a. Likelihood function: f(data|a)

The observational data used here are the familiar Jones data (Jones 1994), consisting of monthly temperature observations on a 5° × 5° grid; we only use data for the period 1961–98. Our analyses are for yearly averages of these data. The Jones grid leads to n = 36 × 72 = 2592 spatial grid points. Some data are missing. Our analysis uses only actual data, that is, we do not impute any missing observations.

Let T_t represent the n vector of “true” annual, gridded temperature anomalies in year t, t = 1, . . . , m. (We perform analyses for four different choices of m.) We assume that these anomalies represent departures from a baseline compatible with that chosen in preparation of the Jones data; namely, the mean of temperatures over the 30-yr period 1961–90. (In general, definitions of baselines appropriate for climate change studies are very important in the interpretation of results.) Let T = (T^′₁, . . . , T^′_m)′ be the m · n vector collecting temperature anomalies over the entire period where ′ denotes transpose. Define vectors of observations Y₁, . . . , Y_m, where Y_t represents “observed” annual temperatures in year t = 1, . . . , m. These vectors may be of various lengths, say n_t, due to missing data. Let Y = (Y^′₁, . . . , Y^′_m)′ be the collection of these vectors.

A fully Bayesian analysis would involve formulation of prior probability models for the quantities a, D, Σ, and perhaps g. In view of the high dimensionality of the problem, this is a daunting task. For purposes of illustration, we follow simpler strategies of (i) formulating plausible estimates of D and Σ, and (ii) selecting fixed fingerprints. We focus on modeling and inference for the amplitude a here and describe our estimation of D and Σ in appendix A.

Both Bayesian and traditional approaches to statistical inference benefit from application of the so-called sufficiency principle (Berger 1985, 126–127; Epstein 1985, 23–25). The idea is to construct functions of the data that summarize all the statistical information contained in the data regarding the unknown parameters. It turns out that for the model (6) a sufficient statistic for a is the generalized least squares estimator (GLS), denoted by â(Y) (e.g., Graybill 1976; Levine and Berliner 1999). This estimator is given by

View Expanded

where

HXGVDXΣX

The quantity â(Y) in (7) is often known as the optimal filter in the climate science literature, though it should be clarified that it is optimal under a mean squared error criterion, and then only within the class of estimators that are both unbiased and linear in the data vector Y. It is also familiar as the most probable value of a, though this Bayesian-like interpretation is founded on a so-called “uniform prior” assumption; we do not use uniform priors in this article.

b. Prior: π(a)

In specifying the prior parameters, note that we should think of a and g as coming in pairs. That is, our prior views concerning the actual values of a depend on the choice of g. The first component of (10) is to model information about a under the qualitative statement that anthropogenic CO₂ forcing has not impacted global temperatures, at least not in a fashion reflected by the fingerprint g. The selection of zero as the prior mean of a in this component of the prior seems quite natural. The variance τ² is interpreted as the variability, arising naturally, we anticipate in fitting a by projecting temperature fields onto the selected fingerprint g. The probability p is then a quantification of the degree of belief in the hypothesis of no anthropogenic impacts.

The second component of the prior represents an implied distribution on a assuming significant anthropogenic effects representable as changes in temperature patterns along the fingerprint g. The quantity μ_A is the hypothesized shift in a, which in turn translates to shifts in temperature anomalies ga. The variance τ²_A reflects anticipated variability in a under CO₂ forcing. This parameter reflects, in part, natural climate variability, though there is no requirement that τ²_A = τ².

Note that the hypotheses here are modeled as distributions rather than as single points (i.e., either a = 0 or a = a_A, exactly). This is in contrast to the simplified prior used in Hasselmann (1998), which is essentially (10) with both τ² and τ²_A set equal to zero, and thus ignores a significant source of uncertainty.

