## 1. Introduction

Investigating causal links between climate forcings and the observed climate evolution over the instrumental era represents a significant part of the research effort on climate. Studies addressing these aspects in the context of climate change have been providing, over the past decades, an ever-increasing level of causal evidence that is important for decision-makers in international discussions on mitigation policy. In particular, these studies have produced far-reaching causal claims; for instance, the latest IPCC report (AR5; IPCC 2014) stated that “It is extremely likely that human influence has been the dominant cause of the observed warming since the mid-20th century” (p. 4). An important part of this causal claim, as well as many related others, regards the associated level of uncertainty. More precisely, the expression “extremely likely” in the latter quote has been formally defined by the IPCC (Mastrandrea et al. 2010; see Table 1) to correspond to a probability of 95%. The above quote hence implicitly means that the probability that the observed warming since the mid-twentieth century was not predominantly caused by human influence but by natural factors is roughly 1:20. Based on the current state of knowledge, that means that it is not yet possible to fully rule out that natural factors were the main causes of the observed global warming. This probability of 1:20, as well as all the probabilities associated with the numerous causal claims that can be found in the past and present climate literature and are summarized in AR5, is a critical quantity that is prone to affect the way in which climate change is apprehended by citizens and decision makers, and thereby to affect decisions on the matter. It is thus of interest to examine the method followed to derive these probabilities and, potentially, to improve it.

Correspondence between language and probabilities in IPCC calibrated terminology (Mastrandrea et al. 2010).

*cause*and, incidentally, of the

*probability to have caused*that we in fact wish to quantify. Hence, beyond these general guidance principles, the actual derivation of these probabilities is left to some extent to the interpretation of the practitioner. In practice, causal attribution has usually been performed by using a class of linear regression models (Hegerl and Zwiers 2011):

*y*is regarded as a linear combination of

*p*externally forced response patterns

*ε*represents internal climate variability and observational error (all variables are vectors of dimension

*n*). The regression coefficient

*f*. After inference and uncertainty analysis, the value of each coefficient

*f*has caused the observed change

*y*) is denoted hereafter

^{1}:

Our proposal to tackle this objective is anchored in a coherent theoretical corpus of definitions, concepts, and methods of general applicability that has emerged over the past three decades to address the issue of evidencing causal relationships empirically (Pearl 2009). This general framework is increasingly used in diverse fields (e.g., in epidemiology, economics, and social science) in which investigating causal links based on observations is a central matter. Recently, it has been introduced in climate science for the specific purpose of attributing weather- and climate-related extreme events (Hannart et al. 2016), which we refer to simply as “extreme events” hereafter. The latter article gave a brief overview of causal theory and articulated it with the conventional framework used for the attribution of extreme events, which is also an important topic in climate attribution. In particular, Hannart et al. (2016) showed that the key quantity referred to as the fraction of attributable risk (FAR) (Allen 2003; Stone and Allen 2005), which buttresses most extreme event attribution (EA) studies, can be directly interpreted within causal theory.

However, Hannart et al. (2016) did not address how to extend and adapt this theory in the context of the attribution of climate changes occurring on longer time scales. Yet, a significant advantage of the definitions of causal theory is precisely that they are relevant no matter the temporal and spatial scale. For instance, from the perspective of a paleoclimatologist studying Earth’s climate over the past few hundred millions of years, global warming over the past 150 years can be considered as a climate event. As a matter of fact, the word “event” is used in paleoclimatology to refer to “rapid” changes in the climate system, but ones that may yet last centuries to millennia. Where to draw the line is thus arbitrary: one person’s long-term trend is another person’s short-term event. It should therefore be possible to tackle causal attribution within a unified methodological framework based on shared concepts and definitions of causality. Doing so would allow us to bridge the methodological gap that exists between EA and trend attribution at a fundamental level, thereby covering the full scope of climate attribution studies. Such a unification would present in our view several advantages: enhancing methodological research synergies between D&A topics, improving the shared interpretability of results, and streamlining the communication of causal claims—in particular when it comes to the quantification of uncertainty, that is, of the probability that a given forcing has caused a given observed phenomenon.

Here, we adapt some formal definitions of causality and probability of causation to the context of climate change attribution. Then, we detail technical implementation under standard assumptions used in D&A. The method is finally illustrated on the warming observed over the twentieth century.

