ISSN (Print) 0473-453X

Discussion Paper No. 1217

ISSN (Online) 2435-0982

THE ROLE OF DATA AND PRIORS IN
ESTIMATING CLIMATE SENSITIVITY

Masako Ikefuji
Jan R. Magnus
Andrey L. Vasnev

November 2023

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

The role of data and priors inestimating climate sensitivity
Masako Ikefuji
OSIPP, Osaka University, JapanJan R. Magnus
Department of Econometrics and Data Science,
Vrije Universiteit Amsterdam and Tinbergen Institute, The Netherlands

Andrey L. Vasnev
University of Sydney, New South Wales, Australia

October 20, 2023

1

Abstract: In Bayesian theory, the data together with the prior produce a
posterior. We show that it is also possible to follow the opposite route, thatis, to use data and posterior information (both of which are observable) toreveal the prior (which is not observable). We then apply the theory toequilibrium climate sensitivity as reported by the Intergovernmental Panelon Climate Change in an attempt to get some insight into the prior beliefs of
the IPCC scientists. It appears that the data contain much less information
than one might think, due to the presence of correlation. We conclude thatthe prior in the ﬁfth IPCC report was too low, and in the sixth report toohigh.
JEL Classiﬁcation: C11, C52, C83, Q54.
Keywords: Revealed prior; climate sensitivity; data uncertainty; combining
information; correlation; IPCC.
Corresponding author: Andrey L. Vasnev, University of Sydney Business
School, Abercrombie Building (H70), Sydney, NSW 2006, Australia
Email addresses: ikefuji@osipp.osaka-u.ac.jp (Ikefuji),
jan@janmagnus.nl (Magnus),
andrey.vasnev@sydney.edu.au (Vasnev).

2

1

Introduction

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 this parameter by β.
In estimating β we rely exclusively on the Intergovernmental Panel on
Climate Change (IPCC) reports. So far, six so-called Assessment Reports
have appeared: the ﬁrst in 1990, the sixth in 2021. In these reports we ﬁnd
estimates and precisions of studies on the ECS and IPCC’s own estimatesand precisions. Our interest is not so much in these estimates themselves
but rather in the process that leads from the underlying studies to IPCC’s
estimates. Adopting a Bayesian framework, we shall think of the underlying
studies as our data and of the published conclusions as our posterior. The
question then arises whether we can reveal the IPCC’s priors (which we do
not observe) from these data and the posterior (which we do observe). This
is indeed possible, and the purpose of this paper is to explain how this canbe achieved, which challenges we encounter on the way, and what we learn
from such an exercise.
The idea of reversing Bayesian thought and — rather than obtain a posterior from data and prior — recover the prior from data and posterior, wasrecently proposed by Ikefuji et al. (2023) and applied to inﬂation forecastingin the UK. Their application was carefully chosen to satisfy two requirements.
First, that inﬂation forecasts can be credibly modeled using a symmetric distribution, such as normality. The assumption of symmetry and normality is,however, only plausible in a limited number of cases. We shall show that ourtheory goes through in the case of monotonic transformations of β, for which
we shall take the log-transformation, which is not symmetric and particularly
appropriate for our application.
Second, the data in Ikefuji et al. (2023) consisted of essentially one pieceof information, i.e. the Phillips curve. But suppose we have two or more
pieces of information as our data, which will typically be the case. How to3then combine these pieces of information into one data distribution? If we
combine too naively, then the resulting precision may be misleadingly high,leading to the theoretical impossibility that the posterior variance is larger(rather than smaller) than the data variance. To resolve this issue we use therecently developed theory in Magnus and Vasnev (2023), where the criticalrole of correlation was discussed and applied to Bank of Japan and European
Central Bank forecasts.
The remainder of this paper is organized as follows. In Section 2 we analyze how to recover the prior from the data and the posterior in the casewhen there is only one parameter of interest, ﬁrst within the framework of
the normal distribution, then extending this framework to the lognormal
distribution. In Section 3 we consider the conclusions of the ﬁfth and sixth
IPCC reports (our posteriors) concerning the ECS, and reformulate these
conclusions using the lognormal distribution. In Section 4 we review thedata sources that underly the IPCC conclusions, and ﬁnd the appropriate
lognormal distributions for each of the data sources. Next, we need to combine these data sources into one data source for each of the two reports. Thisis achieved using Magnus and Vasnev’s (2023) approach through correlations,and we discuss and explain the procedure in Section 5. Without correlation
the data variance is much too small, leading to data that are misleadingly
precise. This, in fact, is one of the ﬁndings of the current paper: when data
sources are (highly) correlated, as is often the case, they contain much lessinformation than casual inspection might indicate. Now that we have thedata and the posterior, we obtain and discuss the revealed prior in Section 6.
The prior turns out to be important and the direction of the prior changes
between the ﬁfth and the sixth report. In Section 7 we study the relationship between the posterior in one period and the prior in the next period(dynamic consistency). Section 8 concludes.

