issue under simple conditions, we shall first assume
normality, which we shall modify later. We consider a parameter vector ofinterest β and suppose that data are generated from a normal distributiony|β ∼ N(Xβ, Ω),(1)where X is a given n × k matrix of rank k and Ω is a positive definite n × nmatrix. A non-Bayesian frequentist would estimate β using the generalized
least-squares (GLS) estimator (which is also the maximum likelihood estimator)
−1 ′ −1b0 = X ′ Ω−1 XXΩ y(2)with varianceΣ0 = X ′ Ω−1 X

−1

.

(3)

A Bayesian, on the other hand, would wish to take prior knowledge about β
into account. Suppose this prior information is given byβ ∼ N(b1 , Σ1 ),6(4)

where Σ1 is positive definite. Then the posterior distribution of β is
β|y ∼ N(b2 , Σ2 ),

(5)

where
−1
Σ2 = Σ−1
1 + Σ0b2 = W b1 + (Ik − W )b0 ,

−1

,

(6)

and W = Σ2 Σ−1
1 is a k × k weight matrix.
Although W is, in general, not symmetric, its eigenvalues are real and
1/2
1/2
lie between zero and one. In fact, letting Z = Σ1 Σ−1with eigenvalues0 Σ1λi (Z) > 0 (i = 1, . . . , k), we see that
−1/2

λi (W ) = λi (Σ1

−1/2

Σ2 Σ1

)=

1
1/2
1/2
λi (Σ1 Σ−1
2 Σ1 )=

1.
1 + λi (Z)(7)

Note that when the prior becomes uninformative, that is when Σ−1→ 0,1
then b2 → b0 and Σ2 → Σ0 .

This is well-established basic Bayesian theory. But now consider the
opposite situation where the data and the posterior are available but not theprior. Can we reveal the prior from the data and the posterior? In general
we can, and in the special case of normality we obtain the prior moments as
b1 = W −1 b2 + (Ik − W −1 )b0 ,−1
Σ1 = Σ−1
2 − Σ0

−1

,

(8)

with
−1
W −1 = Σ1 Σ−1
2 = Σ0 (Σ0 − Σ2 ) ,(9)which assumes implicitly an upper bound to the posterior variance, namely
Σ2 < Σ0 in the usual sense that Σ0 − Σ2 is positive definite. The priormean is thus a ‘weighted average’ of b2 and b0 , but the eigenvalues of W −1
do not lie between zero and one. In fact λi (W −1 ) = 1 + λi (Z) > 1 andλi (Ik − W −1 ) = −λi (Z) < 0 for all i = 1, . . . , k.

The restriction Σ2 < Σ0 does not play a role in the usual Bayesian framework where we go from data plus prior to posterior, because the underlying
variances Σ0 and Σ1 are unrestricted (apart from being positive definite) and7Σ2 will automatically satisfy the restriction. But it does play a role when
we go from data plus posterior to prior, because now the restriction is not
automatically satisfied. This has practical consequences as we shall see later.

4

One parameter of interest under lognormality

In the special but important case where we have only one parameter β ofinterest, we write σ02 , σ12 , and σ22 instead of Σ0 , Σ1 , and Σ2 . From the data(without a prior) we obtain an unbiased estimator of β: b0 ∼ N(β, σ02 ). If we
add a prior β ∼ N(b1 , σ12 ), then we obtain the posterior β ∼ N(b2 , σ22 ), whereb2 =σ02 b1 + σ12 b0,
σ02 + σ12σ22 =σ02 σ12.
σ02 + σ12

(10)

In the reversed case that we are interested in we have an unbiased estimatorb0 ∼ N(β, σ02 ) from the data and the posterior moments of β ∼ N(b2 , σ22 ).
From these two ingredients we obtain the prior as β ∼ N(b1 , σ12 ), whereb1 =σ02 b2 − σ22 b0,
σ02 − σ22σ12 =σ02 σ22,σ02 − σ22

(11)

under the restriction that σ22 < σ02 .
So far we have assumed normality because of its transparency in handling
Bayesian problems. But it happens frequently that the normality assumption
on β is not plausible. Then we need to search for a credible alternative, andthe simplest alternative is normality of a monotonic transformation, say h(β).
Given this transformation the same theory shows that if b0 ∼ N(h(β), σ02 ) isan unbiased estimator of h(β) from the data and h(β) ∼ N(b2 , σ22 ) is theposterior, then the prior can be recovered ash(β) ∼ N(b1 , σ12 ),(12)where b1 and σ12 are given in (11). From the prior distribution of h(β) we

