De mortuis nil nisi bonum and all that, but I realise I only wrote this down in a very abbreviated and perhaps unclear form many years ago, in fact prior to publication of the paper it concerns. I was sad to hear of his untimely death and especially by suicide when he surely had much to offer. But like all innovative researchers, he made mistakes too, and his Dismal Theorem was surely one of them. Since it’s been repeatedly brought up again recently, I thought I should explain why it’s wrong, or perhaps to be more precise, why it isn’t applicable or relevant to climate science in the way he presented it.
His basic claim in this famous paper was that a “fat tail” (which can be rigorously defined) on a pdf of climate sensitivity is inevitable, and leads to the possibility of catastrophic outcomes dominating any rational economic analysis. The error in his reasoning is, I believe, rather simple once you’ve seen it, but the number of people sufficiently well-versed in statistics, climate science and economics (and sufficiently well-motivated to carefully examine the basis of his claim) is approximately zero so as far as I’m aware no-one else ever spotted the problem, or at least I haven’t seen it mentioned elsewhere.
The basic paradigm that underpins his analysis is that if we try to estimate the parameters of a distribution by taking random draws from it, then our estimate of the distribution is going to naturally take the form of a t-distribution which is fat-tailed. And importantly, this remains true even when we know the distribution to be Gaussian (thin-tailed), but we don’t know the width and can only estimate it from the data. The presentation of this paradigm is hidden beyond several pages of verbiage and economics which you have to read through first, but it’s clear enough on page 7 onwards (starting with “The point of departure here”).
The simple point that I have to make is to observe that this paradigm is not relevant to how we generate estimates of the equilibrium climate sensitivity. We are not trying to estimate parameters of “the distribution of climate sensitivity”, in fact to even talk of such a thing would be to commit a category error. Climate sensitivity is an unknown parameter, it does not have a distribution. Furthermore, we do not generate an uncertainty estimate by comparing a handful of different observationally-based point estimates and building a distribution around them. (Amusingly, if we were to do this, we would actually end up with a much lower uncertainty than usually stated at the 1-sigma level, though in this case it could indeed end up being fat-tailed in the Weitzman sense.) Instead, we have independent uncertainty estimates attached to each observational analysis, which are based on analysis of how the observations are made and processed in each specific case. There is no fundamental reason why these uncertainty estimates should necessarily be either fat- or thin-tailed, they just are what they are and in many cases the uncertainties we attach to them are a matter of judgment rather than detailed mathematical analysis. It is easy to create artificial toy scenarios (where we can control all structural errors and other “black swans”) where the correct posterior pdf arising from the analysis can be of either form.
Hence, or otherwise, things are not necessarily quite as dismal as they may have seemed.