2

From posterior to priorSuppose we have data y1 , . . . , yn , possibly correlated and heteroskedastic,
generated from a normal distribution with common mean β, our parameterof interest. In other words, we havey|β ∼ N(βı, Ω),(1)where y = (y1 , . . . , yn )′ , ı is an n × 1 vector of ones, and Ω is a positivedeﬁnite n × n matrix. A frequentist would estimate the scalar parameter β

4using the generalized least-squares estimator b0 with variance σ02 :b0 =ı′ Ω−1 y,ı′ Ω−1 ı

1σ02 = ′ −1 .
ıΩ ı

(2)

A Bayesian, on the other hand, would wish to take prior knowledge about
β into account. If this prior information is given by β ∼ N(b1 , σ12 ), then theposterior distribution of β is β|y ∼ N(b2 , σ22 ) with posterior moments
b2 = wb1 + (1 − w)b0 =
and

(
σ22 =

1
1
+ 2
2
σ0 σ1

σ02 b1 + σ12 b0
σ02 + σ12(3)

σ02 σ12,
σ02 + σ12

(4)

)−1
=

where w = σ02 /(σ02 + σ12 ) = σ22 /σ12 . The expression (4) for the posterior variance implies the restrictions σ22 < σ02 and σ22 < σ12 , and this makes sense:
when two pieces of information (prior and data) are combined then the precision increases. These inequalities, in turn, imply that 0 < w < 1. Whenthe prior becomes uninformative, that is, when σ12 → ∞, then b2 → b0 andσ22 → σ02 .
Now consider the opposite situation where the data and the posterior areavailable but not the prior. Can we reveal the prior from the data and theposterior? Indeed we can, and in the special case of normality we obtain theprior moments as
σ02 b2 − σ22 b0
σ02 − σ22

(5)

σ02 σ22,σ02 − σ22(6)b1 = w−1 b2 + (1 − w−1 )b0 =
and

(
σ12 =

1
1
− 2
2
σ2 σ0

)−1
=

where w−1 = σ12 /σ22 = σ02 /(σ02 − σ22 ). The prior mean is thus a ‘weighted
average’ of b0 and b2 , but w−1 does not lie between zero and one; in fact,
w−1 > 1 and 1 − w−1 < 0. Also, since variances are nonnegative, the expression for σ12 implies an upper bound to the posterior variance, namely σ22 < σ02 .
This restriction does not play a role in the usual Bayesian framework where
we go from data plus prior to posterior, because the underlying variances σ02
and σ12 are unrestricted (apart from being positive) and σ22 will automaticallysatisfy 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. In summary, we canindeed reveal the prior given information on the data and the posterior.
5This simple analysis can be generalized in various directions. First, we
may have not one but several parameters of interest in which case our starting point is y|β ∼ N(Xβ, Ω) for some n × k matrix X; see Ikefuji et al.
(2023). Second, the assumption of normality (implying symmetry) may bereasonable for some applications — as it was in Ikefuji et al. (2023) — but
it is not in our’s. So, a generalization of the framework is required. Recall
the fundamental formula for completing the square,(b0 − β)2 (β − b1 )2(β − b2 )2 (b0 − b1 )2
+
=
+ 2
,
σ02
σ12
σ22
σ0 + σ12(7)where b2 and σ22 are given in (3) and (4). Multiplying both sides by −1/2
and taking exponentials givesf (b0 ; β, σ02 ) f (β; b1 , σ12 ) = f (β; b2 , σ22 ) f (b0 ; b1 , σ02 + σ12 ),
where

{
(
)2 }
x
−
µ
1
1
f (x; µ, σ 2 ) = √
exp −
2
σ
σ 2π

(8)

(9)

denotes the normal density. This, of course, is just Bayes’ formulaf (b0 |β) f (β) = f (β|b0 ) f (b0 ),(10)

and it shows that a normal likelihood plus a normal prior results in a normal
posterior.
Now let h(x) be a monotonic transformation of x, such as h(x) = log x,
and deﬁne the density
{
(
)2 }
′
h
(x)
1
h(x)
−
µ
exp −
.
(11)
g(x; µ, σ 2 ) = √2σσ 2πThen it follows from (7) thatf (b0 ; h(β), σ02 ) g(β; b1 , σ12 ) = g(β; b2 , σ22 ) f (b0 ; b1 , σ02 + σ12 ).

(12)In particular, letting h(x) = log x with h′ (x) = 1/x leads to the lognormal density g(x) for x > 0. Hence, a normal likelihood plus a lognormal
prior results in a lognormal posterior, so that prior and posterior remain
conjugate distributions. Another way of arriving at this result is to realizethat the lognormal distribution is not really a new distribution but rather areparametrization. The parameter of interest remains β, but the analysis isperformed on log β (more generally on h(β)).
6In summary, if we have a likelihood b0 |β ∼ N(log β, σ02 ) and a prior log β ∼N(b1 , σ12 ), then we obtain a posterior log β|b0 ∼ N(b2 , σ22 ). Reversing theprocess, if we have a likelihood b0 |β ∼ N(log β, σ02 ) and a posterior log β|b0 ∼N(b2 , σ22 ), then we obtain a prior log β ∼ N(b1 , σ12 ), where b1 and σ12 are givenby (5) and (6), respectively.
At this point we have two objects that can be regarded as ‘data’: theobservations y and the estimate b0 . The observations y are what one would
usually call data, but in our process the IPCC scientists consider b0 as theirdata, add a prior b1 , and arrive at a posterior b2 . To avoid confusion andfollowing Ikefuji et al. (2023), we shall not refer to b0 as the data but as the
input. In the absence of a prior, the IPCC scientists accept the input as their
only tool: output = input, that is, b2 = b0 . But if the scientists’ prior plays
a role (as of course it does), then output ̸= input and the diﬀerence is theprior, which is the object of our study.