The variation represented by τ² would be best estimated through (unavailable) replications of the climate system under a no CO₂ forcing scenario. As a surrogate, we use CSM control run output. A single 300-yr series was partitioned into 100 batches of equal time duration, each representing “replications” of an unforced, surrogate global climate system over short time periods. The batches are overlapping in that they are constructed by taking a running window of 200 yr. For each batch and our CSM fingerprint pattern g, we computed generalized least squares estimates of a as if the batched data were real observations. That is, in each case, we assumed a model like (6) but with model output playing the role of the data Y and no measurement error covariance matrix D. The sample standard deviation (0.007) of the resulting 100 estimates then provides a plausible first guess at the variance τ. However, since the results batching process of the NCAR CSM runs would vary with sample size, number of batches, and so on, as well as with different CSM runs, we actually use a somewhat larger value τ = 0.02.

Prior parameters under CO₂ forcing were computed analogously to the previous discussion but using batches of the CSM CO₂ forced output. We used the sample mean of the batched results to estimate the prior mean, μ_A. The sample variance provides a guess for τ²_A, though we again used a slightly larger value since we are merely sampling from a particular CSM run.

a. Detection and attribution

We next formalize the specialized inspections of the posterior distribution intended to serve as Bayesian procedures for detection and attribution. One motivation is to offer quantitative measures of direct “practical significance” of results as opposed to restriction to indirect “statistical significance.” We define “no significant climate change” to be the event that a ∈ D, where D is some neighborhood of zero and ∈ is read as “is an element of.” Similarly, define another set, A, a neighborhood of the mean μ_A to represent an attribution set reflecting a considered to be plausible in relation to the physics of CO₂ impacts on climate as reflected in the CSM model (in our case). Recall that the physical meanings of a and g are paired through their product. Hence, the sets D and A may vary with g.

With these definitions, we can use the Bayesian model for inferences regarding detection and attribution. The notions are intuitive:

• Define detection as occurring when the posterior probability that a departs significantly from 0 is large; namely, if

  aDâ(15)
• Define attribution as occurring when both (i) the posterior probability

  aAâ(16)

  and (ii) 1 − p(â) is large.

The two requirements in the above statement of attribution need not be redundant. Consider the following example (the numbers are completely artificial, chosen for ready illustration). Suppose that â|a ∼ N(a, 1) [see (8)]. The prior parameters chosen are p = 0.5, τ² = 1, μ_A = 5, and τ²_A = 25. The observed value of â is −5. Computing (14), we obtain p(â) = 0.045, apparently suggesting that this data offers strong evidence for the anthropogenically forced climate change model. Intuitively, this suggestion seems silly. The observed â = −5 is actually closer to the center, 0, of the no-change prior distribution, than to the center, 5, of the forced prior distribution. However, in a sense, it is true that the forced model offers a better explanation of the data than does the no-change model. The key is the highly different values of the prior variances. Specifically, â is five standard deviations (under the no-change component of the prior) away from 0, but only two standard deviations away from 5 under the forced alternative component. That is, neither alternative offers a good explanation of this data, though in comparing only the two candidates, the forced model is far superior.

These examples, suggesting the potential need for care and inspection of the full posterior rather than simple summaries, may cause some readers to wonder if it might be easier to rely on non-Bayesian approaches. We think not, for a variety of reasons outlined in this article. Further, traditional testing is not immune to unintuitive results, but rather also requires care in interpretations. See Berger (1985) and Levine and Berliner (1999, p. 573) for examples.

Two important questions arise. First, who is to quantify what “small” and “large” mean in our definitions of detection and attribution? Second, who is to define the sets D and A? Note that analogs of the first question also arise in non-Bayesian detection and attribution studies. Namely, who is to decide on cutoffs of achieved statistical significance (in the case of detection) and lack of statistical significance (in the case of attribution)? Regarding the specification of D and A, some might suggest that such specifications are separate from the science of climate change. Even if that is true, we at least provide descriptions of tools for quantitative assessments of the practical impacts of human influences.