## 2. Causal counterfactual theory

While an overview of causal theory cannot be repeated here, it is necessary for clarity and self-containedness to highlight its key ideas and most relevant concepts for the present discussion.

Let us first recall the so-called counterfactual definition of causality by quoting the eighteenth-century Scottish philosopher David Hume: “We may define a cause to be an object followed by another, where, if the first object had not been, the second never had existed.” In other words, an event *E* (*E* stands for effect) is caused by an event *C* (*C* stands for cause) if and only if *E* would not occur were it not for *C*. Note that the word *event* is used here in its general, mathematical sense of a *subset* of a sample space *E* would have occurred in a hypothetical world, termed counterfactual, in which the event *C* would not have occurred. The fundamental approach of causality that is implied by this definition is still entirely relevant in the standard causal theory. It may also arguably be connected to the guidance principles of the conventional climate change attribution framework and to the optimal fingerprinting models, in a qualitative manner. The main virtue of the standard causality theory of Pearl consists in our view in formalizing precisely the above qualitative definition, thus allowing for sound quantitative developments. A prominent feature of this theory consists in first recognizing that causation corresponds to rather different situations and that three distinct facets of causality should be distinguished: (i) necessary causation, where the occurrence of *E* requires that of *C* but may also require other factors; (ii) sufficient causation, where the occurrence of *C* drives that of *E* but may not be required for *E* to occur; and (iii) necessary and sufficient causation, where (i) and (ii) both hold. The fundamental distinction between these three facets can be visualized by using the simple illustration shown in Fig. 1.

*C*and

*E*, and where the notation

*intervention*is applied to the system under causal investigation. For instance PS, the

*probability of sufficient causation*, reads from the above that the probability that

*E*occurs when

*C*is interventionally forced to occur, conditional on the fact that neither

*C*nor

*E*was occurring in the first place. Conversely PN, the

*probability of necessary causation*, is defined as the probability that

*E*would not occur when

*C*is interventionally forced to not occur, conditional on the fact that both

*C*and

*E*were occurring in the first place. While we omit here the formal definition of the intervention

*in silico*experimentation: the implications of this constraint are discussed further.

*factual*probability of the event

*E*in the real world where

*C*did occur and

*counterfactual*probability in the hypothetic world as it is would have been had

*C*not occurred. One may easily verify that Eq. (4) holds in the three examples of Fig. 1 by assuming that the switches are probabilistic and exogenous. In any case, under such circumstances, the causal attribution problem can thus be narrowed down to computing an estimate of the probabilities

*p*. The latter only requires two experiments: a factual experiment

Each of the three probabilities PS, PN, and PNS has different implications depending on the context. For instance, two perspectives can be considered: (i) the *ex post* perspective of the plaintiff or the judge who asks “does *C* bear the responsibility of the event *E* that did occur?” and (ii) the *ex ante* perspective of the planner or the policymaker who instead asks “what should be done w.r.t. *C* to prevent future occurrence of *E*?”. It is PN that is typically more relevant to context (i) involving legal responsibility, whereas PS has more relevance for context (ii) involving policy elaboration. Both these perspectives could be relevant in the context of climate change, and it thus makes sense to trade them off. Note that PS and PN can be articulated with the conventional definition recalled in introduction. Indeed, the “*demonstration that the change is consistent with* (…)” implicitly corresponds to the idea of sufficient causation, whereas “(…) *is not consistent with* (…)” corresponds to that of necessary causation. The conventional definition therefore implicitly requires a high PS and a high PN to attribute a change to a given cause.

PNS may be precisely viewed as a probability that combines necessity and sufficiency. It does so in a conservative way since we have by construction that

## 3. Probabilities of causation of climate change

We now return to the question of interest: for a given forcing *f* and an observed evolution of the climate system *y*, can *y* be attributed to *f*? More precisely, what is the probability *f* has caused *y*? We propose to tackle this problem by applying the causal counterfactual theory to the context of climate change, and more specifically by using the three probabilities of causation PN, PS, and PNS recalled above. This section shows that it can be done to a large extent similarly to the approach of Hannart et al. (2016) for EA. In particular, as in EA, the crucial question to be answered as a starting point consists of narrowing down the definitions of the cause event *C* and of the effect event *E* associated with the question at stake—where the word *event* is used here in its general mathematical sense of subset.

