1. Introduction
A critical issue in climate change impact studies is the assessment of uncertainties associated with future projections. A comprehensive quantification of the different sources of uncertainty is necessary in order to obtain the range of possible future changes, their significance, and their value for adaptation. The identification of the most important uncertainty sources is also important for the allocation of research and development resources (Northrop and Chandler 2014).
Uncertainty sources include scenario uncertainty, model uncertainty, and climate internal variability. Scenario uncertainty corresponds to the uncertain future of greenhouse gas emissions. It is now usually evaluated with climate projections obtained for different representative concentration pathways (RCPs), obtained from various socioeconomic projections (van Vuuren et al. 2011). Model uncertainty is related to the imperfections of the models used to obtain the climate projections (climate models and subsequent impact models when sectorial declinations such as ecology, water resources, hydropower, and so on are considered). Indeed, many different models can be used to represent the Earth system, leading to different climate responses for the same forcing configuration. For a given emission scenario for instance, different climate responses are generally obtained at the scale of the planet from different general circulation models (GCMs) and, for a given scenario/GCM simulation, different climate responses are expected at a regional scale from different regional climate models (RCMs). In addition, for a given GCM/RCM simulation, an ensemble of impact models can be used to represent the uncertainty regarding sectorial processes such as the hydrological response at the catchment scale, snowpack evolution, etc. Model uncertainty is typically estimated using different simulation chains, combining for instance different climate, downscaling, and impact models (Stocker et al. 2013). Climate internal variability originates from the chaotic and nonlinear nature of the climate system (Deser et al. 2012). Unlike scenario and model uncertainty, which could potentially be reduced if our estimates of future emissions and our knowledge and representation of geophysical processes were improved, climate internal variability is purely stochastic and is irreducible (Hawkins and Sutton 2009; Lafaysse et al. 2014; Fatichi et al. 2016).
These different uncertainty sources are typically estimated from multiscenario multimodel ensembles (MMEs) of transient climate projections produced for some control and future periods. Various methods have been proposed for this, usually based on an analysis of variance (ANOVA). An important limitation of most existing ANOVA models applied to MMEs is due to the nature of climate projection datasets available for the analysis. Typically, given the high computational costs of GCMs and RCMs, a large number of possible scenario/GCM/RCM combinations may be missing from the MMEs. For instance, in the EURO-CORDEX dataset (Jacob et al. 2014; Kotlarski et al. 2014), 7 RCMs have been used to produce high-resolution projections over the European domain, from 5 CMIP5 GCMs and for 3 RCP scenarios (Taylor et al. 2012). From the 7 × 5 × 3 = 105 possible scenario/GCM/RCM combinations, only 17 were actually available for the analysis of Jacob et al. (2014). The number is increasing progressively but the matrix of GCM/RCM combinations will probably not fill up entirely by the end of this generation of the EURO-CORDEX dataset.
As indicated above, different approaches based on ANOVA analyses have been proposed to deal with these MMEs. The single-time approach consists in applying the ANOVA on projections available for one given projection lead time (Hingray et al. 2007; Yip et al. 2011; Paeth et al. 2017), independently from the projections available for other lead times. Provided that multiple members (i.e., replicates) are available for each simulation chain, the climate response of the chain is estimated from the multimember mean and its internal variability is estimated from the intermember dispersion. Model uncertainty components are then estimated from the dispersion between the climate responses of the different simulation chains. In the single-time approach, multiple runs are required for each simulation chain, for both the estimation of the chain’s climate response and internal variability. However, in the great majority of available MMEs, multiple GCM runs are only available for a small number of simulation chains. This multiple-run constraint often leads scientists to discard single-run GCM projections from the analysis. Only 6 GCMs out of 32 GCMs, for instance, are considered for this reason in Bracegirdle et al. (2014). Another approach to circumvent this constraint is to avoid separating the variations of the climate variable in the climate response from the fluctuations due to internal variability (e.g., Giuntoli et al. 2015). In this case, however, the contributions of model uncertainty and internal variability to the total uncertainty cannot be evaluated anymore. Another option that has been considered in the literature is to increase the size of MMEs using weather generator models (e.g., Lafaysse et al. 2014; Fatichi et al. 2016).
