A Bayesian network (BN) is a probabilistic graphical model representing probabilistic associations between random variables. The BN is defined by a structure and conditional probability tables (CPT). The structure is defined by nodes and arcs, representing the variables and the relationships between these variables, respectively (Fig. 3). The arrow indicates the influence direction from the parent node to the child node. Each node in the network is associated with a probability table. When a node has no parents, it is associated with a probability table containing a marginal probability distribution, since it is not conditioned on any other variables. When a node has parents, it is associated with a conditional probability table (CPT), which contains a conditional probability distribution that depends on the states of its parent nodes.

We focus on discrete BN, in which the product of the local probability distributions of each node results in the joint probability distribution function of all the variables X= { X1,…Xn in the graph Eq. (1): 1PX1,…Xn=∏i=1npXi|paXi, Where pa(Xi) are the parent nodes of Xi (Pearl, 1988). In this equation, pXi|paXi denotes the local conditional probability distribution of the variable Xi, given its parent nodes in the BN. The joint probability function links all the variables in the BN, therefore, any change can be propagated through the network. This means that the BN can be used in a forward mode where the changes are propagated from the child to parent nodes, but also in an inverse mode, from the parents to child nodes. This flexibility is very useful to explore multiple combinations of events.

Bayesian inference relies on Bayes' theorem to compute posterior probabilities. According to Bayes' theorem, the probability p of an event R given the evidence O is given by Eq. (2): 2pRi|Oj=p(Oj|Ri)p(Ri)p(Oj), The first term of the numerator is the likelihood, which is the probability of seeing the evidence given the event. The second term is the prior probability, which is the probability of the event before the evidence. The denominator, known as the normalization factor, is the marginal probability of the evidence.

Duvat et al. (2021) and the present study focused on four contrasting atoll islands of the Pacific and Indian Oceans (Fig. 1) Malé (Maldives), Fogafale (Tuvalu), Nolhivaranfaru (Maldives), and Tabiteuea (Kiribati). These islands were chosen to cover a contrasted range of geographical settings (Table 3). For example, they include the highly urbanized and densely populated island of Malé in the Maldives and the rural and mostly non-armored island of Tabiteuea in Kiribati.

The main climate change drivers affecting these islands include changes in atmospheric temperatures and rainfall patterns, sea-level rise, increasing sea surface temperature, coral bleaching, and ocean acidification. Other significant drivers are increased distant-source wave height, increased intensity of the most intense tropical cyclones and El Niño/La Niña events, and increased intensity and frequency of marine heat waves (Duvat et al., 2021).

Duvat et al. (2021) assessed the climate-related risk to future atoll habitability under two RCP scenarios. The assessment relied on a comprehensive literature review, available dataset analysis, and expert judgment. First, based on peer-reviewed scientific papers and recent IPCC reports, the authors identified five major Habitability Pillars (HPs): availability of sufficient land; supply of safe freshwater; supply of nutritious food from local and/or imported sources; access to safe settlements and infrastructure; and access to sustainable economic activities (Fig. 4). Then, they defined a set of Risk Criteria (RC) considered as the major factors contributing to the risk to Habitability Pillars in the study islands. For example, coastal erosion and flooding were identified as the main factors contributing to the risk to land.

After that, Duvat et al. (2021) scored from 1 to 5 each risk criterion based on the expected severity of additional climate risk under both RCP and time horizons. For example, if the risk of coastal erosion was expected to be low, the risk level was scored as 1. Conversely, if a low-lying island was expected to experience a high risk of flooding, the risk level was scored as 5. The authors provide a detailed rationale for each score in their supplement (Duvat et al., 2021). The overall risk to habitability was calculated by aggregating the scores of risk criteria and habitability pillars. Six risk levels were considered: undetectable (corresponding to no additional climate risk in the future compared to today's risk level), very low, low, moderate, high, and very high (corresponding to very high additional risk in the future compared to today's risk level). Moreover, each risk level was associated with a confidence level (very low to very high) based on evidence and the level of agreement (see Duvat et al., 2021 for details on the method).