8then obtain the prior distribution of β. The most important example ish(β) = log β in which case the posterior and the prior are both lognormally
distributed; see the Appendix for details. This is the route that we shallfollow.
Defining αm and αv implicitly by
b2 = αm b0 ,σ22 = αv σ02 ,(13)

b1 = κm b0 ,σ12 = κv σ02 ,(14)

we can rewrite (11) as

where
κm =

αm − αv,
1 − αv

κv =αv
1 − αv(15)

measure how far the prior is removed from the data and their effect on theprior mean and variance, respectively. Note that αm is unrestricted but that
αv is restricted by 0 < αv < 1.
The two fractions κm and κv capture the essence of our story. First
consider κv , which relates to the prior variance. What matters here is whether
κv is small (strong prior information) or large (weak prior information). Thisdepends only on αv , not on αm . When αv is close to one, then the variance
σ02 in the data and the variance σ22 in the posterior are approximately equal,
so that the prior has only a small effect. This is represented by a large valueof κv and hence a large value of the prior variance σ12 . The prior is thenuninformative. But when αv is close to zero, then the data variance and theposterior variance are not close at all, and the prior has a big effect. Thisis represented by a small value of κv and hence a small value of the priorvariance σ12 . The prior is then informative.
The situation is quite different with κm . What matters here is not whether
κm is small or large, but rather whether κm is close to one or not. This will
depend on both αm and αv . It is clear that κm = 1 when αm = 1, irrespective

9

of the value of αv . Writing
1 − κm =1 − αm,1 − αv(16)

we see that the deviation of κm from one depends on the deviation of αmfrom one relative to the deviation of αv from one. When αm is close to one
but αv is not, then the mean b0 in the data and the mean b2 in the posterior
are approximately equal, but the variance σ02 in the data and the variance
σ22 in the posterior are not approximately equal. In that case κm ≈ 1 andthe prior mean agrees with the data and the posterior. But when αv is close
to one but αm is not, then the variances σ02 and σ22 are approximately equal,
but the means b0 and b2 are not. In that case κm is large (in absolute value).
Naturally, for people with a very strong prior (σ12 ≈ 0) we have αv ≈ 0, andhence κm ≈ αm and b1 ≈ b2 .

5

The underlying data: Estimating b0 and σ0

In addition to the posterior we need information on the data. Our data consist of n = 15 studies mentioned in the IPCC5 report. Fourteen of these arecontained in Figure 10.20 (p. 925) and Box 12.2 (p. 1110) of the report, andone study (Huber et al., 2011) is included in Figure 2 of Knutti, Rugenstein,and Hegerl (2017) and referred to in various places of IPCC5.
The ith study produces a range (li , ui ) (lower and upper bound) with anassociated probability pi (typically 90% or 95%) for the ECS β, our parameterof interest. Given the range (li , ui ) and the associated probability Pr(li <βi < ui ) = pi we can identify the parameters of the associated lognormal
2
distributions, as follows. Since log βi ∼ N(yi , σ0i) in the ith study, we havePr

log ui − yilog li − yi
< zi <
σ0i
σ0i



= pi ,zi ∼ N(0, 1),(17)

pi + 1,2(18)

and henceyi = log(li ui )1/2 ,σ0i =log(ui /li )1/2,
qi
10Φ(qi ) =where Φ denotes the c.d.f. of the standard-normal distribution. In this way
we end up with fifteen observations yi with associated standard deviations
σ0i ; see Table 2. Our task is to find an estimator b0 ∼ N(log β, σ02 ) based on2
the fifteen data yi ∼ N(log β, σ0i). This task is not trivial as we shall see.
Table 2: Data on climate sensitivity, Fifth Assessment ReportStudypi
1 Lindzen and Choi, 2011
2 Schmittner et al., 2011
land and ocean data
only ocean data

