In this paper we introduce a Bayesian framework, which is explicit about prior assumptions, for using model ensembles and observations together to constrain future climate change. The emergent constraint approach has seen broad application in recent years, including studies constraining the equilibrium climate sensitivity (ECS) using the Last Glacial Maximum (LGM) and the mid-Pliocene Warm Period (mPWP). Most of these studies were based on ordinary least squares (OLS) fits between a variable of the climate state, such as tropical temperature, and climate sensitivity. Using our Bayesian method, and considering the LGM and mPWP separately, we obtain values of ECS of 2.7 K (0.6–5.2, 5th–95th percentiles) using the PMIP2, PMIP3, and PMIP4 datasets for the LGM and 2.3 K (0.5–4.4) with the PlioMIP1 and PlioMIP2 datasets for the mPWP. Restricting the ensembles to include only the most recent version of each model, we obtain 2.7 K (0.7–5.2) using the LGM and 2.3 K (0.4–4.5) using the mPWP. An advantage of the Bayesian framework is that it is possible to combine the two periods assuming they are independent, whereby we obtain a tighter constraint of 2.5 K (0.8–4.0) using the restricted ensemble. We have explored the sensitivity to our assumptions in the method, including considering structural uncertainty, and in the choice of models, and this leads to 95 % probability of climate sensitivity mostly below 5 K and only exceeding 6 K in a single and most uncertain case assuming a large structural uncertainty. The approach is compared with other approaches based on OLS, a Kalman filter method, and an alternative Bayesian method. An interesting implication of this work is that OLS-based emergent constraints on ECS generate tighter uncertainty estimates, in particular at the lower end, an artefact due to a flatter regression line in the case of lack of correlation. Although some fundamental challenges related to the use of emergent constraints remain, this paper provides a step towards a better foundation for their potential use in future probabilistic estimations of climate sensitivity.

In recent years, researchers have identified a number of relationships between observational properties and a future climate change, which was not immediately obvious a priori but which exists across the ensemble of global climate models (GCMs)

Such emergent constraints have been broadly used to constrain properties of the Earth's climate system which are not easily or directly observable. These are usually presented in probabilistic terms, mostly based on ordinary least squares (OLS) methods. For example, studies have explored the constraint on equilibrium climate sensitivity (ECS), which is the global mean equilibrium temperature after a sustained doubling of

Almost all emergent constraint studies have used OLS-based methods to establish the link between variables in the model ensembles. However, whether ECS or another climate parameter was investigated, the theoretical foundations for the calculations have not previously been clearly presented.
An additional problem arising from this is the resulting difficulty in synthesising estimates of climate system properties generated by different statistical methods with different, and often not explicitly introduced, assumptions. These methods include OLS but also alternative Bayesian approaches such as estimates of the climate sensitivity using energy balance models

Two recent papers have also addressed the question of emergent constraints in different ways.

Another Bayesian statistical interpretation of emergent constraints has recently been presented by

We present an alternative Bayesian linear regression approach in which the regression relationship is used as a likelihood model for the problem. This allows the prior over the predictand to be defined separately from and entirely independently of the model ensemble and emergent constraint analysis. Thus, the likelihood arising from the emergent constraint could be used to update a prior estimate of the predictand that arose from a different source.

In Sect.

The general method of emergent constraints seeks a physically plausible relationship in the climate system between two model variables in an ensemble of results from different climate models. Consequently, an observation of one measurable variable (such as past tropical temperatures) could be used to better constrain the other investigated variable, usually unobserved and difficult to measure (such as climate sensitivity). This idea has been used in climate science to estimate quantities of interest such as snow albedo feedback

In this study, we focus on the relationship between equilibrium climate sensitivity, defined here as

The most widely used approach to emergent constraint analysis is to find an observable phenomenon that exhibits some relationship to the parameter of interest and use this as a predictor in a linear regression framework. The ordinary least squares (OLS) method has been widely used because of its simplicity, so we also use it here as a starting point for comparison with alternative statistical methods. In the context of constraining climate sensitivity, the parameter of interest (i.e. the ECS) is considered to be a predicted variable

Integrating the intrinsically frequentist confidence intervals obtained from regression methods used for OLS estimates into a Bayesian framework is challenging. One issue is the misinterpretation of frequentist confidence intervals as Bayesian posterior credible intervals. The former is the representation of the number of random intervals containing the true interval bounds (at 90 % confidence, this would lead to 90 out of 100 random intervals containing the true bounds), while the Bayesian credible interval is an interval which we believe (with the given probability) to contain the truth. For instance, if there is an observed

The (subjective) Bayesian paradigm is based on the premise that we use probability distributions to describe our uncertain beliefs concerning unknown parameters. We use Bayes' theorem to update a prior probability distribution function (PDF) for the equilibrium climate sensitivity via