In this work, we use the risk assessment database, including the scores and confidence levels, as input data for the BN model. Data pre-processing is detailed in the following sections.

In this experiment, we interrogated the BN model about the probability of risk to island habitability given an RCP scenario and a time horizon. Figure 7 shows the probability of risk to habitability under the RCP 2.6 and 8.5 scenarios in 2050 and 2090 for the four islands. In 2050, the risk to habitability is similar in Malé, Tabiteuea, and Nolhivaranfaru. The probability distribution is slightly shifted toward higher levels for RCP 8.5 compared to RCP 2.6, the risk is low to moderate in both scenarios. On the other hand, the risk to habitability is higher for Fogafale compared to other islands, the most likely outcome being moderate risk levels. The contrast between RCP 2.6 and 8.5 is also more marked than in Malé and Tabiteuea. In 2090, the risk to habitability shows more contrast between the two RCP scenarios. In the RCP 2.6 scenario, Malé, Tabiteuea, and Nolhivaranfaru are expected to experience low to moderate risk, and Fogafale is likely to experience moderate to high risk. Under the RCP 8.5 scenario, all islands may experience moderate to high-risk levels.

As expected, these results are consistent with the aggregated risk to island habitability from Duvat et al. (2021) (yellow line), though small differences can be observed. These differences are due to the fact that the aggregated risk in Duvat et al. (2021) results from the combination of their best estimates and does not incorporate confidence levels. Conversely, our approach interprets confidence levels as probability distributions around these best estimates. As shown in Fig. 6, these distributions can be skewed, resulting in slight shifts in the aggregated risk levels. Figure 7 shows that these differences are small, and arguably smaller than the uncertainties resulting from the expert judgment itself. The results include a density curve to smooth the peaks in the histograms. These peaks may be due to deterministic relationships. We populated specific conditional probability tables in a deterministic way, associating a probability of one with the aggregated risk level and a probability of zero with the other levels. The combinations matching the aggregated risk will have a higher probability resulting in peaks in the distribution.

This experiment aims to identify which risk levels could lead to adaptation limits in some islands. Adaptation limits are defined as “the point at which an actor's objectives (or systems needs) cannot be secured from intolerable risks through adaptive actions” (IPCC, 2022). These limits are related to the purple zone on burning embers diagrams, which show the changes in risk to humans and ecosystems as a function of global mean temperature (Zommers et al., 2020). The purple zone in these diagrams indicates a very high risk that can cause irreversible impacts and exceedance of adaptation limits. The burning embers diagrams have four risk categories: undetectable, low, moderate, and very high. In this work, we used the six risk categories defined by Duvat et al. (2021) in which very high risk to habitability is > 80.

We explored the possibility for islands to reach adaptation limits thresholds. To do this, we interrogated the model about the probability of the risk to habitability under different risk criteria levels. Figure 8 illustrates the outcomes for Malé and Fogafale under RCP 8.5 in 2090. In Malé, the results suggest that no risk criteria level could lead to exceeding the adaptation limits threshold (purple line). Conversely, in Fogafale, severe risk criteria (levels 4 or 5) may lead to exceeding this threshold. Even at low risk criteria, the risk to habitability remains high. On this island, implementing adaptation measures that focus only on a specific risk reduction may not be sufficient.

This analysis also allows for identifying the risk criteria with the major contribution to the risk to habitability. This is reflected by the magnitude of the distribution shift. In Malé and Fogafale, variations on the risk level of loss of settlements generate a slight distribution shift and therefore a slight impact on habitability. In contrast, increments in the flooding risk level have a more important contribution. These outcomes could be useful to identify major drivers and target possible adaptation strategies. For example, for these islands, flooding appears as the factor with a major contribution to the risk to habitability. In these cases, flooding reduction measures could be privileged.

The results for the other atoll islands are presented in Sect. S2.1.