95%
67%
90%
67%
90%
only land data
67%
90%
3 Aldrin et al., 201290%
4 Hargreaves et al., 201290%
5 Lewis, 2013
90%
incl. uncertainties
90%
6 Bender et al., 201095%
7 Otto et al., 2013 (2000–2009) 90%
Otto et al., 2013 (1970–2009) 90%
8 Schwartz, 201290%
9 Lin et al., 201090%
10 Libardoni and Forest, 201190%
11 Köhler et al., 201090%
12 Olson et al., 201295%
13 Huber et al. 201167%
14 Holden et al., 2010
67%
90%
15 Palaeosens, 2012
67%
Palaeosens, 201295%

Boundsliui

Lognormal approx.
yi
σ0i

0.5
1.7
1.4
1.5
1.3
2.8
2.2
1.2
1.0
2.0
1.0
1.7
1.2
0.9
1.2
2.8
1.2
1.4
1.8
2.9
2.6
2.0
2.2
1.1

−0.30

0.20

0.68

0.21

0.72
0.72
0.99

0.33
0.44
0.18

0.97

0.22

0.75
0.89
1.17
0.93
0.99
1.09
1.23

0.52
0.43
0.08
0.45
0.40
0.26
0.17

1.15

0.28

1.02

0.47

1.1
2.6
2.8
2.5
2.7
4.1
4.6
3.5
4.2
3.6
3.0
4.1
3.9
5.0
4.9
3.7
5.3
5.2
4.9
4.0
4.4
5.0
4.8
7.0

The IPCC5 report distinguishes between an instrumental period anda palaeoclimatic period. The instrumental period is the short period in
the Earth’s long history where direct instrumental records on climate areavailable, while the palaeoclimatic period is the long period preceding suchrecords. Of our fifteen studies some only use data from the instrumental
11period (studies 1, 3, 5–9, 12, 13), some only from the palaeoclimatic period(2, 4, 11, 14, 15), and some combine different lines of evidence (3, 10, 12).
Studies 3 and 12 appear in both the groups instrumental and combination.
This highlights the first of several problems: comparability.
The second problem is that the first study (Lindzen and Choi, 2011) has
a big impact and is in essence an outlier. The IPCC raises doubts about thereliability of this study (IPCC5, pp. 923–924), but it has not removed thestudy from their report. Not only is the revealed value of yi much lower thanin the other studies, but the effect is much strengthened by the fact that thereported precision is high.
Third, we should take into account that the studies are correlated witheach other. They all estimate the same parameter, probably using similar
highly correlated data sets.

2.0
1.5
1.0
0.0

0.5

Probability density

2.5

Figure 2: The fifteen studies, lognormal distributions

0

2

4

6

8

10

ECS

To gain further insight, let us draw the fifteen lognormal curves; see
Figure 2. From the figure several things become clear. First, that the firststudy (with the highest peak) is an outlier. And second, almost beyond
belief, that what the data tell us is ambiguous in the extreme: The modes
range from 0.71 (1.89 if we exclude the first study) to 3.31, and the medians12from 0.74 (1.98 if we exclude the first study) to 3.41. To make any sense of
these data requires a strong prior.
Not only do we have to find an estimator b0 but also a standard deviation
σ0 . One of the reasons why this is difficult is that σ0 should always be smaller
than each of the individual standard deviations σ0i (more information leads
to higher precision), but larger than the posterior standard deviation σ2 .
These two facts are not in agreement with each other, since σ0 must be
smaller than 0.08 (the smallest of the underlying standard deviations σ0i in
Table 2), while at the same time it must larger than σ2 , which we have set
at 0.53; see Table 1.
In fact, this is an intriguing problem: we combine information and wethus expect the standard deviation to decrease, but in fact it doesn’t. To
illustrate, suppose we have one observation x1 = 50 on an unknown parameter θ with standard deviation 4. We use this as our guideline until a newobservation appears: x2 = 25 with standard deviation 3. A naive approach
would be to conclude that the updated estimate of θ is a weighted average ofx1 and x2 , in this case 34 with standard deviation 2.4. This would conform
statistical theory, at least in the absence of correlation between x1 and x2 .
But it would not convince, because there appears to be a conflict between thetwo estimates. The naive approach works if we draw from the same population, but in our case each study has used different (though correlated) data,
different methods, a different time period, and so on. Hence, there is not
only intra-study noise (captured by the reported standard deviations), butalso inter -study noise, which is not captured and not reported. Each study
behaves like the frog in the well-known Japanese proverb ‘a frog on the bottom of a well,’ which relates the story of a frog living happily at the bottomof a dark well, full of sweet water and buzzing insects, until a turtle tells him
about something outside, called the ocean. The problem is closely related tothe ‘prior-likelihood conflict’ in the Bayesian literature and to the ideas of‘scientific ambiguity’ (Miller, Dietz, and Heal, 2013; Brock and Xepapadeas,2019), ‘negative learning’ (Oppenheimer, O’Neill, and Webster, 2008), andthe combination of disparate lines of evidence (Cooke and Wielicki, 2018).
To understand the problem better let us realize that each of the fifteen
13estimates comes from a different population, so the reported means and standard deviations are conditional on a ‘state’ Si . In fact, therefore, the studies
report the conditional moments E(yi |Si ) and var(yi |Si ) instead of the unconditional moments
E yi = E E(yi |Si )var(yi ) = E var(yi |Si ) + var(E(yi |Si )).