Choosing

The three parameters

In this way, we create a statistical model that can generate a predictive PDF for the tropical temperature change at the LGM or at the mPWP

It is important to note that Eq. (

A prior belief on climate sensitivity (

An additional issue that was briefly mentioned above is that we may like to consider the probability that reality is qualitatively and quantitatively distinguishable from all models. This issue, which was explicitly argued in the context of emergent constraint analysis by

The Bayesian method is more explicit than the standard OLS approach, as the prior assumptions have to be given by the user. This transparency leads to more freedom and control of the statistical model. Moreover, it has a reduced sensitivity to outliers as the prior on the regression coefficients provides a form of regularisation. This should lead to lower variance in the results compared to results with wider priors on the parameters, particularly with small model ensembles.

Additionally, the Bayesian method allows the user to add multiple lines of evidence by updating the chosen prior for

It is not clear if observational errors have always been adequately accounted for in previous emergent constraints research. Our approach provides a natural framework for this, as the likelihood can include the uncertainty of the observational process as we have done. Recent studies have investigated Bayesian ways of integrating uncertainties on proxy reconstructions into the global temperature field (e.g.

Here we only have two dimensions for the Gaussian, these being the scalar predictor (e.g. sensitivity) and predictand (e.g. tropical temperature change). While this approach is a natural and attractive option in many respects, it has the specific drawback (in the context of this work) of using the distribution of model samples as a prior (for both the mean and covariance). Existing literature on emergent constraints does not make this assumption, and this could be seen as a limiting aspect of the method, as it implies that the model ensemble is already a credible predictor even before consideration of the observational constraint. An implication of this approach is that the posterior estimate will be equal to the model distribution in the case that no observational constraint exists, either because there is in fact no relationship between the observation and predictand or when the observational uncertainty is excessively large. The use of a Gaussian prior based on the ensemble range also means that it is difficult for the method to generate posterior estimates that include values significantly outside the model range, even in the case in which the observed value is outside the model spread. We present results generated with a Kalman filter in Sect.

The Bayesian method may be applied to any emergent constraint. In this study, we use the model outputs and data syntheses that have arisen from phases 2 and 3 of PMIP

Models, tropical temperature

Latitudinal distribution of temperature changes relative to pre-industrial values for both simulated climates for various climate models and a proxy reconstruction. Dashed lines are models of the CMIP5 generation, while solid lines are models from the CMIP6 generation. All model distributions correspond to 100-year zonal averages when possible; certain CMIP5 PlioMIP1 models were averaged over 30 years.

Previous results from PMIP2 showed a significant correlation between LGM tropical temperatures and climate sensitivity in the models

Interest in the mPWP (2.97–3.29 million years ago) as a more direct analogy for future climate change has grown during the past years. This is the most recent period with a sustained high level of greenhouse gases and concomitant warmth relative to the pre-industrial period; however, the data are more sparse and uncertain. In Fig.

The Last Interglacial (127 ka, referred to as lig127k in CMIP6) and the mid-Holocene (6 ka) are part of the PMIP simulations and also relatively warm climates. The forcings are, however, seasonal and regional in nature, mostly influencing the patterns of climate change. The global change in temperature and the global climate forcing are both very small, and this coupled with the large uncertainty in paleoclimate data makes these intervals poor candidates for constraining climate sensitivity. We do not explore these intervals further here.

Climate sensitivity has various definitions and there are also a number of different ways of approximating the value in climate models that have not been run to equilibrium. For PMIP3 LGM the model values are mostly based on the regression method of

In addition to the already published results from PMIP2 and PMIP3 we add to our ensembles the results that are currently available from PMIP4 in Sect.

In order to apply the Bayesian linear regression and compute the likelihood

Depending on the strength of the correlation among the dataset, one could expect a sensitivity of the regression to the choice of prior parameters. In the following sections, we first describe the physical arguments behind the choice of priors over

From consideration of energy balance arguments and fundamental physical properties, such as the response of the Earth to an increase in

Summary of the methods and computed posterior sensitivities; n/a indicates “not applicable”.

Deviations from the regression line may be due to different efficacies of other forcing components, especially ice sheets or dust. To take into account the uncertainty on the strength of the response, we performed two additional analyses wherein the prior response was smaller (

The MCMC algorithm samples the posterior distribution of regression parameters, which is represented by the ensemble of predictive regression lines in Fig.

LGM northern tropical (20

To test the robustness of the method and also to compare it with the statistical methods used in previous studies, three cases are investigated in which we use different combinations of the available model ensembles. The results are shown in Fig.