As mentioned in Sect. 4.2, a very high risk to habitability could be reached at specific risk criteria levels. In this section, we use the BN to perform inverse analysis. We explore the conditions that lead to a very high risk to habitability by calculating the probability of each risk criteria level when the risk to habitability is very high.

Figure 9 shows the results for Fogafale under RCP 8.5 scenario in 2090. The probability distributions with and without the constraint of very high risk to habitability are represented by red and gray bars respectively. Under a very high risk to habitability, the probability distributions shift towards levels 4 and 5, indicating a correlation with severe risk criteria. This is the case for multiple variables including decrease in rainwater harvesting and desalination, flooding, reduced fisheries production, and loss of settlements and transport connectivity. The variations between the distributions with and without the habitability constraint reflect the impact of the risk to habitability node on the risk criteria nodes and vice-versa. For example, when we constrain the risk to habitability to very high levels, there is a significant shift in the probability distribution of flooding. This reflects that a variation in habitability will significantly impact flooding and inversely, a variation in flooding will impact the risk to habitability.

In all atoll islands, we observe the same correlation between high risk to habitability with severe risk criteria. These results are presented in Sect. S2.2.

In this experiment, the objective is to assess how the risk to habitability decreases when the key risk factors are reduced, especially those previously identified as major contributors to risk to habitability. To explore this, we assume different risk reduction scenarios that could be achieved through adaptation, such as managed retreat and the implementation of measures to reduce flooding and coastal erosion. The impact of such reductions can be evaluated by calculating the probability of risk to habitability given a risk criterion level ≤ 2 under the RCP = 8.5 in 2090. In both islands, the results show that reducing flooding has a major impact on the risk to habitability Fig. 10. In Malé, reducing flooding to moderate levels could reduce the risk to habitability from high to moderate. These results suggest that an adaptation policy focused on reducing flood risk can already provide substantial benefits to preserve the habitability of Malé Island. Yet, focusing only on flood risk would not be sufficient in all contexts. For example, Fig. 10 shows that reducing multiple risks is necessary for Fogafale to keep the risk to habitability at a moderate level.

The results for the other atoll islands are presented in Sect. S2.3.

BN allows us to derive a best estimate of the risk, but also to quantify the confidence by providing the 17th and 83rd percentile results (Fig. S9) interpreted as likely ranges (to refer to a probability of at least 66 %) according to the IPCC likelihood scale. By 2050, under RCP 2.6 and RCP 8.5 scenarios, the risk to habitability for Malé, Tabiteuea, and Nolhivaranfaru is likely to be low (Fig. 7). Conversely, in Fogafale, a higher risk is more likely. This is due to its very low elevation, limited coastal protection, and high exposure of habitats and critical infrastructures (Duvat et al., 2021). By 2090, the contrast between the RCP scenarios becomes more important. This is attributed to the divergence of climate projections during the second half of the 21st century, as well as the cumulative and cascading impacts of climate change (Duvat et al., 2021). Under the RCP 2.6 in 2090, the risk to habitability is likely to be low-to-moderate across all islands except Fogafale. In contrast, under the RCP 8.5, the risk to habitability is likely to increase from moderate to high or very high levels. In Malé, Duvat et al. (2021) attributed this increase to the expected increase of flooding and degradation of coastal protection. In rural islands, the expected high risk is associated with their high dependence on coastal ecosystem services for food (e.g., reef fish abundance), water supply (e.g., reduction of saltwater intrusion), land stabilization (e.g., sediment supply and wave impact reduction), and economic activities (e.g., tourism, fisheries, agriculture). Under the RCP 8.5, ecosystems are expected to decline due to the exceedance of critical thresholds, including the temperature threshold for tropical reef-building corals (Cooley et al., 2022), as well as regional bleaching thresholds. According to Duvat et al. (2021), risk to habitability is expected to be higher in rural islands due to their limited capacity to manage climate-related impacts through the implementation of coastal adaptation, technology, and imports, compared to urban islands.