Alternatively, in the time series approach, the temporal variation of the climate response of any given chain is assumed to be gradual and smooth, the high-frequency variations of the time series being due to internal variability alone (e.g., Hawkins and Sutton 2009; Hingray and Saïd 2014; Reintges et al. 2017). For each simulation chain, the climate response is estimated with a trend model (e.g., polynomial functions) and the internal variability of the chain, considered constant throughout the time series, is estimated by the variance of the deviations from the climate response. As for the single-time approach, for any projection lead time, model uncertainty components are estimated from the dispersion between the climate responses. Thanks to the discrimination of the raw projections between internal variability and climate response, the time series approach can be applied when only a single member is available for some (or even all) simulation chains. It is thus not subject to the multiple-run constraint. In many MMEs configurations, especially when MMEs are noisy or when a small number of members is available, estimates of uncertainty components obtained with a time series approach are expected to be more precise than with the single-time approach (Hingray et al. 2019).
For both single-time and time series approaches, classical ANOVA methods cannot be applied with missing data (i.e., when several scenario/GCM/RCM combinations are not available). A drastic solution would be to retain only scenario/GCM/RCM runs for which all combinations are available. However, this would lead to a dramatic waste of information, reducing the dataset to very few combinations (e.g., 2 RCMs × 4 GCMs for the EURO-CORDEX dataset; see Verfaillie et al. 2018). Missing-data methods have been developed to reduce biases and increase the efficiency of the estimation in such cases. They use, for instance, an ad hoc data reconstruction algorithm (Déqué et al. 2007) or likelihood-based techniques (Little and Rubin 2014). Under the assumption that missing combinations follow the same decomposition of variance as the data actually at hand, Bayesian inference and data augmentation techniques (Tanner and Wong 1987) can be applied and present the advantage to propagate missing-data uncertainty. As an example, Tingley (2012) applies a Bayesian ANOVA method to infer climate anomalies and to handle missing spatial locations.
In the present study, we introduce a method allowing for the characterization of multiple uncertainty sources, namely scenario, GCM, and RCM uncertainties, as well as internal variability, from an incomplete dataset of climate projections. The impact of missing scenario/GCM/RCM combinations is explicitly treated using a Bayesian ANOVA framework and data augmentation. We show how missing combinations influence the uncertainty of the estimated ANOVA effects, and we compare these estimates to those obtained with a simple empirical estimation approach. This approach, named Quasi-Ergodic Analysis of Climate Projections Using Data Augmentation (QUALYPSO), is illustrated using MMEs of precipitation and temperature projections in four different French mountain massifs located in the Pyrenees and the Alps mountain ranges, at an elevation of 1500 m (Verfaillie et al. 2017, 2018).
Section 2 presents the pretreatments applied to projections, and details the Bayesian framework used for the estimation of the climate response and for the partition of the different uncertainty components. An application of the QUALYPSO method to MMEs of precipitation and temperature projections is presented in section 3. Section 4 discusses the estimation uncertainty and presents a comparison of Bayesian and direct estimates. Section 5 summarizes the results and provides some perspectives.
2. Methodology
We consider a given MME composed of different simulation chains where each chain corresponds to a given GCM/RCM combination for a given emission scenario. Let us note 
a. Estimation of the climate response
For any chain, the time evolution of the climate response to greenhouse gas emissions is assumed to be gradual and smooth, the higher-frequency variations of the time series being due to internal variability of the studied variable alone (the so-called quasi-ergodicity assumption; see Hingray and Saïd 2014). The climate response of a particular simulation chain for a given emission scenario is then estimated from the long-term trend of the climate projections. The internal variability corresponds to the variance of the deviations of projections from the estimated climate response.
The climate response 
b. Estimation of the climate change response
c. Decomposition of the climate change response
d. Bayesian inference
Let 
- The first term is the likelihood of available climate change responses
- The second term is the distribution associated with the missing climate change responses, given the ANOVA model parameters. As indicated in the introduction, we assume that the additive model (7) is valid for missing and nonmissing chains. Similarly to available climate change responses
- The third term is the joint prior distribution of the ANOVA model parameters, for which we consider independent priors, a classical nonrestrictive assumption:
The posterior distribution 
In this study, the posterior distributions of all unknown quantities (parameters and missing climate projections) are sampled sequentially using the Gibbs algorithm. After a burn-in period of 2000 samples, 50 000 draws are retained. Several tests have been proposed to test the convergence of multiple chains (Gelman and Rubin 1992; Brooks and Gelman 1998). In our case, a visual assessment of the chains (not presented here) shows that convergence is reached very quickly, after 100 iterations.
