In July 2004 the IPCC held a Working Group 1 (WG1) Workshop on climate sensitivity, as part of the work plan leading up to AR4. In one session, Myles Allen of Oxford university and a researcher in his group, David Frame, jointly gave a presentation entitled “Observational constraints and prior assumptions on climate sensitivity”. They developed the work presented into what became an influential paper, Frame et al 2005,[i] here, with Frame as lead author and Allen as senior author.
Frame and Allen pointed out that climate sensitivity studies could be – whether or not they explicitly were – couched in a Bayesian formulation. That formulation applies Bayes’ theorem to produce a posterior probability density function (PDF), from which best estimates and uncertainty ranges are derived. The posterior PDF represents, at each value for climate sensitivity (ECS), and of any other parameters (fixed but uncertain variables) being estimated, the product of the likelihood of the observations at that value and the “prior” for the uncertain parameters that is also required in Bayes’ theorem.
Obviously, the posterior PDF, and hence the best estimate and upper uncertainty bound for ECS, depend on the form of the prior. Both the likelihood and the prior are defined over the full range of ECS under consideration. The prior can be viewed as a weighting function that is applied to the likelihood (and can be implemented by a weighted sampling of the likelihood function), but in terms of Bayes’ theorem it is normally viewed as constituting a PDF for the parameters being estimated prior to gaining knowledge from the data-based likelihood.
Frame et al 2005 stated that, unless warned otherwise, users would expect an answer to the question “what does this study tell me about X, given no knowledge of X before the study was performed”. That is certainly what one would normally expect from a scientific study – the results should reflect, objectively, the data used and the outcome of the experiment performed. In Bayesian terms, it implies taking an “Objective Bayesian” approach using a “noninformative” prior that is not intended to reflect any existing knowledge about X, rather than a “Subjective Bayesian” approach – which involves the opposite and produces purely personal probabilities.
Frame and Allen claimed that the correct prior for ECS – to answer the question they posed – depended on why one was interested in knowing ECS, and that the prior used should be uniform (flat) in the quantity in which one was interested. Such a proposal does not appear to be supported by probability theory, nor to have been adopted elsewhere in the physical sciences. Although for some purposes they seem to have preferred a prior that was uniform in TCR, their proposal implies use of a uniform in ECS prior when ECS is the target of the estimate. AR4 pointed this out, and adopted the Frame et al 2005 proposal of using a uniform in ECS prior when estimating ECS. Use of a uniform prior for ECS resulted in most of the observational ECS estimates given in Figure 9.20 and Table 9.3 of AR4 having very high 95% uncertainty bounds.
Consistent with the foregoing thesis, Frame et al 2005 stated that “if the focus is on equilibrium warming, then we cannot rule out high sensitivity, high heat uptake cases that are consistent with, but nonlinearly related to, 20th century observations”. Frame and Allen illustrated this in their 2004 presentation with ECS estimates derived from a simple global energy balance climate model, with forcing from greenhouse gases only. The model had two adjustable parameters, ECS and Kv – here meaning the square root of effective ocean vertical diffusivity. The ‘observable’ variables – the data used, errors in which are assumed to be independent – were 20th century warming attributable to greenhouse gases (AW), as estimated previously using a pattern-based detection and attribution analysis, and effective heat capacity (EHC) – the ratio of the changes in ocean heat content and in surface temperature over a multidecadal period.
Frame and Allen’s original graph (Figure 1) showed that use of a uniform prior in ECS gives a very high 95% upper bound for climate sensitivity, whereas a uniform prior in Feedback strength (the reciprocal of ECS) – which declines with ECS squared – gives a low 95% bound. A uniform prior in the observable variables (AW and EHC) also gives a 95% bound under half that based on a uniform in ECS prior; using a prior that is uniform in transient climate response (TCR) rather than in AW, and is uniform in EHC, gives an almost identical PDF.