Nic Lewis has apparently had a paper published in the Journal of Climate, called Objective Inference for Climate Parameters: Bayesian, Transformation of Variables and Profile Likelihood Approaches. Nic discusses his new paper in a Climate Audit blogpost called Paper justifying AR4’s use of a uniform prior for estimating climate sensitivity shown to be faulty. The paper that Nic is referring to is Frame et al. (2005), which attempted to use prior assumptions to constrain climate forecasts.
The basic issue seems to the following. Frame et al. (2005) wanted to determine the likelihood range for the Equilibrium Climate Sensitivity, . To do this they started with a basic energy balance model (Hansen et al. (1985), Andrews & Allen (2008)) in which each model has a specified sensitivity () and ocean diffusivity ( – which determines the rate at which energy is transferred from the well-mixed ocean layer to the deep ocean). One can then run a whole suite of models which, after integrating over , can be then be used to determine the likelihood of a particular observation (surface temperature, ocean heat content) given a particular sensitivity . By combining this with the probability of that observed quantity and a prior assumption about the sensitivity, one can then determine the probability density function (PDF) of , given the known observations (I’m not an expert at Bayesian inference, so may not have explained that properly).
In the method of Frame et al. (2005), the prior for was assumed to be uniform in the temperature range from 0K to 20K. This means that their prior assumption is – for example – that an equilibrium sensitivity between 2 and 3K is the same as one between 14 and 15K. If you consider their basic result (shown in the figure below) the PDF for has a long tail extending to high values and their 5-95% range is 1.2 – 11.8 K (they’ve just published a correction suggesting that it should actually be 1.2 – 14.5 K).
Nic Lewis’s issus with Frame et al. (2005) is that the use of a uniform prior for climate sensitivity – extending to 20 K – is wrong, and I have to say that I probably agree. It seems highly unlikely that the equilibrium climate sensitivity will be above 10K, or even close. Instead, Lewis argues for using what is called a Jeffrey’s prior, which is apparently “uninformative” (although I’m not quite sure what this means). Consequently his prior (shown in the figure below) has a decreasing probability with increasing and . An obvious problem with this is that it seems to assume that the most likely climate sensitivity is 0K, and there is no physical justification for such a low value.
Lewis therefore uses the same basic method as Frame et al. (2005) but uses a Jeffrey’s prior, rather than a uniform prior. The basic result is in the figure below. The different curves include the Frame et al. (2005) result and different methods used by Lewis. I’ve got a little confused by the different curves, but I think the dark blue is the original Frame et al. (2005) result, and the basic bottom line is that the 5 – 95% range is now 1.2 – 4.5 K (rather than 1.2 to 14.5 K).
So, my basic thoughts on this are that a uniform prior producing a 5 – 95% range of 1.2 – 14.5 K seems wrong. Lewis’s use of a Jeffrey’s prior seems to produce a result that is more consistent with the standard IPCC range (1.5 or 2 – 4.5 K), so I’m not sure why he seems to imply that the Frame et al. (2005) work somehow reflects on the IPCC. I’m also not sure why he felt the need to refer to a 10 year old paper as being faulty, rather than being more constructive (although after the Hawkins vs Mora issue, maybe this is more common in climate science than I’m used to). There are a couple of other issues – in my opinion – with Lewis’s work. The energy balance models are fairly simple. As Andrews & Allen (2008) say
It is important not to over-interpret this model: in particular, the assumptions that λ and C are time and forcing-invariant are over-restrictive.
and they’re unable to incorporate possible inhomogeneities in the aerosol forcings (e.g., Shindell et al. (2014)). Most papers who use these models mention these simplifications, but I couldn’t find any mention of this in Nic Lewis’s paper.
Furthermore, the choice of a Jeffrey’s prior would seem to make a low climate sensitivity much more likely than one might expect. As I understand it, James Annan has already suggested that Lewis is wrong to argue that a Jeffrey’s prior is the most appropriate prior to use. Additionally, Annan & Hargreaves (2009) appear to have already addressed the issues of using a using a uniform prior, as done by Frame et al. (2005). In fact, I found their paper much more physically motivated than Nic Lewis's paper, which seems to be largely a statistical argument. The figure below is probably the key one from Annan & Hargreaves, which shows the PDF for climate sensitivity using a uniform prior and two others, both of which assume that high and low sensitivities are unlikely (which is what we would really expect). The non-uniform priors seem to produce 5 – 95% ranges of about 1.5 – 4.5K, consistent with the IPCC and also, broadly, with Lewis's latest paper.
Anyway, this post has got rather long and a bit convoluted. I don’t have any huge issues with Nic Lewis’s latest work other than I don’t see why the Jeffrey’s prior is somehow optimal, and I’m not sure why he has to imply something with respect to the IPCC when it seems as though his results are broadly consistent with theirs. I’m also not sure why he’s chosen to ignore Annan & Hargreaves (2009) – who he appears to not even cite – when they seem to have pointed out the issue with Frame et al. (2005) about 5 years ago. As usual, if anyone has any thoughts or would like to correct anything I’ve said here (which, given that I’m still trying to get my head around Bayesian statistics, may well be necessary) feel free to do so through the comments.