e. Total uncertainty and uncertainty components
3. Illustration of the Bayesian ANOVA
a. Data
The QUALYPSO method is applied to transient climate simulations of annual precipitation and annual mean temperature available for four French mountain massifs located in the Pyrenees (Pays-Basque and Cerdagne-Canigou) and in the Alps (Mont-Blanc and Haut-Var Haut-Verdon) (see Fig. 1). Total precipitation (rainfall and snow) refers to the mass of water per unit area (kg m−2), which can also be expressed as a water flux [i.e. volume per area and per unit of time (mm)]. These four massifs are a subset of the geographical delineation of French mountain regions for climatological purposes (Durand et al. 2009) and were chosen to maximize the contrast in meteorological and climatological conditions. Mont-Blanc is typical of the internal mountain ranges of the northern French Alps, near the Swiss and Italian borders. Haut-Var Haut-Verdon is typical of the southern French Alps. Pays-Basque is the westernmost massif of the French Pyrenees, at a short distance from the Atlantic Ocean, while Cerdagne-Canigou is the easternmost massif, close to the Mediterranean Sea. These climate projections include 26 simulations obtained with I = 2 RCP emission scenarios (RCP4.5 and RCP8.5; see van Vuuren et al. 2011) and 13 different combinations of J = 5 GCMs and K = 6 RCMs (see Table 1). Simulation chains are composed of historical runs for the periods 1950–2005 (for 12 chains), 1970–2005 (for 8 chains), or 1981–2005 (for 6 chains), and of future runs for the period 2006–2100 (2006–99 for the 6 chains starting in 1981). They have been produced from EUR-11-CORDEX projections (Jacob et al. 2014; Kotlarski et al. 2014) for different elevation bands of the French massifs, using the Adaptation of RCM Outputs to Mountain (ADAMONT) statistical adjustment method (Verfaillie et al. 2017, 2018) and the Système d’analyse fournissant des renseignements atmosphériques à la neige (SAFRAN) meteorological reanalysis as an observation dataset (Durand et al. 2009). In this paper, we consider projections obtained at 1500 m, a typical elevation representative of the mountain environment, allowing comparison between all the four massifs used here, given that this elevation level is common to the four massifs chosen.
Map of France with the locations of the four mountain massifs.
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
Combination of available GCM and RCM climate projections with scenarios RCP4.5 and RCP8.5. More information about the climate models can be found in Verfaillie et al. (2018). (Expansions of acronyms are available online at http://www.ametsoc.org/PubsAcronymList.)
b. Climate responses
The climate response function of each simulation chain is estimated with cubic smoothing splines (de Boor 1978), using the entire time series (i.e., starting in 1950, 1970, and 1981 and ending in 2100). The fit is obtained with the function smooth.spline in R software (https://www.r-project.org/). Further details can be found in Hastie and Tibshirani (1990, chapter 3.4). To obtain smooth trends, a large smoothing parameter is chosen (λ = 1). The corresponding number of degrees of freedom (knots) is 3.7 (79), 3.5 (72), and 3.35 (68) for runs starting in 1950, 1970, and 1981 (and ending in 2100), respectively. Figure 2 shows climate responses 
Climate projections 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
Figure 3 shows the deviations 
Deviations from the climate change response 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
c. Mean climate change response and posterior of the main effects
For each scenario/GCM/RCM combination and each year t, we obtain the climate change response in terms of absolute changes for temperature [Eq. (5)] and relative changes for precipitation [Eq. (6)], compared to the control year c = 1990. For each year t = 1990, …, 2011, we then obtain the posterior distributions of the mean climate change response function for each RCP scenario [
Decomposition of the effects contributing to the variance of the climate projections for absolute temperature changes (°C), and relative precipitation changes (unitless), compared to year 1990, for the Haut-Var Haut-Verdon massif. These results are available for the other massifs in Figs. S1–S3. Posterior distribution of (a) the mean climate change response function for each RCP scenario 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
For both variables, the climate model effects (GCMs or RCMs) logically start from zero for the control period and tend to increase with the projection lead time (GCMs effects actually tend to stabilize during the second part of the century). For the Haut-Var Haut-Verdon massif, for both variables, the effects due to RCMs are of the same order (of magnitude) as those due to GCMs.