3

The posterior distribution

We shall apply the theory of Section 2 to the estimation of equilibrium climate
sensitivity, hereafter β, as presented by the IPCC in several reports. Theirconclusions are our posterior, which we discuss in the current section. These
conclusions are based on many scientiﬁc publications which serve as our
inputs, and we shall discuss these in the next section. Below we review the
IPCC conclusions and the posterior distributions derived from them, ﬁrst forthe ﬁfth then for the sixth IPCC report.

3.1

The ﬁfth IPCC report (posterior)

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 of agreement onvalues across assessed lines of evidence and studies’ (IPCC5, p. 16, footnote).
But later in the same report they do provide a conﬁdence region (as opposed
to a point estimate), 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 an interpretation of terms like ‘extremely unlikely’
and ‘medium conﬁdence’ (IPCC5, p. 36), which diﬀers slightly from the interpretation in the previous Assessment Report (IPCC, 2007, p. 22) by explicitly
7

taking into account the degree of ‘agreement’ in the team about the evidence
provided by each study.1 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 IPCC5 report. There is no unique lognormal distribution that ﬁts our inputs, but b2 = 1.07 and σ2 = 0.53 seemsa reasonable approximation and is also in line with Hwang et al. (2013,
Figure 4) where b2 = 1.071 and σ2 = 0.527.
The selected posterior distribution, plotted in Figure 1 (bold line, in red),satisﬁesPr(1.5 < β < 4.5) = 0.69 (about 67%),
Pr(0 < β < 1.0) = 0.02 (less than 5%), andPr(β > 6.0) = 0.09 (less than 10%),hence close to the IPCC5 conclusions. In addition, the interquartile range isPr(2.04 < β < 4.17) = 50% and there is a 1% probability of β > 10.0. The
skewness of the distribution is well illustrated by the fact that the mode and2
median of β are quite diﬀerent: the mode is eb2 −σ2 = 2.20, while the median
is eb2 = 2.92.

3.2

The sixth IPCC report (posterior)

In the sixth report, hereafter IPCC6, the authors write:1

In fact, the interpretation of these terms is far from easy. Kause et al. (2022) studied how IPCC experts from diﬀerent disciplines interpret the recommended uncertainty
language, and they found that physical science experts were more familiar with the IPCC
instructions than other experts, and followed it more often; that experts’ conﬁdence levels
increased more with perceptions of evidence than with agreement; and that experts’ estimated probability intervals for climate variables were wider when likelihood terms were
presented with ‘medium conﬁdence’ rather than with ‘high conﬁdence’ and when seen incontext of IPCC sentences rather than out of context.

8

Probability density

Black line: inputs
Red line: posterior

ECS
Figure 1: Posterior distribution (in red) of ECS in the ﬁfth report
and the ﬁfteen studies, lognormal distributions‘. . . Based on multiple lines of evidence, the very likely range ofequilibrium climate sensitivity is between 2 (high conﬁdence)and 5 (medium conﬁdence). The (. . . ) best estimate is 3
with a likely range of 2.5 to 4 (high conﬁdence).’(IPCC6, A.4.4, p. 11)











Later (p. 926) they add that it is ‘virtually certain that ECS is larger than
1.5 .’The ‘multiple lines of evidence’ are: the understanding of climate processes, the instrumental record, paleoclimates, and model-based emergent
constraints (IPCC6, p. 11, footnote). The deﬁnition of ‘likely’, ‘very likely’,and ‘virtually certain’ is the same as in the ﬁfth report (IPCC6, p. 4, footnote 4), and hence we obtain the following posterior probabilities:Pr(2.0 < ECS < 5.0) = 0.90,
Pr(2.5 < ECS < 4.0) = 0.67, andPr(ECS > 1.5) = 0.99,



where we interpret the ‘best estimate’ of 3.0 as the median of the ECS.
The posterior in the sixth report diﬀers markedly from the posterior in theﬁfth report, not so much in the mean (which increases a little) but rather in9the variance (which is much lower). The reduction in uncertainty is caused,
not by a single breakthrough or discovery, but by combining evidence frommany diﬀerent sources and a better understanding of their strengths andweaknesses (IPCC6, p. 1024, FAQ 7.3; Sherwood et al., 2020).
Assuming again that the ECS β follows a lognormal distribution, so thatlog β ∼ N(b2 , σ22 ), we seek combinations (b2 , σ2 ) such that the posterior probabilities closely match the above probabilities. If median(β) = 3.0 thenb2 = log 3 = 1.099 andPr(2.0 < β < 5.0) = 0.90 =⇒ σ2 = 0.27,
Pr(2.5 < β < 4.0) = 0.67 =⇒ σ2 = 0.24, andPr(β > 1.5) = 0.99 =⇒ σ2 = 0.30.

Posterior(a) Process understanding(b) Instrumental record(c) Paleoclimates(d) Emergent constraints

Probability density

(d)
(b)

(a)
(c)

ECS
Figure 2: Posterior distribution (in red) of ECS in the sixth report
and the four inputs, lognormal distributions
Thus, a reasonable approximation is provided by b2 = 1.15 and σ2 = 0.27.
The selected posterior distribution, plotted in Figure 2 (bold line, in red),satisﬁesPr(2.0 < β < 5.0) = 0.91 (about 90%),
Pr(2.5 < β < 4.0) = 0.62 (about 67%), andPr(β > 1.5) = 0.997 (about 99%),10and we ﬁnd median(β) = 3.16 and mode(β) = 2.94, hence close to the IPCC6
conclusions. In addition, the interquartile range is Pr(2.63 < β < 3.79) =50%, much tighter than in the IPCC5 report, and there is a 1% probabilityof β > 5.9.