For the PMIP2 ensemble, the correlation between tropical temperature and climate sensitivity was found to be reasonably strong, and in this study the resulting 5 %–95 % range for inferred climate sensitivity is 1.0–4.5 K (Fig.

When all the models of PMIP2 and PMIP3 (see Table

Finally, we consider the PMIP3 models in isolation. For this ensemble no correlation is found, so for the Bayesian method the result is heavily dependent on our prior assumptions. We obtain a 5 %–95 % range here of 0.7–5.5 K (Fig.

The Kalman filtering approach presented by

LGM northern tropical (20

The mPWP tropical (30

As for the LGM, prior parameters have to be defined to perform the BLR with the mPWP data. In principle these may be different to those used for the LGM experiment, since the total positive forcing of the mPWP is not as large as the negative forcing of the LGM, but in practice we have adopted the same priors for our base case, apart from the obvious sign change for

The Bayesian inference method applied above for the LGM model outputs is now applied to the mPWP model outputs (Fig.

The ongoing PMIP4 experiments have produced LGM and mPWP (PlioMIP2) simulations. Here we add those results to our ensembles. There are four model runs available for the LGM and five for the mPWP (see Table

Inclusion of the CMIP6 models into the Bayesian method for the LGM and the mPWP.

For the LGM we have previously combined the PMIP2 and PMIP3 results, and the protocol for PMIP4 is not very different. If we combine all three ensembles we obtain a 5 %–95 % range for the ECS of 0.6–5.2 K using the Bayesian method (Fig.

For PlioMIP1 and PlioMIP2 the situation is a little more complex as the protocol has been redesigned to represent a specific interglacial state rather than a generic warm climate, referred to as a “time slab” in the PlioMIP protocol. Thus, there could be a different regression relationship for these two ensembles. However, when we plot the PlioMIP1 ensemble members (Fig.

As described in Sect.

It is straightforward to first compute the posterior estimate of

Posterior distribution of climate sensitivity computed with a Cauchy prior by combining two assumed independent emergent constraints. The method does not explicitly use both posteriors of the LGM and the mPWP, but it uses the LGM posterior as the mPWP prior. However, the resulting combined posterior will usually be narrower than the two independent posteriors. For the LGM, the posterior is computed by using the latest model versions of PMIP, including PMIP4. For the mPWP, the posterior is computed by using the latest model versions of PlioMIP, including PlioMIP2.

A logical extension of the approach would be to apply it to the ensemble of models in CMIP, wherein multiple emergent constraints exist for the same models. In theory, this should be possible as long as the investigated relationships are physically plausible. This goes beyond the scope of our study, which uses the paleoclimates as an example for the method, and is left for future research.

A major strength of the Bayesian analysis developed here is the way that the prior on the parameter of interest, here climate sensitivity, can easily be specified independently of all other aspects of the analysis.
A uniform prior for

Posterior distributions computed with different priors and datasets.

As previously explored and described by

However, while we may anticipate reality deviating further from the regression line, it is difficult to quantify such deviation. Here, we perform two sensitivity tests for which we define

Past climates are relevant sources of information on the properties of the climate system, specifically the equilibrium climate sensitivity, due to the quasi-equilibrium changes in response to external forcing, which are of similar magnitude as the projected future climate changes. In this study, we have described a new statistical method based on Bayesian inference to approach the question of emergent constraints. We believe this method provides a reasonable representation within the Bayesian paradigm of the underlying structure of emergent constraint principles. This Bayesian method is designed to be as explicit and flexible as possible. Previous work using ordinary least squares has usually applied implicit assumptions. Because of these assumptions, predictions obtained via OLS tend to generate tight posterior ranges, particularly on the lower end and when the correlation is rather weak; this is something that may well be regarded as an artefact of using such a method.

By applying the method to the LGM tropical temperature model ensemble used in

With the new generation of climate models, LGM and mPWP analyses have been widened by the addition of several CMIP6 model outputs. By adding the PMIP4 LGM simulations, we computed a 5 %–95 % interval for climate sensitivity of 0.6–5.2 K. We performed the same analysis by combining PlioMIP1 and PlioMIP2 models and obtained a 5 %–95 % interval of 0.5–4.4 K. However, these results come with some caveats attached. In particular, combining the two model generations of the mPWP could lead to biased results, since the experimental protocol substantially changed in PlioMIP2. An alternative approach is to consider solely the latest version of each model. By doing this we reduce expected redundancy in the ensemble, and so improve our confidence in the result despite the smaller ensemble sizes. This leads to similarly constrained climate sensitivity of 2.7 (0.6–5.2, 5 %–95 %) for the LGM simulations and 2.3 (0.4–4.5, 5 %–95 %) for the mPWP simulations. Although most of the computed ranges are wider than the ranges obtained with OLS or Kalman filtering, the Bayesian framework avoids the underlying assumptions of both methods and, in particular, makes us regard the Kalman filtering approach in its current form as too restrictive for the question of emergent constraints.

