# 9.A.2 Methods of Inference

Detection and attribution questions are assessed through a combination of physical reasoning (to determine, for example, by assessing consistency of possible responses, whether other mechanisms of change not included in the climate model could plausibly explain the observed change) and by evaluating specific hypotheses about the scaling factors contained in a. Most studies evaluate these hypotheses using standard frequentist methods (Hasselmann, 1979, 1997; Hegerl et al., 1997; Allen and Tett, 1999). Several recent studies have also used Bayesian methods (Hasselmann, 1998; Leroy, 1998; Min et al., 2004, 2005; Lee et al., 2005, 2006; Schnur and Hasselmann, 2005; Min and Hense, 2006a,b).

In the standard approach, detection of a postulated climate change signal occurs when its amplitude in observations is shown to be significantly different from zero (i.e., when the null hypothesis H_{D} : a = 0 where 0 is a vector of zeros, is rejected) with departure from zero in the physically plausible direction. Subsequently, the second attribution requirement (consistency with a combination of external forcings and natural internal variability) is assessed with the ‘attribution consistency test’ (Hasselmann, 1997; see also Allen and Tett, 1999) that evaluates the null hypothesis H_{A} : a = 1 where 1 denotes a vector of units. This test does not constitute a complete attribution assessment, but contributes important evidence to such assessments (see Mitchell et al., 2001). Attribution studies usually also test whether the response to a key forcing, such as greenhouse gas increases, is distinguishable from that to other forcings, usually based on the results of multiple regression (see above) using the most important forcings simultaneously in X. If the response to a key forcing (e.g., due to greenhouse gas increases) is detected by rejecting the hypothesis that its amplitude a_{GHG} = 0 in such a multiple regression, this provides strong attribution information because it demonstrates that the observed climate change is ‘not consistent with alternative, physically plausible explanations of recent climate change that exclude important elements of the given combination of forcings’ (Mitchell et al., 2001).

Bayesian approaches are of interest because they can be used to integrate information from multiple lines of evidence, and can incorporate independent prior information into the analysis. Essentially two approaches (described below) have been taken to date. In both cases, inferences are based on a posterior distribution that blends evidence from the observations with the independent prior information, which may include information on the uncertainty of external forcing estimates, climate models and their responses to forcing. In this way, all information that enters into the analysis is declared explicitly.

Schnur and Hasselmann (2005) approach the problem by developing a filtering technique that optimises the impact of the data on the prior distribution in a manner similar to the way in which optimal fingerprints maximise the ratio of the anthropogenic signal to natural variability noise in the conventional approach. The optimal filter in the Bayesian approach depends on the properties of both the natural climate variability and model errors. Inferences are made by comparing evidence, as measured by Bayes Factors (Kass and Raftery, 1995), for competing hypotheses. Other studies using similar approaches include Min et al. (2004) and Min and Hense (2006a,b). In contrast, Berliner et al. (2000) and Lee et al. (2005) use Bayesian methods only to make inferences about the estimate of a that is obtained from conventional optimal fingerprinting.