For temperature, the main effects of the different GCMs and/or RCMs in 2100 are less than ±1°C. The most important GCM effects are obtained for HadGEM2-ES and for CNRM-CM5, which produce slightly larger and smaller warming than the other GCMs, respectively. Similarly, RCA4 warms up to 0.8°C more than the other RCMs and WRF 3.3.1.F up to 0.8°C less. Whatever the simulation chain, these effects are much less than the mean expected warming.
For precipitation, the main effects of GCMs and/or RCMs are up to ±10% in 2100. The most important effects are obtained for GCMs CNRM-CM5 (+10%) and IPSL-CM5A-MR (−10%) and for RCM WRF 3.3.1.F (+10%). These effects are roughly the same as the mean expected change. Similar results are obtained for the other massifs (see Figs. S1–S3). An exception is the Mont-Blanc massif, the alpine region with the highest elevations and altitude gradients, for which the dispersion between the RCM effects is larger than that the dispersion between the GCM effects.
Figure 5 shows the standard deviation σ(t) of the residual terms 
Standard deviation 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
d. Total uncertainty and uncertainty components
As mentioned previously, different types of uncertainty contribute to the total uncertainty of the climate projections, namely uncertainties related to RCP scenarios, GCMs, RCMs, residual variability, and internal variability. For each RCP scenario, the evolution of the corresponding total variance is presented in Figs. 6 and 7. We also present the fraction of total variance explained by each source of uncertainty.
Total uncertainty and fraction of total variance explained by each source of uncertainty for mean annual temperature changes (°C) compared to year 1990 as a function of time. Mean climate change response (white curve) for scenarios (a) RCP4.5 and (b) RCP8.5 [
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
As in Fig. 6, but for relative changes of annual precipitation (unitless).
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
For temperature, the main source of uncertainty is internal variability (in orange in Figs. 6 and 7) until midcentury, which is then exceeded by the uncertainty due to the RCP scenario. For all massifs, differences between the results obtained with RCP4.5 and RCP 8.5 are clear at the end of the century and scenario uncertainty grows up to 60% of the total uncertainty in 2100. Except for the Mont-Blanc massif, the part of total variance due to RCM and GCM uncertainty is no more than 30% and decreases down to less than 20% at the end of the century.
For precipitation, total uncertainty is large regardless of the projection lead time and mainly due to internal variability. Climate model uncertainty (GCMs and RCMs) increases with lead time but always remains less than 30% of total uncertainty (except for the Mont-Blanc massif). Uncertainty related to RCP scenarios remains small and even negligible compared to climate models uncertainty, especially for the Mont-Blanc massif. For this massif, the large fraction related to the RCM uncertainty is mainly explained by the strong departure of the RCM WRF 3.3.1.F, which is much “colder” and “wetter” than the other RCMs (see Figs. S1–S3). The large dispersion between RCM effects for both temperature and precipitation highlights the contribution of the regional model, which is determinant in mountainous regions.
Note that for both variables, the fraction of total variance due to the residual variability (in yellow in Figs. 6 and 7) is small. This confirms that first-order terms in the ANOVA model (7) explain the main part of the variability of the climate change responses, so that further interaction terms are not necessarily needed in the ANOVA model.
e. Significance of changes
The colored intervals in Figs. 6a, 6b, 7a, and 7b represent the total uncertainty, and correspond to the confidence interval of possible future changes at the 90% confidence level. A significant climate change realization is expected when zero (i.e., no change) lies outside this confidence interval.
The time of emergence of a significant warming (Giorgi and Bi 2009), defined here as the first future lead time for which zero lies outside the confidence interval, is found within the 2030–40 period for both emission scenarios, with the exception of the Mont-Blanc massif with the scenario RCP4.5. For precipitation, no significant change is observed, even at the end of the century. Precipitation that may be experienced for a given future period could therefore be higher or lower than what has been observed for the control period. This is mainly due to the large internal variability for this variable.