4The distribution of the inputs

In addition to the posterior we need information on the inputs. These inputs
are not ‘raw’ data but outcomes of scientiﬁc publications, each with its own
merits and viewpoints. In the ﬁfth report we have ﬁfteen inputs, one of which
we shall consider an outlier and disregard; in the sixth report we have four.
We discuss these inputs and the resulting distributions below.

4.1

The ﬁfth IPCC report (inputs)

Our inputs in the IPCC5 report consist initially of m = 15 studies. Fourteenof these studies are contained in Figure 10.20 (p. 925) and Box 12.2 (p. 1110)
of the report, and one study (Huber et al., 2011) is included in Figure 2 ofKnutti et al. (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(b0i , σ0i) in the ith study, we have)(log ui − b0i
log li − b0i< zi <= pi ,zi ∼ N(0, 1),
(13)
Pr
σ0i
σ0i
and henceb0i = log(li ui )1/2 ,σ0i =log(ui /li )1/2,
qi

Φ(qi ) =pi + 1,2(14)

where Φ denotes the c.d.f. of the standard-normal distribution. In this way
we end up with ﬁfteen inputs b0i with associated standard deviations σ0i ; see
Table 1.
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
period (studies 1, 3, 5–9, 12, 13), some only from the palaeoclimatic period
11Table 1: Inputs and lognormal approximations, ﬁfth report
Study
pi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

Lindzen and Choi, 2011
95%
Schmittner et al., 201190%
Aldrin et al., 201290%
Hargreaves et al., 2012
90%
Lewis, 2013
90%
Bender et al., 201095%
Otto et al., 201390%
Schwartz, 201290%
Lin et al., 201090%
Libardoni and Forest, 2011 90%
Köhler et al., 201090%
Olson et al., 201295%
Huber et al. 2011
67%
Holden et al., 2010
90%
Palaeosens, 201295%

Boundsliui

Lognormal approx.
b0i
σ0i

0.5
1.4
1.2
1.0
2.0
1.7
0.9
1.2
2.8
1.2
1.4
1.8
2.9
2.0
1.1

−0.30
0.68
0.72
0.72
0.99
0.97
0.75
0.89
1.17
0.93
0.99
1.09
1.23
1.15
1.02

1.1
2.8
3.5
4.2
3.6
4.1
5.0
4.9
3.7
5.3
5.2
4.9
4.0
5.0
7.0

0.20
0.21
0.33
0.44
0.18
0.22
0.52
0.43
0.08
0.45
0.40
0.26
0.17
0.28
0.47

(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 should be considered an outlier. The IPCC raises doubts
about the reliability of this study (IPCC5, pp. 923–924), but it has notremoved the study from their report. Not only is the revealed value of b0i
much lower than in the other studies, but the eﬀect is much strengthened by
the fact that the reported precision is high.
To gain further insight, Figure 1 presents the ﬁfteen lognormal curves
corresponding to the ﬁfteen studies. From the ﬁgure several things become
clear. First, that the ﬁrst study (with the highest peak) is an outlier, andthat we should therefore disregard this study from our analysis, thus ending
up with m = 14 studies. Second, that what the inputs tell us is ambiguous
and uncertain. The modes range from 0.71 (1.62 if we exclude the ﬁrst study)
to 3.32, and the medians from 0.74 (1.97 if we exclude the ﬁrst study) to 3.42.

12

4.2

The sixth IPCC report (inputs)

In earlier IPCC reports, the assessment of ECS relied on either CO 2 -doubling
experiments using global atmospheric models coupled with mixed-layer ocean
models or on standardized CO2 -quadrupling experiments using fully coupled
ocean-atmosphere models or Earth system models. In the sixth report, theassessment of ECS is based on multiple lines of evidence, with Earth system models representing only one of several sources of information (IPCC6,
p. 993). These ‘multiple lines of evidence’ are divided in four classes: process understanding, the instrumental record, paleoclimates, and emergent
constraints.
Each class is analyzed separately and is based on a number of underlying
studies (IPCC6, Section 7.5, pp. 992–1011). Our input can thus be viewed
as a two-step procedure: ﬁrst from underlying studies to a distribution of
β in each of the four classes, then from these four class distributions tothe ﬁnal input distribution. To estimate the input distribution in two steps
would, in principle, be possible, but it would require a two-level aggregation
theory where at the moment a one-level aggregation theory is already a major
challenge as we shall see in Section 5.
Table 2: Probabilities for the four inputs, sixth report

median0.67

Probability
0.90
0.95

(a)

Process
understanding

3.4

2.5–5.1

2.1–7.7(b)

Instrumental
record

2.5–3.5

> 2.2

> 1.8

> 1.6

(c)

Paleoclimates

3.3–3.4

< 4.5

> 1.5

< 8.0

(d)

Emergent
constraints

2.4–3.3

1.5–5.0

Hence, we shall take the m = 4 intermediate conclusions of the IPCC as
our inputs. These conclusions are presented in Table 2, which summarizes
Table 7.13 in IPCC6. The table contains the median and probabilities of 67%(likely), 90% (very likely), and 95% (extremely likely) attached to various

13ranges in each class.2 In contrast to the IPCC5 report, there seems to bebroad agreement across multiple lines of evidence.
Table 3: Lognormal approximations based on the four inputs, sixth report

σ0i

mode

Quantiles
5% 50% 95%

1.22 0.36

2.98

1.87

3.39

6.12

(b) Instrumental
record

1.03

0.39

2.41

1.47

2.80

5.32

(c)

1.20 0.61

2.29

1.22

3.32

9.06

1.05

2.51

1.58

2.86

5.17

b0i
(a)

Process
understanding

Paleoclimates(d) Emergent
constraints

0.36

2
If we assume, as in Section 4.1, that log βi ∼ N(b0i , σ0i) for the ith input,
then we can approximate the values of b0i and σ0i from the given probabilistic
ranges in Table 2. These approximations are presented in Table 3 together
with the implied mode, and 5%, 50% (the median), and 95% quantiles. The
four implied distributions will serve as our inputs.