(19)

To work with E(yi |Si ) and var(yi |Si ) instead of E E(yi |Si ) and E var(yi |Si )may be reasonable, but to ignore the variation in the conditional meanvar(E(yi |Si )) is not reasonable. This explains, at least in part, why thereported variance seriously underestimates the true variance.
Since we have no knowledge of the states Si , we take a different routebased on credible scenarios. Since theory prescribes that σ0 > 0.53, weconsider three values of σ0 : 0.6, 0.7, and 0.8. In addition, since ȳ = 0.87
if we include the first study and ȳ = 0.95 if we exclude it, we consider four
values of b0 : 0.8, 0.9, 1.0, and 1.1.
Table 3: Estimates of b0 and σ0 from the data

b0

σ0

mode

quantiles
5% 50%
95%

0.8

0.6
0.7
0.8

1.55
1.36
1.17

0.83
0.70
0.60

2.23
2.23
2.23

5.97
7.04
8.30

0.9

0.6
0.7
0.8

1.72
1.51
1.30

0.92
0.78
0.66

2.46
2.46
2.46

6.60
7.78
9.17

1.0

0.6
0.7
0.8

1.90
1.67
1.43

1.01
0.86
0.73

2.72
2.72
2.72

7.29
8.60
10.13

1.1

0.6
0.7
0.8

2.10
1.84
1.58

1.12
0.95
0.81

3.00
3.00
3.00

8.06
9.50
11.20

Thus we shall consider twelve scenarios, where each scenario obeys thetheoretical restrictions and the combination of scenarios covers what we be14

lieve is a credible range of the data parameters. Table 3 presents the modeand three quantiles (5%, 50%, and 95%) for the implied distributions. We
see that the median ranges from 2.23 to 3.00, which is a little narrower thanin the fourteen studies where the median ranges from 1.98 to 3.41 (if we
exclude the first study where the median is 0.74). The scenarios also allow
for a wide range of (right) tail behavior. The value β ∗ of β which defines
the right tail (that is, where 5% of the distribution lies to the right of β ∗ )ranges from 5.97 to 11.20. Given the data (particularly, ȳ = 0.95) and therestriction σ0 > σ2 = 0.53, our preferred estimate is (b0 , σ0 ) = (1.0, 0.7) (inbold), but we shall consider the whole spectrum of twelve scenarios when we
derive and analyze the priors in the next section.

6

Analysis of the prior moments

Given the moments (b0 , σ0 ) from the data and (b2 , σ2 ) from the posterior,
we can now discuss the revealed prior moments. The prior mean is givenby (11),b2 − αv b0,
αv = σ22 /σ02 ,
(20)
b1 =
1 − αvand thus depends only on the posterior mean b2 (which we set at 1.07), thedata mean b0 (which ranges from 0.8 to 1.1), and on the ratio of σ2 = 0.53
and σ0 (which ranges from 0.6 to 0.8).
In contrast, the prior standard deviation, also given in (11), is
σ2
σ0 σ2
=√
,
σ1 = p 22
1 − αv
σ0 − σ2(21)

which, given σ2 = 0.53, depends only on σ0 or alternatively on αv .
Table 4 shows the revealed prior means and standards deviations foreach of the twelve scenarios (four values of b0 , three values of σ0 ). Given(b2 , σ2 ) = (1.07, 0.53) and the twelve selected pairs (b0 , σ0 ), we present theinduced values of κm , κv , b1 , and σ1 .
Since b2 must lie in-between b0 and b1 , it follows that b1 > 1.07 if b0 < 1.07
and b1 < 1.07 if b0 > 1.07. The lower is b0 , the higher must be b1 . If b0 = 0.8
15