4. Discussion
a. Internal variability
In our application, the contribution of internal variability to total variability of the ensemble is large, for both variables and all regions. Indeed, in our case, internal variability includes a high-frequency (interannual) variability at the annual scale and a variability corresponding to lower frequencies (e.g., decadal scales), which could possibly be separated (Solomon et al. 2011; Seitola 2016). However, as shown by Vidal et al. (2016), for instance, a smaller contribution would have been obtained by applying QUALYPSO to time-slice averages instead of annual values (e.g., 20-yr mean precipitation instead of annual precipitation), as done in many studies (see, e.g., Yip et al. 2011; Lafaysse et al. 2014). Furthermore, it is important to note that spatial aggregation also removes an important part of the variability, which explains that regional projections lead to a larger internal variability than global means (Kendon et al. 2008; Hawkins and Sutton 2009).
b. Uncertainty of the estimation
The main output of Bayesian methods is the posterior distributions obtained for the different unknown quantities to be estimated. The distribution obtained for each quantity provides a direct assessment of the uncertainty associated to the estimation. We focus on uncertainty related to the estimation of ANOVA model parameters (i.e., mean climate change response, scenarios, and climate model effects). Figure 8 represents the standard deviation of the posterior distribution obtained for the mean climate change response of each RCP scenario, for the GCM and RCM effects. These standard deviations are related to the width of the credible intervals provided in Fig. 4 (the width of the 95% is about 4 times the standard deviation when posterior distributions are Gaussian). For all effects and for both temperature and precipitation, the estimation uncertainty roughly increases with time and directly follows, by construction, the evolution of the standard deviation σ(t) obtained for the residual errors of the ANOVA model [see Fig. 5 and Eqs. (A2), (A9), (A11), and (A13) in the appendix].
Uncertainty of the estimation (standard deviation of the posterior distribution) for (a) the mean climate change response for each RCP scenario 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
The magnitude of the estimation uncertainty depends mostly on the size and setup of the dataset available for the estimation. For climate model effects for instance, it depends on the number of available RCM/GCM combinations. As an illustration, the estimation uncertainty of the GCM effect is similar for all GCMs, except for IPSL-CM5A-MR for which it is significantly larger. Conversely to the other GCMs, IPSL-CM5A-MR was not used to drive the RCM CCLM 4.8.17, for which the effect is well estimated since it is available for four GCMs. Similarly, the same estimation uncertainty is obtained for the RCMs ALADIN 53, RACMO 2.2E, and REMO 2009. These three RCMs share the same configuration. They are driven by only one GCM, which has also been used to run the RCMs CCLM 4.8.17 and RCA 4. The estimation uncertainty is lower for RCMs CCLM 4.8.17 and RCA 4 as they are driven by 4 and 5 GCMs, respectively. The largest estimation uncertainty obtained for the RCM WRF 3.3.1.F is finally due to the fact that it was driven by the GCM IPSL-CM5A-MR only, which has the largest estimation uncertainty among all GCM effects.
As illustrated above, missing scenario/RCM/GCM combinations logically determine the estimation uncertainty of the different parameters of the analysis (climate model effects, mean climate change response, etc.). The data augmentation approach accounts for this and propagates the uncertainty due to missing data in the analysis.
c. Comparison of Bayesian and direct estimates
In numerous climate impact studies, the mean climate change response is estimated by the empirical mean of all available climate experiments. When many scenario/RCM/GCM combinations are missing, this empirical estimate might be significantly different from the mean response that would be obtained with the complete ensemble.
Let 
Mean climate change response for each RCP scenario 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
When the mean response is estimated by the empirical mean of available scenario/GCM/RCM combinations, each available climate experiment has the same weight in the estimation. However, some climate models are more represented than other ones. For example, RCMs CCLM 4.8.17 and RCA 4 concern 8 and 10 climate experiments, respectively, among the 
This configuration corresponds more or less to the MMEs presented previously for the temperature projections (see Fig. 4). In the present case, however, we consider a different set of available experiments, as indicated in Table 2. Clearly, this set is unbalanced in the sense that GCM1 and RCM1 are overrepresented. This situation is quite similar to real-world applications for which one or two GCMs/RCMs are more used than other existing climate models. Figure 10a shows the generated climate projections 
Combinations of available GCM and RCM climate projections for the synthetic experiment.
