Observational constraints on the equilibrium climate sensitivity have been generated in a variety of ways, but a number of results have been calculated which appear to be based on somewhat informal heuristics. In this paper we demonstrate that many of these estimates can be reinterpreted within the standard subjective Bayesian framework in which a prior over the uncertain parameters is updated through a likelihood arising from observational evidence. We consider cases drawn from paleoclimate research, analyses of the historical warming record, and feedback analysis based on the regression of annual radiation balance observations for temperature. In each of these cases, the prior which was (under this new interpretation) implicitly used exhibits some unconventional and possibly undesirable properties. We present alternative calculations which use the same observational information to update a range of explicitly presented priors. Our calculations suggest that heuristic methods often generate reasonable results in that they agree fairly well with the explicitly Bayesian approach using a reasonable prior. However, we also find some significant differences and argue that the explicitly Bayesian approach is preferred, as it both clarifies the role of the prior and allows researchers to transparently test the sensitivity of their results to it.

While numerous explicitly Bayesian analyses of the equilibrium climate sensitivity have been presented

The paper is organised as follows. In Sect.

Let us assume we have a measuring process that produces an observational estimate

Following on from this measurement model, there is a simple syllogism (i.e. a logical argument) that seems common in many areas of scientific research, which runs as follows: since we know a priori that

This syllogism is intuitively appealing but incorrect. It appears to arise from the misinterpretation of frequentist confidence intervals as being Bayesian credible intervals. We should note that calculating and presenting the interval

We start by noting that probabilistic statements concerning the true value

Bayes' theorem is a simple consequence of the axioms of probability: the joint density

The error in the syllogism is to interpret

We now present a simple example in which the syllogism leads to poor results in a physically based scenario with continuous data. We take as given that the timing error of a handheld stopwatch is

An observed time of

In order to make sensible use of this observation, we can instead perform a simple Bayesian updating procedure. The distribution

While it is formally invalid, we must acknowledge that this syllogism does actually work rather well in many cases. In particular, if the likelihood

Thus, in practice the syllogism can often be interpreted as a Bayesian analysis in which a uniform prior has been implicitly used, and in cases in which this is reasonable it will generate perfectly acceptable results. Statements to this effect have occasionally appeared in some papers wherein a non-Bayesian analysis has been presented as directly giving rise to a posterior PDF. It may therefore seem that the terms “fallacy” and “confusion” are somewhat melodramatic: this convenient shortcut is often harmless enough. However, this cannot be simply asserted without proof: there are many examples of procedures for generating frequentist confidence intervals in which the results cannot plausibly be interpreted as Bayesian credible intervals

Some have attempted to retrospectively defend the use of this syllogism with the claim that the uniform prior is necessarily the correct one to use, generally via the belief that this represents some sort of pure or maximal state of ignorance. However, it is well established (and indeed is sometimes used as a specific point of criticism) that there is no such thing as pure ignorance within the Bayesian framework. See

It is a fundamental assumption of this paper that in the cases presented below, in which researchers have presented observational estimates of temperature change

Most probabilistic estimates of the equilibrium climate sensitivity which have explicitly presented a Bayesian framework have used a prior which is uniform in sensitivity

We now consider three areas in which observational constraints have been used to estimate the equilibrium climate sensitivity. Firstly, we consider paleoclimatic evidence, which relates to intervals during which the climate was reasonably stable over a long period of time and significantly different to the pre-industrial state. We then consider analyses of observations of the warming trend over the 20th century (strictly, extending into the 21st and 19th century). Finally, we consider analyses of interannual variability.

A common paradigm for estimating the equilibrium climate sensitivity

The interval which has been examined in the most detail in this manner is probably the Last Glacial Maximum at 19–23 ka

The method adopted by

Using values based broadly on those used in

As mentioned in Sect.

When applied to the numerical estimates provided above, the PDF sampling method of

Prior and posterior estimates for the climate sensitivity arising from paleoclimatic evidence. Dashed lines show priors, and solid lines are posterior densities. The thick cyan line shows the posterior estimate arising from the method of sampling observational PDFs, with the corresponding prior shown in Fig.

Now we present alternative calculations which take a more standard and explicitly Bayesian approach. We start by writing the model in the form

Although the method of sampling observational PDFs described in Sect.

Using Eq. (

Implicit prior used in the paleoclimate estimate. The contour plot shows the joint prior in

In order to perform a more conventional Bayesian updating procedure using Eq. (

Perhaps the most common approach to estimating

As in Sect.

We adopt the distributions used by

Given the similarities between Eqs. (

Implicit prior used in the 20th century estimate.

Priors and posteriors in explicit Bayesian estimates using 20th century data. Dashed lines show priors, and solid lines are posterior densities. The thick cyan line shows the posterior estimate arising from the method of sampling observational PDFs, with its implicit prior shown in Fig.

Alternative priors and their resulting posteriors after Bayesian updating using Eq. (

Priors and posteriors over

Finally, we consider a method which has been used to estimate the climate sensitivity via interannual variation in the radiation balance and temperature

As noted by

In Fig.

We have shown how various calculations which have presented probabilistic estimates of the equilibrium climate sensitivity

In many cases, the implied prior for

Our calculations suggest that the PDF sampling method can generate acceptable results in some cases, agreeing fairly well with a fully Bayesian approach using reasonable priors. However, this is not always the case. We recommend that researchers present their analysis in an explicitly Bayesian manner as we have done here, as this allows the influence of the prior and other uncertain inputs to be transparently tested.

All codes used in this paper can be found in the Supplement.

The supplement related to this article is available online at:

Both authors contributed to the research and writing.

We are grateful to Andrew Dessler and two anonymous referees for helpful comments on the paper. We acknowledge the modelling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI), and the WCRP's Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 multi-model dataset. Support for this dataset is provided by the Office of Science, US Department of Energy.

This paper was edited by Michel Crucifix and reviewed by two anonymous referees.