Table 4: Estimates of prior moments b1 and σ1 for given
posterior moments (b2 , σ2 ) = (1.07, 0.53)
data
b0
σ0

κv

prior
b1
σ1

mode

quantiles
5% 50%
95%

κm

0.8

0.6
0.7
0.8

2.54
1.79
1.60

3.55
1.34
0.78

2.03
1.43
1.28

1.13
0.81
0.71

2.12
2.17
2.18

1.18
1.10
1.12

7.61
4.19
3.60

48.84
15.91
11.53

0.9

0.6
0.7
0.8

1.86
1.44
1.34

3.55
1.34
0.78

1.67
1.30
1.20

1.13
0.81
0.71

1.48
1.90
2.02

0.83
0.96
1.04

5.33
3.66
3.33

34.24
13.91
10.66

1.0

0.6
0.7
0.8

1.32 3.55
1.16 1.34
1.12 0.78

1.32
1.16
1.12

1.13
0.81
0.71

1.04
1.66
1.87

0.58
0.84
0.96

3.74
3.20
3.08

24.01
12.16
9.86

1.1

0.6
0.7
0.8

0.88
0.94
0.95

0.96
1.03
1.05

1.13
0.81
0.71

0.73
1.45
1.73

0.41
0.74
0.89

2.62
2.80
2.85

16.83
10.64
9.12

3.55
1.34
0.78

then we need a high prior mean, leading to a prior median of 3.6℃ or more.
If the standard deviation in the data is relatively large (σ0 = 0.8), then thestandard deviation in the prior will be relatively small. But even then theright tail is substantial with a 5% probability that the ECS will exceed 10℃
or worse. In our bolded scenario there is a 5% probability that the ECS will
exceed 12℃.
As discussed in Section 4, the story can be summarized in terms on the
key parameters κm and κv . What matters regarding κm is whether or not it isclose to one. When the mean b0 in the data and the mean b2 in the posterior
are approximately equal, but the variance σ02 in the data and the variance
σ22 in the posterior are not approximately equal, then κm ≈ 1 and the priormean agrees with the data and the posterior. But when the variances σ02
and σ22 are approximately equal, but the means b0 and b2 are not, then κm islarge (in absolute value). This is well illustrated in Table 4 where κm = 1.12
is close to one when b0 = 1.0 is close to b2 = 1.07, while σ0 = 0.8 is not closeto σ2 = 0.53. At the other extreme, κm = 2.54 is not close to one, because

