Title
Posterior Confidence Intervals in Linear Calibration Problems: Calibrating the Thompson Ice Core Index
Speaker
Dr. Hu McCulloch, The Ohio State University
Abstract
In calibration problems, an exogenous state variable and an endogenous response variable or proxy are both observed in a set of calibration observations. We wish to make inferences about the unobserved state variable from an additional observation on the response variable under a diffuse prior. Hoadley (1970) argued that an informative prior is required in order to obtain a proper posterior distribution. Hunter and Lamboy (1981) proposed a solution, but were sharply criticized at the time. This paper presents a new derivation of the Hunter-Lamboy posterior distribution under a diffuse prior that meets these objections. At the same time, it is shown that Hoadley's approach was based on a subtle inconsistency in the application of Bayes' Rule.
The reinstated Hunter-Lamboy posterior is applied to the problem of calibrating the ice core index of Thompson et al. (2003) to instrumental temperatures. It is found, contrary to the famous claim of Gore (2006), that this index is in fact uninformative about the question of whether Medieval Warm Period was warmer or cooler than the present.
It is shown that the "classical" confidence intervals proposed by Fieller (1954) are a good approximation to the posterior confidence intervals when the calibration slope coefficient is highly significant relative to the desired confidence interval tail probability. However, when the slope is only marginally significant, the Fieller intervals become increasingly distorted and then meaningless. The proposed posterior confidence intervals simply become wider as the slope loses significance, but remain bounded.
Extensions to multiple proxies, sequentially organized data and prior restrictions on the calibration slope coefficient are outlined but not implemented.
Read his full paper: Calibrating the Thompson ice core index [pdf]
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