(a) Synthetic climate change projections 
Citation: Journal of Climate 32, 8; 10.1175/JCLI-D-18-0606.1
Bayesian estimates of the mean climate change response 
5. Conclusions and outlooks
This work presents the development and application of a Bayesian approach, named QUALYPSO, which uses data augmentation to assess the different sources of uncertainty in incomplete multiscenario multimodel ensembles (MMEs) of climate experiments. In a first step, the climate response of each available simulation chain is estimated with a trend model (i.e., cubic splines here) fitted to raw climate projections. Climate change responses can then be obtained and residuals from the climate change response are used to estimate the internal variability of the chain. The other uncertainty components of the projections (scenario uncertainty and climate model uncertainty) are estimated with a Bayesian ANOVA model applied to the climate change responses of available simulation chains. The ANOVA model provides an estimate of the mean climate change response of the MME, as well as an estimate of the main effects of the different emission scenarios and climate models (GCMs and RCMs). The different parameters of the ANOVA model and the missing quantities associated to the missing simulation chains are jointly estimated using data augmentation techniques. For illustration, we apply QUALYPSO to MMEs of climate projections (mean annual temperature and total annual precipitation) produced for four different French massifs at 1500-m elevation. Projections are available for 13 GCM/RCM combinations and two emission scenarios.
QUALYPSO presents many advantages over more classical estimation approaches. Along with the estimation of missing data, it provides an assessment of the estimation uncertainty and adequately propagates the uncertainty due to missing scenario/GCM/RCM combinations. With the explicit treatment of missing climate experiments, it is then expected to produce unbiased estimates of all parameters, in contrast to direct empirical estimates, typically obtained as the average over all available projections. The Bayesian approach also exploits all available climate experiments, avoiding a dramatic loss of information when standard single-time or time series approaches are applied (in this case, a classical solution is to select a complete subset of climate experiments). The QUALYPSO methodology can be applied to any kind of climate variable and any kind of MMEs of climate projections. Additional effects due to some impact model (e.g., a hydrological model or a snow model; see Vidal et al. 2016; Verfaillie et al. 2018) or to some other explanatory factor (e.g., spatial effects; see Geinitz et al. 2015; Tingley 2012) could also be easily included in the analysis.
A key step of QUALYPSO is the estimation of the climate response for each simulation chain. In the present analysis, the climate responses are modeled by cubic splines. Many alternatives are possible and have been considered in previous works. For example, Hawkins and Sutton (2009) and Hingray and Saïd (2014) apply fourth-order polynomial functions to temperature changes. Cubic splines are expected to provide a flexible and robust fit of the climate response. This estimation can also be improved using multiple runs of each chain, if available (e.g., Kendon et al. 2008; Deser et al. 2012; Hingray et al. 2019). In all cases, an improvement of the method would be to consider the estimated climate response of each chain as uncertain, and to account for this uncertainty in the Bayesian ANOVA approach.
This work was supported by the French National program LEFE (Les Enveloppes Fluides et l’Environnement) and by the French National Research Agency, via the CDP-Trajectories project, in the framework of the “Investissements d’avenir” program (ANR-15-IDEX-02). This study also benefited from the outcomes of the ADAMONT project (GICC program and ONERC) and of the Interreg project POCTEFA/Clim’Py. Irstea, IGE and CNRM/CEN are part of Labex OSUG@2020. Deborah Verfaillie’s work has been funded by the European project EUCP (H2020-SC5-2016-776613). We also thank Matthieu Lafaysse (Univ. Grenoble Alpes, Université de Toulouse, Météo-France, CNRS, CNRM, CEN, Grenoble, France), Samuel Somot, Aurélien Ribes and Michel Déqué (CNRM, Météo-France, CNRS, Université de Toulouse, Toulouse, France) and Wilfried Thuiller (LECA, Grenoble, France) for interesting discussions on this subject. We also thank the editor and two anonymous reviewers for their constructive comments, which helped us to improve the manuscript.
APPENDIX
Full Conditional Distributions for the Decomposition of Climate Change Response
As indicated in section 2, QUALYPSO applies a Bayesian approach in order to infer the different unknown quantities (parameters of the ANOVA model and missing scenario/RCM/GCM combinations). The joint posterior distribution of all unknown quantities is sampled using the Gibbs algorithm, which necessitates to sample iteratively the full conditional posterior distributions of all unknown quantities. In this section, we specify conjugate priors for the parameters, we provide the full conditional posteriors, and we propose values for the hyperparameters, following Tingley (2012). In the remainder of this section, 
a. Mean climate change response function μ
Note that the expression for 
b. Variance of the residual terms 
c. Effect of the GCM 
d. Effect of the RCM 
e. Effect of the RCP 
f. The missing values 
g. Hyperparameters
The standard choices for the hyperparameters are the following:
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