16b0 = 0.8 is not close to b2 = 1.07, while σ0 = 0.6 is close to σ2 = 0.53.
The prior standard deviation satisfies σ1 > σ2 = 0.53 (as it must), but σ1
can be larger or smaller that σ0 . In our set up, σ1 can only take three values:
0.71 (small), 0.81 (medium), or 1.13 (large), depending on whether σ0 equals
0.8 (large), 0.7 (medium), or 0.6 (small). Regarding κv what matters iswhether it is small (κv = 0.78, strong prior information) or large (κv = 3.55,
weak prior information). When the standard deviations σ0 in the data andσ2 in the posterior are approximately equal, then we see from Table 4 thatthe prior has only a small effect. This is represented by a large value of κv(3.55) and hence a large value of the prior variance σ1 (1.13). The prior isthen uninformative. But when the standard deviation in the data and theposterior are not close, then the prior has a big effect. This is represented
by a small value of κv (0.78) and hence a small value of the prior standard
deviation σ1 (0.71). The prior is then informative.
So, what do these results tell us about the unknown prior of the IPCC?
And in particular, how strong is this prior in view of the fact that a strong
prior seems to be required to make any sense of the data.
Let us compare our prior with the conclusions (the posterior) of the previous IPCC report (IPCC, 2007). One would expect that the posterior of theprevious study serves as the prior in the next study, at least approximately.
The fourth report concludes about the ECS:
‘It is likely to be in the range 2.0℃ to 4.5℃ with a best estimate of about 3.0℃, and is very unlikely to be less than 1.5℃.
Values substantially higher than 4.5℃ cannot be excluded, but
agreement of models with observations is not as good for thosevalues.’(IPCC, 2007, p. 12)
If we fit a lognormal distribution with mean 1.1 and standard deviation 0.4,
then these statements translate to a median of 3.0 andPr(2.0 < ECS < 4.5) = 0.69 > 0.67 (likely),
Pr(ECS < 1.5) = 0.04 < 0.10 (very unlikely), andPr(ECS > 4.5) = 0.16,
17which seems about right.
The posterior mean 1.10 from the previous report is a little lower than theprior mean in the current report (say, 1.16), but it is comparable. The sameis true for the implied median, which is 3.0 in the previous report, and now alittle higher, say 3.2. The main difference is in the standard deviation, which
is 0.4 in the posterior distribution of the previous report and about 0.7 in theprior distribution of the current report. It appears that the IPCC scientists
have agreed a priori on a value for the ECS between 3℃ and 4.0℃, while
judging the occurrence of a real disaster much more likely than the posterior
in the previous report predicts. For example, while Pr(ECS > 6.0) equals
4.2% for the posterior in the previous report and 8.7% for the posterior in thecurrent report, the revealed prior gives a probability range between 14.8%(b0 = 1.05, σ0 = 0.71) and 58.3% (b0 = 2.03, σ0 = 1.13). These high numbers
reflect the a priori view of the IPCC scientists; they are not based on new
information becoming available in the current report.

0.4

Figure 3: Distributions of β based on data b0 , prior b1 , posterior b2 ,and previous report b∗2

b2

0.2

b0

0.1

b1

0.0

Probability density

0.3

b*2

0

2

4

6

8

10

ECS

In Figure 3 we plot the four representative lognormal distributions of βfor the data (b0 = 1.00, σ0 = 0.70), the prior (b1 = 1.16, σ1 = 0.81), the18

posterior (b2 = 1.07, σ2 = 0.53), and the posterior from previous report(b∗2 = 1.10, σ2∗ = 0.40). These four graphs summarize our story. The smaller
the variance of the lognormal distribution, the higher the top. The posteriorb∗2 from the previous reports has the smallest variance, hence the highest top
and the smallest probability of a really large temperature. There is a large
discrepancy between the posterior b∗2 of the previous report and the prior b1of the current report, showing dynamic inconsistency. The main difference
between the conclusions in the previous and the current report is that uncertainty has increased. Why has uncertainty increased in the presence ofmany new scientific studies, often claiming rather precise predictions? Thiscan only be because the IPCC’s prior judgement has become more cautious
and less optimistic, perhaps rightly so, than the previous posterior justifies.

7

Conclusions and further work

In this paper we have tried to reveal the prior, given information about thedata and the posterior. We do so in the context of the normal distribution and an extension (reparametrization) based on a monotonic transformation of the normal distribution, in our case the lognormal distribution. The(log)normality assumption can be relaxed without changing the underlying
theory, but the expressions would be become less transparent.
We developed the theory and we applied this theory to climate sensitivity, more precisely to the prior views of the IPCC panel towards the valueof this parameter. It appears that the IPCC scientists have agreed a priori
on a value for the ECS between 3℃ and 4.0℃, while judging the occurrenceof a real disaster much more likely than the posterior in the previous report
predicts. Our application assumes that the IPCC behaves in a Bayesian way
and thinks in terms of priors and posteriors. This may not be so. But it
doesn’t matter because we are all Bayesians in the sense that we all havepriors. Scientific Bayesians have both explicit and implicit priors; scientificfrequentists only have implicit priors. But we all have priors and the purposeof this paper has been to make these priors explicit. Of course, the field of
application of the ideas developed in this paper is not confined to climate is19