Our proposition of detection and attribution analyses are different from conventional, two-step approaches. This is primarily due to the points of the previous paragraph. The Bayesian framework is not limited to simply computing direct analogs of classical procedures. For example, Hasselmann (1998) wrote “. . . it is not possible in the Bayesian approach to separate formally between detection and attribution.” We disagree, and offered such a separation.

b. Classes of priors

The prior (10) assumes that μ_A, τ², and τ²_A are known quantities. In reality, we would be uncertain about these parameters. (This also is true for p; one could seldom argue that p should be 0.7253, but not 0.73. However, study of the impact of varying p is quite easy.) In the Bayesian paradigm, a hierarchical view, in which prior distributions for these quantities are incorporated into the analysis, could be adopted. An alternative is the robust Bayesian viewpoint. The idea is that our information seldom is refined enough to identify a single prior distribution. Rather, our prior information specifies a class of priors, denoted by Γ. We then study ranges of posterior measures (probabilities, estimates, etc.) as the prior varies over Γ. See Berger (1985, chapter 4) for overviews and examples of both the hierarchical and robust Bayesian viewpoints. Those descriptions include more general classes of priors (relaxation of reliance on Gaussian priors, etc.) and analyses.

As in (10), consider priors of the form π(a) = pn(0, τ²) + (1 − p)n(μ_A, τ²_A). For ease in conveying the idea, we assume that τ² is fixed, but we have substantial uncertainty in μ_A and τ²_A. Similar calculations without this imposition are quite accessible, but our preliminary analyses for formulating estimates of the prior parameters suggest that this simplification is reasonable.

Define a collection of priors:

πaμ_lμ_Aμ_uτ_lτ_Aτ_u(18)

for specified bounds μ_l < μ_u and τ_l < τ_u. One example of a robust Bayesian calculation is to find the range of p(â) as the prior varies. During these calculations, â, p, σ², and τ² are fixed. Hence, from (14) we see that the maximum of p(â) is of the form

View Expanded

and the minimum is of the form

View Expanded

where the values M(â) and m(â) are to be determined. The mathematics for these optimizations are outlined in appendix B. The results have a useful interpretation in terms of odds ratios. The quantity (1 − p)/p is the prior odds in favor of the anthropogenically forced component of the distribution of a. The values M(â) and m(â) are extremal values of the corresponding odds as reflected by the data. The products of these values and the prior odds are extremals of posterior odds ratios. We return to this interpretation when presenting numerical results in the next section.

Computing bounds for Pr(a ∈ D|â) and Pr(a ∈ A|â) are more complicated than those for p(â). However, these bounds are easily found using simple numerical optimization methods.

a. Basic analysis

Our prior parameter estimates, resulting from the“batching” of CSM output described earlier, are τ = 0.02, μ_A = 0.17, and τ_A = 0.05. (We actually inflated standard deviations obtained from those model data analyses a bit.) For these prior parameter values and each of the four time periods, Table 1 presents the means and standard deviations [see (12) and (13)] for the two components of the posterior distribution (11). Table 1 also includes the posterior weight p(â) [see (14)] on the no change posterior model, assuming the prior weight p = 0.5. Figure 2 displays the prior and posterior distributions of a for p = 0.5. The plots also display the likelihood functions derived from (8). Note that the posterior distributions are bimodal for time periods a and b, but not time periods c and d. The bimodality is a consequence of the mixing of the two components in the posterior distribution (11). In the latter two time periods, the posterior mixture probability p(â) is essentially zero. Thus the posterior distribution in these time periods is solely the component of the posterior distribution under CO₂ forcing. These results generally suggest that the data most resemble the component of the posterior distributions of a corresponding to anthropogenic forcing; this behavior seems overwhelming for the most recent time periods.

Figure 3 displays the probability p(â) of the “no change” component of the posterior model, as a function of the prior probability p. These probabilities are of course related. For example, the more confident one is in the hypothesis of “no climate change,” the more difficult it is for data to sway that opinion. The posterior probability p(â) is of moderate size for time period a. In fact, for p = 0.5, p(â) is about 0.70, providing more weight for the no change component of the posterior distribution. For time period b, the posterior probabilities p(â) lie well below 0.5 for most of the range of p, thus placing substantial weight on the “anthropogenic forcing” component of the posterior distribution. We have not included corresponding plots for the remaining time periods c and d. The evidence from these periods is so overwhelmingly supportive of the forced model that p(â) (and its upper bound) is essentially zero unless p is absurdly large (0.99 or so).