Table 4: Probabilities of the four inputs based on thelognormal approximations, sixth report

(a)
(b)
(c)
(d)

Target = 0.67

Target = 0.90

Target = 0.95

Pr(2.5 < β < 5.1) = 0.67
Pr(β > 2.2) = 0.73
Pr(β < 4.5) = 0.69

Pr(2.1 < β < 7.7) = 0.90
Pr(β > 1.8) = 0.87
Pr(β > 1.5) = 0.90
Pr(1.5 < β < 5.0) = 0.90

Pr(β > 1.6) = 0.92
Pr(β < 8.0) = 0.93

The precision of our approximations is veriﬁed in Table 4, where we calculate the implied probabilities for the ranges presented in Table 2. The
2The IPCC writes about the ‘best’ or the ‘central’ estimate, and they also occasionallymention the median but never the mode of a distribution. For this reason we interpretthe ‘best’ estimate as the median rather than as the mode, but some ambiguity remains.

14

probabilities in Table 4 should be close to those in Table 2, and they are.
Figure 2 presents the four lognormal curves corresponding to the four lines ofevidence. The distributions are closer to each other and less ambiguous thanthe distributions in Figure 1. The modes range from 2.29 to 2.98 (rather
than from 1.62 to 3.32 in IPCC5) and the medians range from 2.80 to 3.39
(rather than from 1.97 to 3.42 in IPCC5).

5Combining the inputs

Combining information from various sources into a single piece of information is a nontrivial exercise. One needs to know the nature of the sources,
how relevant and reliable they are, and if and to what extent they are correlated. We shall concentrate on the eﬀect of correlation, ﬁrst in general, thenspeciﬁcally for our application.
In the general case, we consider a sequence of correlated random variables
x = (x1 , x2 , . . . , xm )′ with common mean µ, so that E(x) = µ ı and var(x) =σ 2 V , where ı denotes the vector of ones and V is positive deﬁnite. Without
loss of generality we normalize V by imposing the restriction tr(V ) = m.
The generalized least-squares estimators of µ and σ 2 areµ̂ =ı′ V −1 x,
ı′ V −1 ıσ̂ 2 =(ı′ V −1 ı)(x′ V −1 x) − (ı′ V −1 x)2,m ı′ V −1 ı

(15)while τ 2 = var(µ̂) is estimated by(ı′ V −1 ı)(x′ V −1 x) − (ı′ V −1 x)2σ̂ 2.
τ̂ 2 = ′ −1 =ıV ım (ı′ V −1 ı)2(16)

Important is the diﬀerence between σ 2 and τ 2 . The parameter σ 2 is thevariance of the process, while τ 2 measures the precision of the estimator µ̂.
As the number of observations increases we are able to estimate µ more andmore precisely (as measured by τ 2 ), but the variance of the process (measuredby σ 2 ) is not aﬀected by the number of observations. The behavior of thetwo estimators σ̂ 2 and τ̂ 2 as a function of V is also very diﬀerent. We areinterested in σ̂ 2 , the estimator of tr(var(x))/m, the average variance of thexi .
When the xi are uncorrelated then σ̂ 2 is bounded and ﬁnite, but whenthe xi are correlated then this is no longer the case. To see why, let us write
V −1 = |V |−1 CV , where CV denotes the cofactor matrix of V . In the expression for µ̂ the determinant |V | cancels as it appears in both the numerator
and the denominator, but in the expression for σ̂ 2 it does not cancel and15

we are left with one factor |V | in the denominator. If |V | goes to zero thenthe ratio goes to ∞. This heuristic argument was recently made precise in
Magnus and Vasnev (2023) and applied to forecast combinations of GDP and
CPI by the Bank of Japan and the European Central Bank. They conclude
that ignoring possible correlation in the observations can lead to estimatesof σ 2 which are (much) too small.
We shall adopt the ideas in Magnus and Vasnev (2023) to aggregate them = 14 inputs in IPCC5 and the m = 4 inputs in IPCC6. Our task is to ﬁnd2
an estimator b0 ∼ N(log β, σ02 ) based on the fourteen inputs b0i ∼ N(log β, σ0i)in Section 4.1 and, similarly, an estimator b0 based on the four inputs b0i in
Section 4.2. Thus we need to ﬁnd two lognormal distributions which canbe thought of as reasonable representations of the two sets of inputs. These
distributions must have a larger variance than the red curves in Figures 1
and 2, because theory prescribes that the posterior variance is smaller than
the input variance (and also than the prior variance).
Let V0 denote the diagonal matrix containing the diagonal elements of
−1/2
−1/2
is the correlation matrix associated with V . In
V V0
V . Then, P = V0
the absence of correlation we have P = Im and we obtain b0 = 1.07 (ﬁfthreport), b0 = 1.11 (sixth report), and
σ0 = 0.27 < 0.53 = σ2 (ﬁfth report)
σ0 = 0.10 < 0.27 = σ2 (sixth report).
These input variances violate the restriction σ0 > σ2 and are therefore too
small. But the assumption of zero correlation is unrealistic anyway. We
expect the inputs to be correlated in our case, probably highly correlated. 3
To take possible correlation into account we shall assume equicorrelation, sothat the correlation between each pair of inputs is the same, say ρ. The
correlation matrix P then takes the form