sues. It can be used more generally to determine people’s biases, for example
politicians or scientists.
There are at least two further closely related questions worth investigating. First, suppose that our observations stretch over several, say two,
periods. In both periods we have data and posteriors, and we can recover
the priors in periods 1 and 2. Dynamic consistency requires that the priorin period 2 is the posterior in period 1. But is it? If it is, then the agent
is consistent or rational in this Bayesian updating scheme. But if it isn’t,
then the agent is not consistent. One can easily imagine a situation where
the agent remains too loyal to their original prior, which one may call ‘priorstubbornness’ or ‘bunching’. This stubbornness may be politically motivatedand is related to the theory of learning. It may continue until some boundhas been reached (a tipping point), after which the prior is adjusted andmoves to a new level. In fact, it should be possible to derive a measure forsuch stubbornness.
Second, we may consider the situation where there is not one but several,
say two, agents over several, say two, periods. The data available to thetwo agents are the same, but their posteriors are not, and hence their priors
are also different. Under what conditions would their priors converge in thissetting?

Appendix: The lognormal distribution
Recall the fundamental formula for ‘completing the square’,(y − β)2 (β − a1 )2(β − a2 )2 (y − a1 )2
+
=
+ 2
,
σ02
σ12
σ22
σ0 + σ12
where
a2 =

σ02 a1 + σ12 y,
σ02 + σ12

σ22 =

(22)

σ02 σ12.
σ02 + σ12Multiplying both sides by −1/2 and taking exponentials givesf (y; β, σ02) f (β; a1 , σ12 ) = f (β; a2 , σ22 ) f (y; a1 , σ02 + σ12 ),
20

(23)

where11
exp −f (x; µ, σ ) = √
2
σ 2π
2



x−µ
σ

2

denotes the normal density. This, of course, is just Bayes’ formulaf (y|β) f (β) = f (β|y) f (y)and shows that a normal likelihood plus a normal prior results in a normal
posterior.
Now let g denote the lognormal density
1h′ (x)exp −g(x; µ, σ ) = √
2
σ 2π
2



h(x) − µσ2(x > 0).

where h(x) = log x and consequently h′ (x) = 1/x, Then it follows immediately from (22) that, for β > 0,f (y; h(β), σ02) g(β; a1, σ12 ) = g(β; a2, σ22 ) f (y; a1, σ02 + σ12 ).

(24)

Hence, a normal likelihood plus a lognormal prior results in a lognormal posterior, so that prior and posterior are conjugate distributions. The function
h can be any monotonic transformation of x, not necessarily log x.
Another way of arriving at this result is to realize that the lognormal
distribution is not really a new distribution but rather a reparametrization.
The parameter of interest remains β, but the analysis is performed on log β(more generally on h(β)). So we have a normal likelihood with mean log β
and a normal prior on log β, resulting in a normal posterior on log β.

Acknowledgements
We are grateful to Peter Boswijk, Siem Jan Koopman, Gideon Magnus, Mark
Steel, and Rick van der Ploeg for helpful discussions. Earlier versions of thispaper were presented at the Vrije Universiteit Amsterdam (April 2020), the
NESG Conference (May 2020), and at ISER, Osaka University (December

21

2020). This work was financially supported by the JSPS KAKENHI Grants
Number JP20H01477 and the program of the Joint Usage/Research Center
for Behavioral Economics at ISER, Osaka University.

References
Aldrin, M., M. Holden, P. Guttorp, R. B. Skeie, G. Myhre, and T. K.
Berntsen (2012). Bayesian estimation of climate sensitivity based ona simple climate model fitted to observations of hemispheric temperatures and global ocean heat content. Environmetrics, 23, 253–271.
Bender, F. A.-M., A. M. L. Ekman, and H. Rodhe (2010). Response to theeruption of Mount Pinatubo in relation to climate sensitivity in the
CMIP3 models. Climate Dynamics, 35, 875–886.
Brock, W., and A. Xepapadeas (2019). Regional climate policy under deep
uncertainty: Robust control, hot spots and learning. DEOS Working
Papers #1903, Athens University of Economics and Business.
Cooke, R. M. and B. Wielicki (2018). Probabilistic reasoning about measurements of equilibrium climate sensitivity: Combining disparate lines
of evidence. Climatic Change, 151, 541–554.
Hargreaves, J. C., J. D. Annan, M. Yoshimori, and A. Abe-Ouchi (2012).
Can the Last Glacial Maximum constrain climate sensitivity? Geophysical Research Letters, 39, L24702.
Holden, P. B., N. R. Edwards, K. I. C. Oliver, T. M. Lenton, and R. D.
Wilkinson (2010). A probabilistic calibration of climate sensitivity andterrestrial carbon change in GENIE-1. Climate Dynamics, 35, 785–806.
Huber, M., I. Mahlstein, M. Wild, J. Fasullo, and R. Knutti (2011). Constraints on climate sensitivity from radiation patterns in climate models. Journal of Climate, 24, 1034–1052.