### a. Counterfactual setting

For the cause event *C*, a straightforward answer is possible: we can follow the exact same approach as in EA by defining *C* as “presence of forcing *f*” (i.e., the factual world that occurred) and *f*” (i.e., the counterfactual world that would have occurred in the absence of *f*). Indeed, forcing *f* can be switched on and off in numerical simulations of the climate evolution over the industrial period, as in the examples of Fig. 1 and as in standard EA studies. Incidentally, the sample space *f*, including the observed one *y*. In other words, all forcings other than *f* are held constant at their observed values as they are not concerned by the causal question.

In practice and by definition, the factual runs of course always correspond to the historical experiment (HIST), using the Climate Model Intercomparison Project’s (CMIP) terminology as described by Taylor et al. (2012). The counterfactual runs are obtained from the same setting as historical but switching off the forcing of interest. For instance, if the forcing consists of the anthropogenic forcing then the counterfactual runs correspond to the historicalNat (NAT) experiment, that is, _{2} forcing, then the counterfactual runs corresponds to the “all except CO_{2}” experiment. However, no such runs are available in CMIP5 (https://cmip.llnl.gov/cmip5/docs/historical_Misc_forcing.pdf; see section 6 for discussion). Last, it is worth underlining that the historicalAnt experiment, which combines all anthropogenic forcings, thus corresponds to the counterfactual setting associated with the natural forcings. Therefore, runs from the historicalAnt experiment are relevant for the attribution of the natural forcings only; they are not relevant for the attribution of the anthropogenic forcings under the present counterfactual causal theory.

These definitions of *C* and *f* only), the latter experiments are required to be counterfactual (i.e., all forcings except *f*). We elaborate further on this remark in section 6.

### b. Balancing necessity and sufficiency

*E*, we propose to follow the same approach as in EA, where

*E*is usually defined based on an ad hoc climatic index

*Z*exceeding a threshold

*u*:

*E*implies choosing an appropriate variable

*Z*and threshold

*u*that reflect the focus of the question while keeping in mind the implications of the balance between the probabilities of necessary and sufficient causation. We now illustrate this issue and lay out some proposals to address it.

_{2}emissions caused global warming?”. Following the above, the event “global warming” may be loosely defined as a positive trend on global Earth surface temperature, that is,

*Z*is the global surface temperature linear trend coefficient and the threshold

*u*is zero. In that case,

*E*nearly always occurs in the factual world

*global warming*as a trend exceeding an intermediate value

*Z*exceeding the optimal threshold

Probabilities of causation in three different climate attribution situations: (a)–(c): factual PDF (red line) and counterfactual PDF (blue line) of the relevant index *Z*, observed value *z* of the index (vertical black line) and (d)–(f): PN, PS, and PNS for the event *u*, showing attribution of (left) the Argentinian heatwave of December 2013, (middle) twentieth-century temperature change, and (right) precipitation change over the satellite era (Marvel and Bonfils 2013).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Probabilities of causation in three different climate attribution situations: (a)–(c): factual PDF (red line) and counterfactual PDF (blue line) of the relevant index *Z*, observed value *z* of the index (vertical black line) and (d)–(f): PN, PS, and PNS for the event *u*, showing attribution of (left) the Argentinian heatwave of December 2013, (middle) twentieth-century temperature change, and (right) precipitation change over the satellite era (Marvel and Bonfils 2013).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Probabilities of causation in three different climate attribution situations: (a)–(c): factual PDF (red line) and counterfactual PDF (blue line) of the relevant index *Z*, observed value *z* of the index (vertical black line) and (d)–(f): PN, PS, and PNS for the event *u*, showing attribution of (left) the Argentinian heatwave of December 2013, (middle) twentieth-century temperature change, and (right) precipitation change over the satellite era (Marvel and Bonfils 2013).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Let us now consider the question “Did anthropogenic CO_{2} emissions cause the Argentinian heatwave of December 2013?” (Hannart et al. 2015). Here, the event can be defined as *Z* is surface temperature anomaly averaged over an ad hoc space–time window. Like in the previous case, the causal evidence again shifts from necessary and not sufficient (Figs. 2a,d) when *u* is equal to the observed value of the index *z* = 24.5°C (unusual event in both worlds but much more so in the counterfactual one) to sufficient and not necessary when *u* is small (usual event in both worlds but much more so in the factual one). Like in the previous case, a possible approach here would be to balance both quantities by maximizing PNS in *u* as in Eq. (6). However, this would lead here to a substantially lower threshold that no longer reflects the rare and extreme nature of the event “heatwave” under scrutiny. Furthermore, this would yield a well-balanced but fairly low level of causal evidence *u* matching with the observed value *z* should be used in order to maximize it. In contrast with extreme events, long-term changes are prone to be associated with much powerful causal evidence that simultaneously involves necessary and sufficient causation, and may yield high values for PN, PS, and PNS. PNS is thus an appropriate summary metric to consider for the attribution of climate changes, in agreement with D&A guidance principles (Hegerl et al. 2010). An optimal intermediate threshold can be chosen by maximizing PNS.