b. Detection and attribution results

Figures 4 and 5 present the posterior probabilities of detection and attribution regions, Pr(a ∈ D|â) and Pr(a ∈ A|â), respectively, as functions of p, for the time periods a and b. We selected D = [0 − 0.05, 0 + 0.05] and A = [μ_A − 0.05, μ_A + 0.05]. Figure 4 shows moderate probabilities of the detection region across the range of p for time period a suggesting little or no evidence of a detected signal. In fact, when p = 0.5, Pr(a ∈ D|â) is about 0.6. For the same data, Fig. 5 displays small posterior probabilities of the attribution region across the range of p. However, there is some suggestion of a detected change based on time period b in that Pr(a ∈ D|â) < 0.5 for most of the range of p. However, since the corresponding posterior probability Pr(a ∈ A|â) is on the order of 0.3–0.4 across the range of p, there is little evidence for attribution to the CO₂ forcing represented in our CSM-based fingerprint. These moderate values may be expected after examining Fig. 2. In that figure, the likelihood function lies in between the two components of the prior mixture and the two components of the posterior distribution overlap significantly, particularly for time period a.

For time periods c and d, the data strongly support the CO₂ forcing model. In particular, the posterior probabilities Pr(a ∈ D|â) are essentially zero (less than 6 × 10⁻³ for c; less than 10⁻⁶ for d) for reasonable prior probabilities p. The posterior probabilities Pr(a ∈ A|â) are constant across the range of p with values 0.92 (period c) and 0.73 (period d). We thus do not include these probabilities for time periods c and d in Figs. 4 and 5. The constant values arise because the posterior weights 1 − p(â) on the forced posterior component is essentially one for all p. (We explain the apparent decrease in this probability based on period d below.)

c. Robust Bayesian results

To illustrate robust Bayesian results for a plausible class of priors, we specified Γ based on the ranges μ_A ∈ [0.15, 0.19] and τ_A ∈ [0.05, 0.07] (recall that our baseline prior had μ_A = 0.17, τ_A = 0.05).

Table 2 displays the computed values of M(â) and m(â) needed to find the extrema of p(â) described in (19) and (20). These extrema are plotted as functions of p for the first two time periods in Fig. 3. For time period b, the results are quite strong. For example, we have that for p = 0.5, 0.041 ⩽ p(â) ⩽ 0.104; even for the large value p = 0.75, 0.115 ⩽ p(â) ⩽ 0.258.

For the two later time periods c and d the very strong evidence against the no change hypothesis remains even over this class of priors. The values of m and M for these time periods are overwhelmingly supportive of the CO₂ forced distribution. The posterior probability of the no change model is essentially zero (to three decimal points) over the entire class for the third data subset, unless one has an absurdly high prior odds on the no change model. For example, one would need a prior odds in favor of no change to about 6300-to-1 to be indifferent on the hypotheses; that is, for the posterior range on p(â) to include 0.5. The results for the last data subset are even more extreme. However, we do not claim that based on the fourth time period, we have proven that starting with an indifferent prior (p = 0.5), the odds in favor of the CO₂ forced model are at least 43 524-to-1. Such extreme odds and corresponding probabilities are computed as tail events of the Gaussian distributions assumed in this article. While these Gaussian distributions appear plausible, they can imply probabilities of tail events that are subject to round-off error and are not testable or believable. Further, these probabilities are based on all the other assumptions made concerning other parameter specifications. Hence, while we find these results compelling, the actual values obtained should not be considered exact when they are too extreme to be experienced. However, rounding off 43 524-to-1 to 1000-to-1 or even 100-to-1 remains of serious interest.

Figures 4 and 5 present the upper and lower bounds on the detection and attribution probabilities, Pr(a ∈ [−0.05, 0.05]|â) and Pr(a ∈ [μ_A − 0.05, μ_A + 0.05]|â), respectively for time periods a and b. Notice that the ranges for the probability of the detection region are tight for each of the two time periods. Though not displayed here, the latter two time periods c and d show similar behavior with probability ranges bounded very close to zero. This suggests that our interpretation of these posterior probabilities is “robust” with respect to priors varying within Γ. Our conclusion is that there is very strong evidence of a detected signal based on the last three time periods.