1 ρ ρ ... ρρ 1 ρ . . . ρ (0 ≤ ρ < 1).
(17)
P =  .. .. ..
.. 
. . .
.ρ ρ ρ ... 1
Since P depends on only one parameter, we can now investigate the eﬀect ofcorrelation on our input moments.
3

IPCC6 (p. 1006) is also concerned with the question on how to combine the inputs,
but with a view to narrow the range of ECS values. See Annan and Hargreaves (2006),
Stevens et al. (2016), and Sherwood et al. (2020) for a Bayesian perspective.

16

Table 5: Combined inputs, ﬁfth and sixth reports

ρ
0.00
0.50
0.80
0.90
0.95
0.99
1.00

Fifth report
b0
σ0
τ
1.07
1.20
1.22
1.22
1.23
1.23
1.23

0.27
0.32
0.49
0.68
0.96
1.98
∞

0.04
0.06
0.06
0.06
0.06
0.06
0.06

Sixth report
b0
σ0
τ
1.11
1.10
1.07
1.05
1.03
1.01
1.00

0.10
0.13
0.21
0.30
0.41
0.89
∞

0.04
0.09
0.14
0.17
0.19
0.21
0.22

The input moments are presented in Table 5, where we let the correlation
ρ increase from 0 to 1.4 The restriction σ0 > σ2 is only satisﬁed when ρ > 0.83
(ﬁfth report) and ρ > 0.88 (sixth report), conﬁrming that the estimation
results in the underlying inputs are highly correlated. When there is nocorrelation, then b0 increases from 1.07 to 1.11 between the ﬁfth and sixth
report. But when we take correlation into account, then b0 decreases from
1.23 to 1.03 (at ρ = 0.95). Since the posterior b2 increases from 1.07 to 1.15
it will need a strong prior to achieve this change of direction.
While b0 is quite stable for ρ ≥ 0.9, this is not the case for σ0 . The
precision in the input distribution depends heavily on the value of ρ, thatis, on the degree in which the inputs are correlated with each other. In thelimit, when ρ → 1 and σ0 → ∞, the inputs provide no additional informationand hence the posterior is equal to the prior. We conclude that the inputs
provide less information (σ0 is larger) in the presence of correlation, and given
our assumptions we can quantify how much less. We do not know what thevalue of ρ is and hence, when we reveal the priors, we shall work with three
scenarios: ρ = 0.90, 0.95, and 0.99.

6

The revealed prior

Given the moments (b2 , σ2 ) from the posterior and (b0 , σ0 ) from the input, we
can now reveal the implied prior moments. Recall from Section 2 that, given
a posterior log β|b0 ∼ N(b2 , σ22 ) and an input b0 |β ∼ N(log β, σ02 ), the prior
4

The results are obtained from (15) and (16) for ρ < 1. For ρ = 1 the matrix P issingular, but we can calculate the limiting values from the theory developed in Magnus
and Vasnev (2023).

17

distribution is log β ∼ N(b1 , σ12 ), where b1 and σ12 are given by (5) and (6),respectively:σ02 b2 − σ22 b0
b2 − (1 − w)b0
b1 =
(18)
=
σ02 − σ22
w
and
σ12 =

σ02 σ22σ22,=σ02 − σ22w

(19)

where w = 1−σ22 /σ02 , as deﬁned in Section 2. The prior mean b1 thus depends
only on the posterior mean b2 (which we set at 1.07 and 1.15 respectively),
the input mean b0 (which is quite stable at approximately 1.23 and 1.03
respectively), and on the weight w (which is not stable and depends heavily
on the assumed correlation). In contrast, the prior standard deviation σ1
depends only on the posterior standard deviation σ2 (which we set at 0.53
and 0.27 respectively) and on the weight w.
Table 6: Input, prior, and posterior

Input

Prior

PosteriorρFifth report

Sixth report

0.90
0.95
0.99
1.00

b0
σ0
w
1.22 0.68 0.39
1.23 0.96 0.69
1.23 1.98 0.93
1.23
∞ 1.00b0
σ0w
1.05 0.30 0.16
1.03 0.41 0.58
1.01 0.89 0.91
1.00
∞ 1.00

0.90
0.95
0.99
1.00

b1
σ1
0.83 0.85
1.00 0.64
1.06 0.55
1.07 0.53

b1
1.66
1.24
1.16
1.15

b2
1.07

b2
σ2
1.15 0.27

σ2
0.53

σ1
0.67
0.36
0.28
0.27

Table 6 shows the revealed prior means and standard deviations for bothreports in the range 0.9 ≤ ρ ≤ 1.0, together with the corresponding moments
for the input and the posterior. We observe that there is essentially nodiﬀerence in the prior at ρ = 0.99 and ρ = 1.00. In the latter case, the input
variance is ∞ and hence prior = posterior and the inputs (the underlying
academic studies) play no role at all. Although there are situations where the18

posterior is completely determined by the prior (and hence the data, howeverconvincing, are disregarded), we do not believe this is a credible descriptionof the process by which the IPCC scientists reach their conclusions. On theother hand, for ρ < 0.83 (ﬁfth report) and ρ < 0.88 (sixth report) the weightw becomes negative and the theoretical constraint σ2 < σ0 is violated. Let
us therefore concentrate on the two cases ρ = 0.90 and ρ = 0.95.