22

Hwang, C., F. Reynès, and R. S. J. Tol (2013). Climate policy under
fat-tailed risk: An application of Dice. Environmenal and Resource
Economics, 56, 415–436.
IPCC (2007). Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of theIntergovernmental Panel on Climate Change. Cambridge University
Press, Cambridge, United Kingdom and New York, NY, USA.
IPCC (2013). Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of theIntergovernmental Panel on Climate Change. Cambridge University
Press, Cambridge, United Kingdom and New York, NY, USA.
Knutti, R., M. A. A. Rugenstein, and G. C. Hegerl (2017). Beyond equilibrium climate sensitivity. Nature Geoscience, 10, 727–736.
Köhler, P., R. Bintanja, H. Fischer, F. Joos, R. Knutti, G. Lohmann, and V.
Masson-Delmotte (2010). What caused Earth’s temperature variations
during the last 800,000 years? Data-based evidence on radiative forcing
and constraints on climate sensitivity. Quaternary Science Reviews, 29,
129–145.
Lewis, N. (2013). An objective Bayesian, improved approach for applying
optimal fingerprint techniques to estimate climate sensitivity. Journalof Climate, 26, 7414–7429. doi:10.1175/JCLID-12-00473.1.
Libardoni, A. G., and C. E. Forest (2011). Sensitivity of distributions ofclimate system properties to the surface temperature dataset. Geophysical Research Letters, 38, L22705. Correction in: Geophysical Research
Letters (2013), 40, 2309–2311. doi:10.1002/grl.50480.
Lin, B., L. Chambers, P. Stackhouse Jr., B. Wielicki, Y. Hu, P. Minnis, N.
Loeb, W. Sun, G. Potter, Q. Min, G. Schuster, and T.-F. Fan (2010).
Estimations of climate sensitivity based on top-of-atmosphere radiation
imbalance. Atmospheric Chemistry and Physics, 10, 1923–1930.
23

Lindzen, R. S., and Y. S. Choi (2011). On the observational determination of climate sensitivity and its implications. Asia-Pacific Journal of
Atmospheric Sciences, 47, 377–390.
Miller, A., S. Dietz, and G. Heal (2013). Scientific ambiguity and climate
policy. Environmental and Resource Economics, 55, 21–46.
Olson, R., R. Sriver, M. Goes, N. M. Urban, H. D. Matthews, M. Haran, and
K. Keller (2012). A climate sensitivity estimate using Bayesian fusionof instrumental observations and an Earth system model. Journal ofGeophysical Research: Atmospheres, 117, D04103.
Oppenheimer, M., B. C. O’Neill, and M. Webster (2008). Negative learning.
Climatic Change, 89, 155–172.
Otto, A., F. E. L. Otto, O. Boucher, J. Church, G. Hegerl, P. M. Forster,N. P. Gillett, J. Gregory, G. C. Johnson, R. Knutti, N. Lewis, U.
Lohmann, J. Marotzke, G. Myhre, D. Shindell, B. Stevens, and M. R.
Allen (2013). Energy budget constraints on climate response. Nature
Geoscience, 6, 415–416.
Palaeosens Project Members (2012). Making sense of palaeoclimate sensitivity. Nature, 491, 683–691.
Schmittner, A., N. M. Urban, J. D. Shakun, N. M. Mahowald, P. U. Clark,
P. J. Bartlein, A. C. Mix, and A. Rosell-Melé (2011). Climate sensitivity estimated from temperature reconstructions of the last glacial
maximum. Science, 334, 1385–1388. Response to comment in: Science(2012), 337, 1294.
Schwartz, S. E. (2012). Determination of Earth’s transient and equilibrium climate sensitivities from observations over the twentieth century: Strong dependence on assumed forcing. Surveys in Geophysics,
33, 745–777.

24