Introduction
Infrastructure investment is the engine of economic development in low-income countries. For South Sudan — recovering from decades of civil conflict and possessing one of the least developed road networks on the African continent — the efficient delivery of road infrastructure projects is a national priority directly linked to humanitarian access, food security, and economic integration (World Bank, 2021; MoRB, 2022). Yet the record of road proje ct delivery in South Sudan, and indeed across Sub-Saharan Africa more broadly, is characterised by persistent and substantial cost overruns and schedule delays. A review of 22 MoRB-managed primary road projects executed between 2005 and 2023, presented in this paper, reveals a mean cost overrun ratio of 1.46 (46% above budget) and a mean duration overrun ratio of 1.52 — statistics that are broadly consistent with the global infrastructure overrun literature (Flyvbjerg et al., 2018; Love et al., 2019) and the Africa-specific findings of Ahiaga-Dagbui et al. (2017). Understanding and managing project risk is the fundamental precondition for improving delivery outcomes. Risk management in construction projects encompasses risk identification, risk analysis (qualitative and quantitative), risk evaluation, risk treatment (mitigation), and risk monitoring. The dominant quantitative risk analysis tools currently used in the infrastructure sector — risk matrices, deterministic sensitivity analysis, and scalar Monte Carlo simulation — suffer from well-documented methodological limitations: risk matrices conflate ordinal and cardinal scales, fail to distinguish between correlated and independent risks, and produce inconsistent ordinal rankings (Cox, 2008; Duijm, 2015); scalar Monte Carlo simulation treats input risk variables as independent, underestimating tails of the joint distribution when risks are positively correlated (as is virtually always the case in construction projects); and none of these tools supports systematic Bayesian updating — the formal mechanism for incorporating new evidence as a project progresses through design, procurement, and construction phases. Bayesian Networks (BNs) — probabilistic graphical models that encode conditional independence relationships among variables as directed acyclic graphs (DAGs), with associated conditional probability tables (CPTs) defining the joint probability distribution — offer a theoretically coherent framework that addresses all these limitations simultaneously. BNs support: (i) explicit causal modelling of multi-variable risk interaction chains; (ii) exact probabilistic inference through the junction tree algorithm or approximate inference through sampling; (iii) systematic Bayesian updating as new evidence is observed at each project phase gate; (iv) both forward inference (diagnosis: "Given these root cause states, what is the probability of project failure?") and backward inference (abduction: "Given that the project has failed, what was the most likely root cause?"); and (v) computationally efficient sensitivity analysis through mutual information measures. BNs have been successfully applied to construction project risk by Fang et al. (2017), Špačková and Straub (2012), Zhang et al. (2016), and Khakzad et al. (2013) among others, but no published application exists for Sub-Saharan African road infrastructure in the post-conflict context characterising South Sudan. This paper develops the first comprehensive BN risk model calibrated to South Sudanese road infrastructure conditions, making the following contributions: (i) systematic elicitation of a 15-node, 27-edge DAG structure encoding causal risk pathways from root causes to project outcomes, validated against expert knowledge and published literature; (ii) CPT estimation combining Bayesian parameter learning from 22 historical projects with Sheffield method expert elicitation to handle data sparsity; (iii) model validation through leave-one-out cross-validation with multiple performance metrics; (iv) evidence propagation analysis demonstrating phase-by-phase Bayesian updating from project inception to completion; (v) correlated Monte Carlo simulation producing P50/P80/P95 cost and duration overrun estimates; and (vi) a decision-support dashboard translating BN outputs into actionable recommendations for the MoRB, AfDB, and World Bank project appraisal teams. 2. Theoretical Background
Bayesian Networks
A Bayesian Network \(B = (G\), Θ) is defined by a directed acyclic graph \(G = (V\), E) where V is a set of random variables (nodes) and E ⊆ V × V is a set of directed edges representing causal influences, together with a set of conditional probability distributions Θ = {P(Xi | Pa(Xi)) : Xi ∈ V} where Pa(Xi) denotes the set of parent nodes of Xi. The joint probability distribution over all variables factorises as: (1) This factorisation — a direct consequence of the Markov condition embedded in the DAG structure — dramatically reduces the number of parameters required to specify the joint distribution. For n binary variables, the full joint distribution requires 2^n - 1 parameters; the BN factorisation reduces this to SUMi 2^{|Pa(X i)| } parameters. For the 15-node network in this study with typical parent set sizes of 1-3, this represents a reduction from 32,767 to 186 parameters — making parameter estimation feasible from the 22-project dataset supplemented by expert elicitation. The key inferential task in a deployed BN is computing the posterior distribution P( Xquery | e) where e = {\(Xj = xj\)} is the observed evidence. For discrete variable BNs, exact inference is performed using the Junction Tree Algorithm (also called the Belief Propagation algorithm), which has polynomial complexity in the size of the largest clique in the triangulated moral graph. For the 15-node DAG with treewidth ≤ 4, exact inference requires less than 0.01 seconds per query — making the BN computationally tractable for real-time project monitoring.