The BN method has the advantage of enabling users to make complex requests. Typically, these complex requests involve queries that assume specific priors and compute the probabilities using the BN in reverse mode. In other words, the use of BN here simplifies the investigation of “what if” scenarios. The identification of major risk factors presented in Sect. 4.3 is an example of such inverse analysis, which could be useful to target high-risk factors and thereby identify potential adaptation measures. Figure 9 shows that severe flooding and loss of settlements are highly correlated with a very high risk to habitability. We used this information to target risk reduction scenarios that could potentially be achieved through adaptation measures proposed in the experiment 4.4. This analysis also highlights that a high risk to habitability can arise from the interaction of multiple factors. Therefore, integrated risk assessments and multi-risk adaptation strategies are needed.

Finally, the risk reduction scenario analysis highlights the potential of BNs as decision-support tools, as suggested by multiple authors (Ferreira et al., 2019; Jäger et al., 2018; Rachid et al., 2021). Duvat et al. (2021) evaluated climate risk considering the current level of adaptation, classified as moderate. Examples of currently implemented measures are food imports and water desalination to counter local resources decline, and hard protection to reduce flooding and coastal erosion. In our BN analysis, we explored different risk reduction scenarios that could potentially be achieved through context-specific adaptation measures. Examples include accommodation, vertical adaptation, ecosystem-based measures, and hard protections (Mycoo et al., 2022). However, we did not assess the effectiveness or the practicalities of their local implementation. Instead, this risk reduction analysis could be useful to target potential adaptation options. In future work, a wider range of local parameters including the elevation of the atoll island, and exposure to extreme events could be considered in the context of a wider framework involving social perception, economic feasibility, co-benefits, and trade-offs.

The presented method has limitations. The first one was the translation of expert judgments into probabilities. To do this, we selected specific beta distributions that best reflect the expert knowledge (best estimate and confidence level). We conducted several tests to select beta distributions dispersed enough to represent the expert knowledge without overfitting the model. However, the selection of a specific distribution introduces a degree of subjectivity (in addition to that related to the nature of the data). To evaluate the impact of our choices, we carried out a sensitivity test using an alternative set of beta distributions (detailed in the supplement). For this alternative set, we associated a higher weight to the confidence level. This is reflected by a higher probability associated with the risk level assessed by experts. For example, an assessed risk level associated with low confidence was represented by a probability of 30 % in the initial set, and a probability of 40 % in the alternative set. The sensitivity test shows slight differences between the two sets. The BN analysis using the alternative set of beta distributions results in less dispersed distributions and slightly different median values. However, we obtained the same risk categories in both cases, suggesting that the results are relatively robust against the probabilistic interpretation of expert judgments.

Other limitations are related to (1) the BN structure, (2) the discretization of the variables, and (3) inference methods. Regarding (1), we used a structure similar to that of Duvat et al. (2021), where the scores are essentially added to compute an aggregated risk level. Duvat et al. (2021) did not consider interactions between habitability pillars, and therefore, we did not include these interactions in the structure of our BN. The habitability pillar Risk to land has an impact on local freshwater and food land-based supply, settlements and infrastructure, and economic activities. Not considering these interactions could result in an underestimation of the risk to habitability. In future work, we will consider these relationships, but it must be considered that since BNs are based on direct acyclic graphs, their structure cannot explicitly account for positive or negative feedback. This can be a limitation in our approach if risk or adaptation problems with large positive or negative feedback are considered. Concerning (2), the discretization of continuous variables could be another source of uncertainty. Depending on the states of the variables, the discretization could lead to a loss of information (Rohmer, 2020). For example, the use of too large or too short categories may overlook subtle variations in the system, reducing the accuracy and relevance of the results. The use of hybrid BNs including continuous and discrete variables (Beuzen et al., 2018) could be helpful to minimize these uncertainties. Regarding (3), in our BN analysis, we used the likelihood weighting method for inference. This method has some limitations, including low precision for estimating low-probability scenarios (Scutari, 2010), which were not in the scope of this study. To improve computational efficiency, alternative sampling methods will be considered in future research work (Yuan and Druzdzel, 2006).