### c. Building an optimal index

In the above example where *global warming* is the focus of the question, the variable of interest *Z* to define the event can be considered as implicitly stated in the question, insofar as the term “global warming” implicitly refers to an increasing trend on global temperature. However, in the context of climate change attribution, we often investigate the cause of “an observed change *y*” with no precise a priori regarding the characteristics of the change that are relevant w.r.t. causal evidencing. Furthermore, *y* may be a large dimensional space–time vector. Thus the definition of the index *Z* in this case is more ambiguous.

We argue that in such a case, the physical characteristics of *y* that are implicitly considered relevant to the causal question are precisely those that best enhance the existence of a causal relationship in a PNS sense. This indeed corresponds to the idea of “fingerprinting” used thus far in climate change attribution studies (as well as in criminal investigations, hence the name): we seek a fingerprint—that is, a distinctive characteristic of *y* that would never appear in the absence of forcing *f* (i.e., *p* and

As an illustration, Marvel and Bonfils (2013) focus on the attribution of changes in precipitation, and subsequently address the question “Has anthropogenic forcing caused the observed evolution of precipitation at a global level?”. Arguably, this study illustrates our point in the sense that it addresses the question by defining a *fingerprint* index *Z* that aims precisely at reflecting the features of the change in precipitation that are thought to materialize frequently (if not systematically) in the factual world and yet are expected to be rare (if not impossible) in the counterfactual one, based on physical considerations. In practice, the index *Z* defined by the authors consists of a nondimensional scalar summarizing the main spatial and physical features of precipitation evolution w.r.t. dynamics and thermodynamics. The factual and counterfactual PDFs of *Z* are then derived from the HIST and NAT runs respectively (Fig. 2c). From these PDFs, one can easily obtain an optimal threshold *Z* by applying Eq. (6) (Fig. 2f). This yields *have about as likely as not caused the observed evolution of precipitation.*

*Z*that aims at maximizing PNS. However, a quantitative approach can also help in order to define

*Z*optimally, and to identify the features of

*y*that best discriminate between the factual and counterfactual worlds. Indeed, the qualitative, physical elicitation of

*Z*may be difficult when the joint evolution of the variables at stake is complex or not well understood a priori. Furthermore, one may also wish to combine lines of evidence by treating several different variables at the same time in

*y*(i.e., precipitation and temperature; Yan et al. 2016). Let us introduce the notation

*Y*is the space–time vectorial random variable of size

*n*of which observed realization is

*y*, and

*ϕ*is a mapping from

*ϕ*among the set all possible indexes

*optimal fingerprint*w.r.t. forcing

*f*. The maximization performed in Eq. (7) also suggests that our approach shares some similarity with the method of Yan et al. (2016), insofar as the variables of interest are in both cases selected mathematically by maximizing a criterion that is relevant for attribution [i.e., potential detectability in Yan et al. (2016); PNS in the present article] rather than by following qualitative, physics- or impact-oriented, considerations.