Probability density

Probability density

Input
Prior
Input
Posterior

Posterior

Prior

ECS

ECS

Figure 3: Input, prior, and posterior for ρ = 0.90,
ﬁfth report (left) and sixth report (right)

Posterior

Prior

Probability density

Probability density

Input

Posterior

Input

ECS

Prior

ECS

Figure 4: Input, prior, and posterior for ρ = 0.95,
ﬁfth report (left) and sixth report (right)
The table conﬁrms what we already know from the theory, namely thatwhen ρ increases then w and σ0 also increase, while σ1 decreases. This
reﬂects the fact that more correlation in the inputs reduces the precision of
the combined input and makes the prior more important.
The prior mean b1 is much larger in the sixth than in the ﬁfth report andthe prior standard deviation σ1 is much smaller. Speciﬁcally, b1 increased
from 0.83 to 1.66 (ρ = 0.90) and from 1.00 to 1.24 (ρ = 0.95), implying an
increase in the prior median of β from 2.30 to 5.27 (ρ = 0.90) and from 2.72
to 3.45 (ρ = 0.95). At the same time, the standard deviation σ1 decreased
from 0.85 to 0.67 (ρ = 0.90) and from 0.64 to 0.36 (ρ = 0.95). Hence, the19

IPCC scientists changed their prior views between the ﬁfth and the sixth
report, and they held these views with more conviction.
It is remarkable that b1 < b2 in the ﬁfth report, while b1 > b2 in thesixth report. This means that in the ﬁfth report the IPCC scientists usedtheir prior views to tune the input (= data) information down, arriving at
median(β) = 2.92 in the posterior while median(β) = 2.72 (at ρ = 0.95)
in the prior. In contrast, in the sixth report the prior views are tuned up
arriving at median(β) = 3.16 in the posterior while median(β) = 3.45 (atρ = 0.95) in the prior.
In Figures 3 and 4 we draw the lognormal distributions of β at ρ = 0.90
and ρ = 0.95, respectively. The ﬁgures illustrate that as ρ increases the priorbecomes more important, and that the prior distribution shifts to the rightand becomes more concentrated (lower variance) when we compare the ﬁfth
and sixth reports.

7

Dynamic consistency

One would expect that the posterior in one period serves as the prior in thenext period, at least approximately. When this happens, the decision makeris rational and the process is said to be ‘dynamically consistent.’ In our case,dynamic consistency requires that the prior in the sixth report is equal tothe posterior in the ﬁfth report. But is it?
The posterior in the ﬁfth report is b2 = 1.07 (σ2 = 0.53), while theprior in the sixth report is b1 = 1.24 (σ1 = 0.36) at ρ = 0.95. Hence, themedian of the posterior in the ﬁfth report is 2.92 (the mode is 2.20) and the
median of the prior in the sixth report is 3.45 (the mode is 3.04). These arelarge diﬀerences, both in terms of location (median, mode) and in terms ofprecision.
One possible explanation of this large diﬀerence is that there is indeed
a lack of dynamic consistency. One can easily imagine a situation where
the decision maker remains too loyal to their original prior, which one may
call ‘prior stubbornness’ (bunching). This stubbornness may be politicallymotivated and it could continue until some bound has been reached (a tipping
point) after which the prior is adjusted and moves to a new level.
But there is also another possible explanation. There is a gap of eight
years between the IPCC5 and IPCC6 reports. During this period the IPCC
experts were exposed to climate news (ﬂoods, ﬁres, rising temperatures),they attended conferences, they read newspapers, etc. Their beliefs were
therefore inﬂuenced and updated before they started working on IPCC6. Inother words, an adjustment to their priors occurred.
20

To analyze this possibility we perform a thought experiment. In therelationship between input, prior, and posterior we have so far encounteredinput+prior = posterior (the standard Bayesian method) and also posterior−
input = prior (our perspective to reveal the prior). We now rewrite ourequations based on posterior − prior = input, assuming that the decision
maker behaves dynamically consistent. That is, we create a hypothetical
input by considering the IPCC5 posterior as our prior and the IPCC6 prioras our posterior. To move from this new prior to the new posterior we needa new input, which is constructed fromb0 =σ12 b2 − σ22 b1,
σ12 − σ22σ02 =σ12 σ22,σ12 − σ22

(20)

using (3) and (4) or, alternatively, (5) and (6). The revealed hypothetical
input thus represents the input that would be required to bridge the gap
between the posterior from IPCC5 and the prior from IPCC6.
Table 7: Consistency gap and hypothetical input at ρ = 0.95
Sourcebσmedian

mode

IPCC5
Input
Prior
Posterior Prior

1.23
1.00
1.07

0.96
0.64
0.53

3.41
2.72
2.92

1.36
1.81
2.20

IPCC6
Input
Prior
Posterior

Posterior

1.03 0.41
1.24 0.36
1.15 0.27

2.80
3.45
3.16

2.36
3.04
2.94

Input

1.37

3.95

3.14

Gap

Experiment

0.48

In Table 7 we copy the posterior in the ﬁfth report and the prior in thesixth report (at ρ = 0.95) from Table 6, and then use (20) to calculate theinput that would be required to bridge the gap and bring about dynamic
consistency. In this experiment the IPCC5 posterior thus plays the role ofprior and the IPCC6 prior plays the role of posterior.
When we compare the hypothetical input with the actual input in thesixth report (also displayed in Table 7), we see that the hypothetical input
distribution is located to the right of the actual input distribution. In fact,21

both the median and the mode of the hypothetical input are much larger
than of the actual input: for the median we have 3.95 ≫ 2.80 and for themode 3.14 ≫ 2.36.

