Combining independent Bayesian posteriors into a confidence distribution, with application to estimating climate sensitivity

Combining estimates for a fixed but unknown parameter to obtain a better estimate is an important problem, but even for independent estimates not straightforward where they involve different experimental characteristics. The problem considered here is the case where two such estimates can each be well represented by a probability density function (PDF) for the ratio of two normally-distributed variables. Two different statistical methods - objective Bayesian and frequentist likelihood-ratio - are employed and compared. Each probabilistic estimate of the parameter value is represented by a fitted three-parameter Bayesian posterior PDF providing a close approximation to the ratio of two normals, that can legitimately be factored into a likelihood function and a noninformative prior distribution. The likelihood functions relating to the parameterised fits to the probabilistic estimates are multiplicatively combined and a prior is derived that is noninformative for inference from the combined evidence. An objective posterior PDF that incorporates the evidence from both sources is produced using a single-step approach, which avoids the order-dependency that would arise if Bayesian updating were used. The frequentist signed root likelihood-ratio method is also applied. The probability matching of credible intervals from the posterior distribution and of approximate confidence intervals from the likelihood-ratio method is tested, showing that both methods provide almost exact confidence distributions. The approach developed is applied in the important case of the Earth's equilibrium climate sensitivity, by combining an estimate from instrumental records with an estimate representing largely independent paleoclimate proxy evidence, resulting in a median estimate of 2.0 °C and a 5%-95% confidence/credibility interval of (1.1, 4.5) ∘C.