## 4. Implementation under the standard framework

We now turn to the practical aspects of implementing the approach described in section 3 above, based on the observations *y* and on climate model experiments. We detail these practical aspects in the context of the standard framework briefly recalled in section 1, namely multivariate linear regression under a Gaussian setting. Note that the assumptions underlying the latter conventional framework could be challenged (e.g., pattern scaling description of model error and Gaussianity). However, the purpose of this section is not to challenge these assumptions. It is merely to illustrate in detail how these assumptions can be used within the general causal framework proposed. Furthermore, the details of the mathematical derivation shown in this subsection cannot be covered exhaustively here in order to meet the length constraint. However, some important steps of the derivation are described in appendix A, and the complete details and justification thereof can be found in the references given in the text.

### a. Generalities

*p*and

*ϕ*and

*u*. For this purpose, it is convenient to derive beforehand the factual and counterfactual PDFs of the random variable

*Y*, denoted

*μ*and

*in silico*experimentation as the only option. While the increasing realism of climate system models renders such an

*in silico*approach plausible, it is clear that modeling errors associated to their numerical and physical imperfections should be taken into account into

*μ*,

Structural chart of the statistical model introduced in section 4, showing the underlying hierarchy of parameters (i.e., unobserved quantities; circles) and data used for inference (i.e., observed quantities; squares).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Structural chart of the statistical model introduced in section 4, showing the underlying hierarchy of parameters (i.e., unobserved quantities; circles) and data used for inference (i.e., observed quantities; squares).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Structural chart of the statistical model introduced in section 4, showing the underlying hierarchy of parameters (i.e., unobserved quantities; circles) and data used for inference (i.e., observed quantities; squares).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

### b. Model description

*Y*is Gaussian with mean

*β*thereby aim at representing the error associated to models, we prefer to treat

*β*as a random variable instead of a fixed parameter to be estimated. The latter factors are also assumed to be Gaussian:

*β*is

*ω*is a scalar parameter that will be determined further in this section. Combining Eqs. (10) and (11) yields (see appendix A)

*Y*to the covariance of

*Y*. We believe such a mean-covariance translation is relevant here, since the pattern scaling assumption is meant to represent a source of uncertainty. Furthermore, the covariance

*Y*, denoted

*μ*, is equal to the sum of the latter individual responses. Under the additivity assumption prevailing in the conventional framework,

*μ*thus corresponds to the model response under the scenario where the

*p*forcings are present. Hence,

*μ*can be obtained by estimating directly the combined response as opposed to estimating the individual responses

*μ*is indeed advantageous from a sampling error standpoint, as will be made clear immediately below.

*Y*in Eq. (12) involves three quantities,

*μ*,

*i*,

*i*; and

*r*is the number of combined forcings runs. Combining Eqs. (12) and (13), and after some algebra, this becomes (appendix A)

*μ*and

*t*distribution with mean

*μ*, variance

*ν*degrees of freedom. Equation (17) implies that taking into account the sampling uncertainty on

*Y*. Instead, it only affects the shape of the PDF of

*Y*, which has thicker tails than the Gaussian distribution. With these parameterizations, our model for

*Y*is thus a parametric Student’s

*t*model with two parameters

*y*based on likelihood maximization. The log-likelihood of the model has the following expression:

*f*, this procedure thus yields the PDF of

*Y*in the factual world:

*f.*” Some changes also need to be made regarding the covariance. Indeed, since forcing

*f*is absent in the counterfactual world, the model error covariance component

*f*, must be taken out of the covariance. Furthermore, if the number of counterfactual runs

*r*, the sampling uncertainty

*μ*also has to be modified. The PDF of

*Y*in the counterfactual world can thus be written

*f*is anthropogenic forcing, the counterfactual experiment NAT is usually available in CMIP runs, allowing for a straightforward derivation of

_{2}is the forcing of interest, the counterfactual response to all forcings except CO

_{2}emissions can be approximated by subtracting the CO

_{2}individual response

### c. Derivation of the probabilities of causation

*Y*in hand, an approximated solution to the maximization of Eq. (7) can be conveniently obtained by linearizing

*ϕ*, yielding a closed mathematical expression for the optimal index

*Y*onto the vector

*G*and

*N*random realizations of

*Y*from the Student

*t*distributions

*t*as a compound Gaussian chi-squared distribution. Samples of

*Z*are then immediately obtained by projecting onto

### d. Reducing computational cost

*y*is large, the above described procedure can become prohibitively costly if applied straightforwardly, due to the necessity to derive the inverse and determinant of