Probability density

Input (IPCC6)

Posterior
Prior
Input (Gap)

ECS
Figure 5: The consistency gap at ρ = 0.95
The gap is illustrated in Figure 5 where we display the prior and the posterior of our experiment, and the implied input which we wish to comparewith the IPCC6 input (also displayed). The ﬁgure conﬁrms what we foundin the table, namely that the gap is too large to explain the dynamic inconsistency. One can imagine that IPCC6 scientists felt the pressure causedby the increase of climate exposure in the media. There has been politicalpressure and an abundance of qualitative information — information thatwas not reﬂected in the underlying scientiﬁc reports underlying IPCC6. But
the consistency gap is too large too be credibly explained by political andmedia pressure alone.
If the gap experiment does not explain the dynamic inconsistency, thenwhat does? Perhaps the IPCC6 scientists were not fully conﬁdent with theconclusions from IPCC5, because they now had access to multiple lines of evidence which were not available in IPCC5. They would then not necessarily
accept the IPCC5 posterior as their prior. Another possibility is prior stubbornness, as mentioned earlier. Interestingly, it seems that the opposite has
occurred here. We know from the previous section that the prior in IPCC522

is smaller than the posterior,
prior < posterior < input(IPCC5),that is, scientiﬁc evidence (the input) indicated a higher value of the ECS
than the IPCC scientists’ prior views.5 But in IPCC6 the situation is reversed:
input < posterior < prior(IPCC6).
The inconsistency is apparently caused, at least in part, by overcompensation. The prior in IPCC5 was too low, so the IPCC scientists adjusted theirprior upward in IPCC6, perhaps too much.

8

Concluding remarks

Let us translate our ﬁndings back to the language of the IPCC. As discussed
in Section 3.1, IPCC5 concludes that ECS is likely in the range 1.5 to
4.5 , that it is positive, extremely unlikely less than 1.0 , and very unlikely
greater than 6.0 . This reﬂects the posterior distribution as presented by
the IPCC itself. In Section 5 we deduce the combined input and approximate
its distribution, from which we ﬁnd (at ρ = 0.95) that













ECS is likely in the range 1.4 to 8.6 , is positive, extremely
unlikely less than 0.7 , and very unlikely greater than 11.6 .

Then, in Section 6, we present the prior which tells us that

ECS is likely in the range 1.5 to 5.0 , is positive, extremely
unlikely less than 1.0 , and very unlikely greater than 6.2 .

In the same way, IPCC6 concludes (our posterior, Section 3.2) that the very
likely range of ECS is between 2.0 and 5.0 , that the best estimate is 3.0
with a likely range of 2.5 to 4.0 , and that it is virtually certain that ECS
is larger than 1.5 . From the distribution corresponding to the combined
input we ﬁnd that



















the very likely range of ECS is between 1.4 and 5.5 , the best
estimate is 2.8 with a likely range of 1.9 to 4.2 , and it isvirtually certain that ECS is larger than 1.1 .

Then, combining posterior and input, the induced prior tells us that
5

The inequality holds for b and the median, but not for the mode, where the inequality
is reversed.

23








the very likely range of ECS is between 1.9 and 6.2 , the best
estimate is 3.4 with a likely range of 2.4 to 4.9 , and it isvirtually certain that ECS is larger than 1.5 .



These verbal summaries correspond exactly to the estimates obtained in theprevious sections, and they conﬁrm that the prior plays an important rolein the IPCC decision process when drawing conclusions from the data (ourinputs). In the ﬁfth report the prior is more optimistic than the input with aprior median of 2.7 versus an input median of 3.4, while in the sixth report theprior is more pessimistic with a prior median of 3.5 versus an input medianof 2.8. The ECS is a critical parameter in the climate debate but it remains
diﬃcult to estimate, and it seems that the scientiﬁc evidence (the inputs)
and the prior views held by the IPCC scientists are not well synchronized.
In addition to an in-depth analysis of the estimation process of the ECS
in the ﬁfth and sixth reports of the IPCC, the current paper makes twotheoretical contributions. First, we show how the prior can be revealed fromthe input and the posterior, also when the underlying distributions are skewed
in which case the normal distribution is unsuitable. Second, we show how
inputs can be combined, and the critical role of correlation in this aggregation
process. The underlying studies (the inputs) are naturally correlated and
if we ignore the correlation we overestimate the precision of our combined
estimator: under correlation the data contain much less information thana naive investigator might assume. These two theoretical contributions will
also be relevant in other applications where priors can be revealed.
Priors as discussed in the current paper are the human ﬁlter between
data and conclusions. Such a ﬁlter is present in any scientiﬁc project. Whenthe data and the conclusions are publicly available, which is often the case,then our methodology makes it possible to infer the underlying prior, whichshould make scientiﬁc and policy reports more transparent.

Acknowledgements
We are grateful to Rick van der Ploeg and Takashi Yamagata for helpful
discussions. This work was ﬁnancially supported by the JSPS KAKENHI
Grant Number JP21H00700 and the program of the Joint Usage/ResearchCenter for Behavioral Economics at ISER, Osaka University.

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27