The attribution region probabilities are also robust over Γ. Based on time period a, the posterior probabilities of A lie within a range of zero to 0.3, suggesting no ability to attribute a climate change signal to anthropogenic forcing. For time period b, the attribution probabilities may fall slightly above 0.5 depending on prior parameter specification on Γ. The range over which the posterior probability of A may fall is tight for periods c and d as well, being (0.8, 0.92) and (0.45, 0.95) respectively. These ranges are constant across p and are thus not shown in Fig. 5. Hence our conclusion that the data most resemble the component of the posterior distribution corresponding to anthropogenic forcing is reasonable over this range of prior distributions. The larger range, (0.45, 0.95), on the last time period is a consequence of the large estimated signal, â = 0.22, actually being greater than the upper bound on μ_A used to define Γ. Since these data do intuitively support strong projection onto our fingerprint, we still consider this good evidence for attribution. In fact, had both our range for μ_A been more conservative and our definition of A been larger to include large a, the resulting probabilities of A would be much greater. (Recall that such increases are actually in accord with our prior beliefs, though we did not vary the prior on a with the time period considered.)

a. An alternate fingerprint

We performed an analogous analysis using the CO₂ fingerprint (see Fig. 6) developed in Santer et al. (1995). This fingerprint uses the GRANTOUR/CCM1 experimental configuration (Taylor and Penner 1994). Table 3 presents the GLS estimates â of a and associated standard deviations σ for the time periods a–d. Note that these results are similar to those based on our CSM fingerprint, except that the estimated amplitudes for the Santer et al. fingerprint are consistently smaller. However, recall our earlier alert that amplitudes and fingerprints should be viewed as arising in pairs. This means that we should redefine our basic prior, class of priors, and detection and attribution regions for use in analyzing the Santer et al. fingerprint. One option is to repeat the batching ideas used earlier. A second option is to view the adjustments needed as a problem of standardization to place each fingerprint on a common scale for a fair comparison. We chose the second option.

Let g denote our CSM fingerprint, w denote the Santer fingerprint, and a_g and a_w denote the corresponding amplitudes. We consider renormalization of the fingerprints by their L₂ norms (sum of squared values): ‖g‖₂ = 90 and ‖w‖₂ = 100. Standardizing each fingerprint by the square root of its norm, we suggest that if we are to compare projections of the data

View Expanded

then we expect that a_w ≈ 0.95 a_g.

A simple transformation of the prior developed for a_g takes us to a comparable prior for a_w: namely, μ_A = (0.95)(0.17) = 0.162, τ_A = (0.95)(0.05) = 0.0475, and τ = (0.95)(0.02) = 0.019. Since this argument is quite ad hoc and leaves us less certain about where we anticipate a_w to be, we thought it reasonable to use a larger class of priors for a robust Bayesian analysis: namely, we assumed that Γ_w be generated by the ranges μ_A ∈ [0.13, 0.21] and τ_A ∈ [0.05, 0.08]. (Compare these with μ_A ∈ [0.15, 0.19] and τ_A ∈ [0.05, 0.07] used for our fingerprint.)

For brevity, we present results only for detection and attribution analyses. We again associate detection with small probabilities Pr(a_w ∈ [−0.05, 0.05]|â), and attribution with large probabilities Pr(a_w ∈ [μ_A − 0.05, μ_A + 0.05]|â) (recall, now the baseline μ_A = 0.162). Figures 7 and 8 are plots of these probabilities and ranges as functions of p for the time periods a and b. For time periods c and d, the probabilities of the detection region are tightly bounded around zero for reasonable values of p and thus not presented in Fig. 7. For these same time periods, the probabilities of the attribution region are constant across the range of p with values 0.65 and 0.95 and constant bounds from the robust analysis (0.05, 0.98) and (0.90, 0.96), respectively.

The results appear somewhat less definitive for the Santer fingerprint than for the CSM fingerprint. In particular, the ranges on probabilities appear larger, but of course the class Γ_w is larger than its counterpart. However, there is one major exception: time period d now yields very strong and robust attribution. Note that the estimate of a_w from this period does fall in the range of means defined by Γ_w. Indeed, our overall conclusion is in concert with that of our CSM-based fingerprint: namely, there is strong evidence in the recent data for detection and attribution to CO₂ forcing as reflected in either fingerprint.