Nevertheless, our results obtained by analysing the LGM or the mPWP in isolation are broadly consistent with results obtained by other statistical methods used in previous studies. The differences between the way the information is obtained from the paleo-record for the mPWP and the LGM and the different dominant climate features of the intervals suggest it may be reasonable to consider these estimates to be statistically independent, given climate sensitivity. It is then possible to combine them within the same Bayesian framework to compute a narrower range of climate sensitivity. By doing so, we evaluated the climate sensitivity to be 2.5 K (0.8–4.0, 5 %–95 %). However, this approach requires independence between the different combined emergent constraints.

It is, in principle, straightforward to include other independent emergent constraints into our Bayesian framework. As well as evidence from historical or present-day analyses, other past climates are starting to be explored by modellers and may be potential candidates for future analyses, such as the Eocene, the Miocene, and the last deglaciation. Over the next couple of years we expect new outputs for models from CMIP6 and new data analyses to become available, which will enable these preliminary analyses to be compared with results from expanded LGM and mPWP ensembles and improved data estimates.

In Sect.

For the case of the mPWP, we defined the priors

To illustrate this approach, we use the R implementation bayesLMConjugate (where LM stands for linear model) of the package spBayes

The priors provided on the regression parameters in the MCMC method then need to be modified for comparison with the conjugate approach, i.e. conditioned on

Posterior distributions of the three parameters

Although the choice of conjugate priors would simplify the computation, NUTS (or MCMC methods in general) have the advantage of allowing an explicit and flexible choice of priors for the users. Having such flexibility is a vital element for the analysis presented in this paper.
The example taken here to illustrate the Bayesian framework, i.e. the relationship of

The Python codes used for the different statistical methods are available from the Bolin Centre Code Repository at

The BLR method was conceived by JDA and JCH. TM put the project together. The code for the Bayesian framework and for the OLS was written by MR. The code for the Kalman filter was written by JDA and translated to Python by MR. The code for the conjugate prior approach was written by MR. The statistical analyses were performed by MR. The climate sensitivities of the CMIP6 models were computed by CF. The paper was written by MR, JDA, JCH, NS, and TM. RO provided the LGM outputs of MIROC-ES2L. UM and MLK provided the LGM outputs of MPI-ESM1.2-LR. GL and XS provided the LGM outputs of AWI-ESM-1-1-LR. QZ and QL provided the mPWP outputs of EC-EARTH3.3.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Paleoclimate Modelling Intercomparison Project phase 4 (PMIP4) (CP/GMD inter-journal SI)”. It is not associated with a conference.

We thank the two anonymous reviewers for their insightful comments. We are very grateful for the extensive work of the scientists involved in the different PMIPs and their efforts to publish and share their data with us. We acknowledge the Bolin Centre for Climate Research at Stockholm University for giving us access to the code repository, which allows us to freely share our code. We acknowledge the STFC Centre for Environmental Data Analysis (CEDA) in collaboration with the InfraStructure for the European Network for Earth System Modelling (IS-ENES) and the Natural Environment Research Council via the National Centre for Atmospheric Science (NCAS) for making the CMIP5 and CMIP6 dataset available on the ESGF portal. We acknowledge Alan Haywood and Richard Rigby for giving us access to the PlioMIP1 dataset. We acknowledge the Statistical Research Group at the Department of Mathematics at Stockholm University and its director Jan-Olov Persson for his useful comments. Rumi Ohgaito acknowledges support from the Integrated Research Program for Advancing Climate Models (TOUGOU programme) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. The simulations using MIROC models were conducted on the Earth Simulator of JAMSTEC. Gerrit Lohmann and Xiaoxu Shi acknowledge support from the PACMEDY project of the Belmont Forum and the PalMod project through BMBF. The simulations using AWI-ESM-1-1-LR were conducted on the German Climate Computing Centre (DKRZ). Qiong Zhang acknowledges support from Swedish Research Council VR grants 2013-06476 and 2017-04232. The MPI-M contribution was supported by the German Federal Ministry of Education and Research (BMBF) as a Research for Sustainability initiative (FONA) through the project PalMod (FKZ: 01LP1504C). The analysis and storage of data were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Centre at Linköping University (NSC).

This research has been supported by the European Research Council (highECS (grant no. 770765) and CONSTRAIN (grant no. 820829)).The article processing charges for this open-access publication were covered by Stockholm University.

This paper was edited by Julien Emile-Geay and reviewed by two anonymous referees.