*n*, the matrix

*p*and matrix

## 5. Illustration on temperature change

Our methodological proposal is applied to the observed evolution of Earth’s surface temperature during the twentieth century, with the focus being restrictively on the attribution to anthropogenic forcings. More precisely, *y* consists of a spatial–temporal vector of size *n* = 54, which contains the observed surface temperatures averaged over 54 time–space windows. These windows are defined at a coarse resolution: Earth’s surface is divided into six regions of similar size (three in each hemisphere) while the period 1910–2000 is divided into nine decades. The decade 1900–10 is used as a reference period, and all values are converted to anomalies w.r.t. the first decade. The HadCRUT4 observational dataset (Morice et al. 2012) was used to obtain *y*. With respect to climate simulations, the runs of the IPSL-CM5A-LR model (Dufresne et al. 2013) for the NAT, ANT, HIST, and PIcontrol experiments were used (see appendix C for details) and converted to the same format as *y* after adequate space–time averaging.

Following the procedure described in section 4, we successively derived the estimated factual response *r* HIST runs, the estimated counterfactual response

An assessment of the relative importance of the four components of uncertainty was obtained by deriving the trace of each component (i.e., the sum of diagonal terms) normalized to the trace of the complete covariance. Climate variability is found to be the dominant contribution, followed by model uncertainty, observational uncertainty, and sampling uncertainty (not shown). The split between model and observational uncertainty is to some extent arbitrary as we have no objective way to separate them based only on *y*; that is, the model could be equivalently formulated as

The optimal vector *Y* are captured by this optimal mapping, the coefficients

Illustration of twentieth-century temperature change: optimal mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Illustration of twentieth-century temperature change: optimal mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Illustration of twentieth-century temperature change: optimal mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

*u*. The maximum of PNS determines the desired probability of causation:

*have virtually certainly caused the observed evolution of temperature*, according to our approach. More precisely, the probability that the observed evolution of temperature is not caused by anthropogenic forcings is then one in ten thousand (1:10 000) instead of one in twenty (1:20). Therefore, the level of causal evidence found here is substantially higher than the level assessed in the IPCC report. This discrepancy will be discussed in section 6.

Illustration of twentieth-century temperature change: results. (a) Factual PDF (red line) and counterfactual PDF (blue line) of the global mean index, observed value (thin vertical black line), and PNS as a function of the threshold *u* (thick black line). (b) As in (a), but for the space–time pattern index. (c) Scatterplot of factual (red dots) and counterfactual (blue dots) joint realizations of the global mean index (horizontal axis) and of the space–time pattern index (vertical axis). (d) As in (a), but for the optimal index

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Illustration of twentieth-century temperature change: results. (a) Factual PDF (red line) and counterfactual PDF (blue line) of the global mean index, observed value (thin vertical black line), and PNS as a function of the threshold *u* (thick black line). (b) As in (a), but for the space–time pattern index. (c) Scatterplot of factual (red dots) and counterfactual (blue dots) joint realizations of the global mean index (horizontal axis) and of the space–time pattern index (vertical axis). (d) As in (a), but for the optimal index

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Illustration of twentieth-century temperature change: results. (a) Factual PDF (red line) and counterfactual PDF (blue line) of the global mean index, observed value (thin vertical black line), and PNS as a function of the threshold *u* (thick black line). (b) As in (a), but for the space–time pattern index. (c) Scatterplot of factual (red dots) and counterfactual (blue dots) joint realizations of the global mean index (horizontal axis) and of the space–time pattern index (vertical axis). (d) As in (a), but for the optimal index

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

## 6. Discussion

### a. Comparison with previous statements

The probabilities of causation obtained by using our proposal may depart from the levels of uncertainty asserted by the latest IPCC report, and/or by previous work. For instance, when *y* corresponds to the evolution of precipitation observed over the entire globe during the satellite era (1979–2012), we have shown in section 3 that, using the dynamic–thermodynamic index built by Marvel and Bonfils (2013), the associated probability of causation *likely* implicitly means

In contrast with the situation prevailing for precipitation, when *y* corresponds to the observed evolution of Earth’s surface temperature during the twentieth century, and in spite of using a very coarse spatial resolution, we found a probability of causation

First, the probability of causation defined in our approach is of course sensitive to the assumptions that are made on the various sources of uncertainty, all of which are here built into

PNS as a function of the inflation factor applied to all uncertainty sources: global mean alone (light green line), space–time pattern (dark green line), and total (thick black line).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

PNS as a function of the inflation factor applied to all uncertainty sources: global mean alone (light green line), space–time pattern (dark green line), and total (thick black line).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

PNS as a function of the inflation factor applied to all uncertainty sources: global mean alone (light green line), space–time pattern (dark green line), and total (thick black line).

