Objective Inference for Climate Parameters: Bayesian, Transformation of Variables and Profile Likelihood Approaches

2 1 Abstract 2 3 Insight is provided into the use of objective Bayesian methods for estimating 4 climate sensitivity by considering their relationship to transformations of variables in the 5 context of a simple case considered in a previous study, and some misunderstandings 6 about Bayesian inference are discussed. A simple model in which climate sensitivity (S) 7 and effective ocean heat diffusivity (K v) are the only parameters varied is used, with 20th 8 century warming attributable to greenhouse gases (AW) and effective ocean heat capacity 9 (HC) being the only data-based observables. Probability density functions (PDFs) for 10 AW and HC are readily derived that represent valid, independent, objective Bayesian 11 posterior PDFs, provided the error distribution assumptions involved in their construction 12 are justified. Using them, a standard transformation of variables provides an objective 13 joint posterior PDF for S and K v ; integrating out K v gives a marginal PDF for S. Close 14 parametric approximations to the PDFs for AW and HC are obtained, enabling derivation 15 of likelihood functions and related noninformative priors that give rise to the objective 16 posterior PDFs that were computed initially. Bayes' theorem is applied to the derived AW 17 and HC likelihood functions, demonstrating the effect of differing prior distributions on 18 PDFs for S. Use of the noninformative Jeffreys' prior produces an identical PDF to that 19 derived using the transformation of variables approach. It is shown that quite similar 20 inference for S to that based on these two alternative objective Bayesian approaches is 21 obtained using a profile likelihood method on the derived joint likelihood function for 22 AW and HC. 23 24 3 1. Introduction 25 Estimates of climate sensitivity often use global energy balance or other simple climate 26 models with a limited number of adjustable parameters, and compare modeled and 27 observed values of multidecadal warming and other climate variables. Such estimates 28 play an important role in assessment of climate sensitivity (Hegerl et al., 2007; Bindoff et 29 al., 2013). Most of these studies use a Bayesian framework as a basis for assessing 30 uncertainty and developing a probability density function (PDF) for climate sensitivity. 31 This paper addresses the methodological challenge of selecting the appropriate Bayesian 32 prior distributions for climate sensitivity and other parameters employed in simple 33 climate model analyses. 34 Deriving valid probabilistic estimates for …