b. Comparison to traditional results

We provide traditional, non-Bayesian detection and attribution results. By non-Bayesian detection, we mean rejection of the null hypothesis that a = 0, when compared to the alternative hypothesis a > 0. Such a test rejects when the significance probability or p value for the test is small. By non-Bayesian attribution, we mean failure to reject (or “acceptance” of) the null hypothesis that a = μ_A, when compared to the alternative hypothesis a ≠ μ_A. Intuitively this is indicated by a relatively large significance probability. Note that the results given in Table 4 are generally in accordance with simple summaries of our Bayesian results. One exception is the apparent strong, traditional detection results for both fingerprints based on the full data for the time period 1961–98. Our companion Bayesian analyses are more conservative. This is a result of the fact that the non-Bayesian calculation only considers the plausibility of the hypothesis a = 0 exactly, while the our Bayesian formulation studies the plausibility of a lying near zero. Such conservatism through more attention to uncertainty suggests that one may derive comparatively more confidence in our Bayesian detections when they do occur. The situation is somewhat reversed in the context of attribution. The traditional test considers the plausibility of the the value a = μ_A exactly, while our Bayesian formulation essentially considers events of the form “a lies near μ_A.” Nevertheless, the use of robust Bayesian calculations in conjunction with inspection of practical significance of results suggest both the inferential value and robustness of the Bayesian viewpoint.

c. Computation

d. Extensions

An intriguing and important issue involves the selection of fingerprints. In our analysis we viewed the fingerprint g as fixed. Of course, we might also quantify our uncertainty about g (e.g., Allen and Tett 1999). (Before doing so, we must entertain the question of the existence of a g.) In developing a joint prior on a_g and g, a two-step thought process may be a value. Any joint prior, π(g, a_g) can be factored as π(g, a_g) = π(a_g|g)π(g). This means one can build the prior on the fingerprints, and then a prior on amplitudes conditional on fingerprints. For example, we might envision a collection of models or theories, each of which produces particular forms of fingerprints, and then implied distributions on the a_g. One can use the Bayesian strategy to assess the models (see Leroy 1998), as well as to combine information from them. In the statistics literature, these tracks are known as model selection and model averaging, respectively.

The specification and interpretations of modeled quantities are intricately related. We have already alluded to the intertwining of fingerprints g and amplitudes a. In addition, the meaning of climate variability, expressed through Σ must be gauged against the specific that the primary model is essentially a signal-plus-noise model. That is, Σ does not exist in isolation but rather is defined to reflect variation around a particular specification of the mean (taken here to be zero) of anomalies. The difficulty is that the anomalies are in turn defined as departures from some, typically ad hoc, baseline. These apparently subtle issues merit more careful attention to enable unequivocal interpretation of statistically estimated amplitudes.

e. Objectivity versus subjectivity

Bayesian statistics is by nature subjective; people with different priors may arrive at different conclusions. This issue has a long history and many interesting points of discussion (Berger 1985; Bernardo and Smith 1994), especially since traditional statistics methods are often thought to be objective. In complex settings it is difficult to argue that only objectivity really guides analysis; a variety of assumptions are made about the modeling. Indeed, substantial prior information is used in deciding on methods of analysis and data sources to be used, as well as in the final interpretation of results. Rather than “sweeping them under the carpet” (a phrase due to I. J. Good in this context), the Bayesian seeks to make clear what the assumptions are and quantifies them. This enables comparatively simple assessments of model assumptions and permits inspections of the sensitivity of results to these assumptions. Of course, we admit that there is essentially no such thing as a perfect Bayesian, and in that spirit, alluded to the robust Bayesian viewpoint.

However, even if we accept that classical statistical results are objective, we reiterate that they are only indirect answers to the fundamental questions typically of interest. Further, what about climate change science should be objective? The public is not interested in truly objective predictions regarding climate change, and so on. Rather, the most informed, intelligent predictions ought to be the goal. This statement should not be misinterpreted to suggest that analysts should base analyses on their personal hidden agendas, bounties on “proofs” of anthropogenic climate change or the reverse, and so on. Rather than being objective, analyses should be impartial (a word used by Laplace in this context), but informed by the best information and scientific opinion available. Again, there may be no such thing as a perfect and impartial Bayesian. Nevertheless, Bayesians suggest that the most effective way to seek impartiality as a community is for analyses to include description and quantification of the prior information used.