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

Besides the effect of inflating the individual variances, it is important to note that the probability of causation may also be greatly reduced when the correlation coefficients of the covariance

*.*” More precisely, the inference on the scaling factors

*β*and the associated uncertainty quantification are obtained by projecting the observation

*y*as well as the patterns

To assess whether or not these theoretical remarks hold in practice, we revisited our illustration and quantified the impact on

As in Fig. 5, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

As in Fig. 5, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

As in Fig. 5, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

As in Fig. 4, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

As in Fig. 4, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

As in Fig. 4, but for the mapping

Citation: Journal of Climate 31, 14; 10.1175/JCLI-D-17-0304.1

### b. Counterfactual experiments

Our methodological proposal has an immediate implication w.r.t. the design of standardized CMIP experiments dedicated to D&A: a natural option would be to change the present design “forcing *f* only” into a counterfactual design “all forcings except *f.*” Indeed, *μ* (i.e., historical experiment) and the counterfactual response *f* experiment). Under the assumption that forcings do not interact with one another and that the combined response matches with the sum of the individual responses, the difference *f* only experiment). While this hypothesis is well established for temperature at large scale (Gillett et al. 2004), it appears to break down for other variables (e.g., precipitation; Shiogama et al. 2013) or over particular regions (e.g., the southern extratropics; Morgenstern et al. 2014) where forcings appear to significantly interplay. Such a lack of additivity would inevitably damage the results of the causal analysis. It is thus important in our view to better understand the domain of validity of the forcing-additivity assumption and to evaluate the drawbacks of the present “one forcing only” design versus its advantages. Such an analysis does require “forcing *f* only” experiments, but also “all forcings except *f*” experiments in order to allow for comparison. This effort would hence justify including in the Detection and Attribution Model Intercomparison Project (DAMIP) set of experiments an “all forcings except *f*” experiment—which is presently absent even in the lowest priority tier thereof—at least for the most important forcings such as anthropogenic CO_{2}.

### c. Benchmarking high probabilities

Section 5 showed that the proposed approach may sometimes yield probabilities of causation that are very close to one. How can we communicate such low levels of uncertainty? This question arises insofar as the term “virtual certainty” applies as soon as PNS exceeds 0.99 under the current IPCC language (Table 1). Thus, this terminology would be unfit to express in words a PNS increase from 0.99 to 0.9999, say—even though such an increase corresponds to a large reduction of uncertainty by a factor of 100. One option to address this issue is to use instead the uncertainty terminology of theoretical physics, in which a probability is translated into an exceedance level under the Gaussian distribution, measured in numbers of *σ* from the mean (where *σ* denotes standard deviation), that is, *F* the CDF of the standard Gaussian distribution. Under such terminology, “virtual certainty” thus corresponds to a level of uncertainty of 2.3*σ*, while *σ*. It is interesting to note that the level of uncertainty officially requested in theoretical physics to corroborate a discovery as such (e.g., the existence of the Higgs boson) is 5*σ*. By applying such standards, one may actually consider that

### d. Alternative assumptions

The mathematical developments of section 4 are but an illustration of how our proposed causal approach, as framed in section 3, can be implemented when one uses the conventional assumptions of pattern scaling and Gaussianity associated to the standard linear regression setting. In that sense, section 4 thus shows that the proposed causal framing is perfectly compatible with the conventional linear regression setting: it should be viewed as an extension of, rather than an alternative to, the latter setting. Nevertheless, it is important to underline that the application of the causal framework of section 3 is by no means restricted to the conventional linear regression setting. One may, for instance, challenge some aspects of the latter (e.g., the pattern scaling description of model error) and formulate an alternative parameterization of the covariance

### e. Attribution as a classification problem

Last, it should be noted that the maximization of Eq. (7) can be viewed as a binary classification problem. Indeed, as illustrated in Fig. 5, solving Eq. (7) is equivalent to building a function of observations that allows us to optimally discriminate between two “classes”: the factual class and the counterfactual class. Under this perspective, PNS is related to the percent of correct classification decisions made by the classifier and is thus a measure of its skill.

Viewing the fingerprinting index

## 7. Summary and conclusions

We have introduced an approach for deriving the probability that a forcing has caused a given observed change. The proposed approach is anchored into causal counterfactual theory (Pearl 2009), which has been introduced recently in the context of event attribution (EA). We argued that these concepts are also relevant, and can be straightforwardly extended to the context of climate change attribution. For this purpose, and in agreement with the principle of *fingerprinting* applied in the conventional detection and attribution (D&A) framework, a trajectory of change is converted into an event occurrence defined by maximizing the causal evidence associated to the forcing under scrutiny. Other key assumptions used in the conventional D&A framework, in particular those related to numerical models error, can also be adapted conveniently to this approach. Our proposal thus allows us to bridge the conventional framework with the standard causal theory, in an attempt to improve the quantification of causal probabilities. Our illustration suggests that our approach is prone to yield a higher estimate of the probability that anthropogenic forcings have caused the observed temperature change, thus supporting more assertive causal claims.

## Acknowledgments

We gratefully acknowledge helpful comments by Aurélien Ribes and three anonymous reviewers. This work was supported by the French Agence Nationale de la Recherche grant DADA (AH, PN), and the grants LEFE-INSU-Multirisk, AMERISKA, A2C2, and Extremoscope (PN). The work of PN was completed during his visit at the IMAGE-NCAR group in Boulder, Colorado, United States.

## APPENDIX A

### Derivation of the PDF of *Y*

*β*:

*β*of the two terms under the integral in the right-hand side of Eq. (A1), it is clear that the PDF of the left-hand side is also Gaussian. Thus, instead of computing the above integral, it is more convenient to derive the mean and variance of this PDF by applying the rule of total expectation and total variance:

*μ*, we apply the Bayes theorem to derive the PDF of

*μ*conditional on the ensemble

*r*simulated responses in

*μ*and covariance

*μ*, we proceed by following the same reasoning as above for integrating out

*β*. Since the resulting PDF is clearly Gaussian, it suffices to derive its mean and variance:

**x**because

**x**appears in the covariance of

*μ*and

**x**. However, we use this time an informative conjugate prior on

*a*is a scalar parameter that drives the a priori variance. Furthermore, the mean and variance parameters

*n*-variate gamma function and

*Y*, in order to obtain

## APPENDIX B

### Optimal Index Derivation

*half-space*events, which are defined by

*ϕ*a vector of dimension

*n*, and

*u*is a threshold. Let us consider PNS as a function of

*ϕ*and

*u*:

*t*PDF from the calculations of section 4. Note that this approximation is made restrictively here for deriving an optimal index. Once this index is obtained, it is the then the true Student

*t*PDF of

*Y*that will be used to derive the desired value of PNS. Therefore, the implication of this approximation is to yield an index that is suboptimal and thereby underestimates the maximized value

*F*is the standard Gaussian CDF. The first-order condition in

*u*,

*ϕ*,

## APPENDIX C

### Data Used in Illustration

As in Hannart (2016), observations were obtained from the HADCRUT4 monthly temperature dataset (Morice et al. 2012), while GCM model simulations were obtained from the IPSL CM5A-LR model (Dufresne et al. 2013), downloaded from the CMIP5 database. An ensemble of runs consisting of two sets of forcings was used, the natural set of forcings (NAT) and the anthropogenic set of forcings (ANT) for which three runs are available in each case over the period of interest and from which an ensemble average was derived. On the other hand, a single preindustrial control run of 1000 years is available and was thus split into 10 individual control runs of 100 years. Temperature in both observations and simulations were converted to anomalies by subtracting the time average over the reference period 1960–91. The data were averaged temporally and spatially using a temporal resolution of 10 years. Averaging was performed for both observations and simulations by using restrictive values for which observations were nonmissing, for a like-to-like comparison between observations and simulations.

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The notation