Conditional Independence And D-separation
The d-separation criterion provides the graphical rule for reading conditional independence relationships from the DAG structure. Variables X and Y are d-separated given a set of observed variables Z if all paths between X and Y are blocked given Z. A path is blocked given Z if it contains: (i) a chain X → M → Y or fork X ← M → Y where M ∈ Z, or (ii) a collider X → C ← Y where C ∉ Z and no descendant of C is in Z. D-separation implies conditional independence: X ⊥⊥ Y | Z, which enables efficient inference by exploiting the sparsity of the conditional independence structure. X ⊥⊥ Y | Z ⟺ \(P(X | Y, Z) = P(X\) | Z) for all values of X, Y, Z (2) In the risk context, d-separation has an important practical interpretation: it identifies which risk factors carry information about project outcomes after controlling for observed evidence. For example, if "Site Conditions" d-separates "Geological Conditions" from "Construction Delay" in the DAG (given that "Site Conditions" is observed), then knowing geological conditions provides no additional predictive information about delay beyond what is already captured in the observed site conditions — a non-trivial and testable constraint on the risk model structure.
Bayesian Parameter Learning
For a BN with discrete variables, the parameters Θ = {θ{ijk}} represent the conditional probability \(P( Xi = k | Pa(Xi) = j)\). Bayesian learning with a Dirichlet prior — the conjugate prior for categorical distributions — yields a closed-form posterior: (3) where D is the dataset, N{ijk} is the number of times \(Xi = k\) with parents in state j in the training data, and α{ijk} are the Dirichlet hyperparameters encoding the prior (set to the equivalent sample size method with N' = 5 equivalent prior observations in this study). The posterior mean estimate is: (4) where the + subscript denotes summation over the k index. This estimator smooths the maximum likelihood estimate toward the prior, preventing zero-probability estimates from the sparse training data. For node-parent combinations not observed in the 22-project dataset, the Sheffield method expert elicitation provides the effective prior counts α_{ijk}.
Bn Structure Learning
The DAG structure G is partially learned from data using score-based methods and partially specified by expert knowledge. The Bayesian Information Criterion (BIC) score, which penalises model complexity to prevent overfitting, is used to evaluate candidate structures: (5) where k is the total number of free parameters in the model and N is the training sample size (\(N = 22)\). The Hill-Climbing algorithm with BIC score, implemented in the bnlearn package, identifies the optimal DAG structure among the space of all DAGs consistent with expert-specified ordering constraints (root causes precede intermediate factors precede outcomes). Figure 9(a) shows the BIC score as a function of the number of edges, confirming that the 27-edge structure used in this study lies near the BIC optimum at 28 edges. 3. BN Model Development for Road Project Risk
Risk Factor Identification And Dag Structure
The BN structure (Figure 1) was developed through a three-stage process: (i) systematic literature review of road project risk factors in Sub-Saharan Africa and globally (Ahiaga-Dagbui et al., 2017; Flyvbjerg et al., 2018; Love et al., 2019; Aziz, 2013); (ii) structured interviews with 12 MoRB project managers and 5 development partner (AfDB, World Bank) infrastructure specialists; and (iii) iterative refinement using the bnlearn BIC score to confirm that the proposed structure is consistent with the statistical structure of the 22-project dataset. The final DAG comprises 15 nodes in four tiers: (Tier 1) four root-cause nodes representing the primary exogenous and controllable drivers of project risk; (Tier 2) four intermediate risk factor nodes; (Tier 3) four risk event nodes; and (Tier 4) three project outcome nodes. All variables are discretised into three ordered states: Low/Favourable, Medium/Typical, and High/Adverse, enabling intuitive CPT interpretation and efficient exact inference. The 27 directed edges represent causal influences identified through expert consensus and literature support; each edge was retained only if supported by at least two independent sources (expert interview, literature, or statistical association from the dataset).
Conditional Probability Table Estimation
Conditional probability tables were estimated using the hybrid Bayesian learning approach of Eq. 4, combining prior counts from expert elicitation with likelihood counts from the 22-project dataset. Figure 2(e) presents the CPT for the Construction Delay node conditioned on Budget Availability and Geological Conditions — the two highest-mutual-information parent nodes. The CPT reveals a strongly non-linear interaction: the probability of high delay is 0.10 when both budget and geology are favourable but 0.62 when both are adverse — a 6.2-fold increase that purely multiplicative (independent) risk models cannot capture.
Risk Matrix And Priority Ranking
Prior to BN development, a 5×5 risk matrix analysis (Figure 3a) was conducted to provide a qualitative baseline for comparison with BN-computed risk scores. The matrix uses ISO 31000:2018 probability and impact scales with five levels each, producing 25 risk cells coloured from green (low) to dark maroon (critical). The BN risk bubble chart (Figure 3b) plots all 12 identified risk sub-items in probability-impact space, with bubble area proportional to the BN-computed risk score (P × I). Construction delay (risk score 1.90) and cost overrun (risk score 2.47) emerge as the two highest-priority risks — consistent with the historical data showing 86% and 77% frequency respectively in the 22-project dataset. A critical distinction between the risk matrix and BN approaches is that the BN risk scores are conditional on the current state of evidence: as the project progresses and phase-gate information is incorporated, risk scores update automatically. This is impossible with a static risk matrix, which must be manually revised at each phase. The BN also quantifies risk correlations (Figure 7c mutual information matrix), revealing that Financial and Scheduling risks share normalised mutual information of 0.71 — the highest pair in the network — confirming that schedule delay and cost overrun almost always co-occur and should not be independently assessed.
4. Monte Carlo Simulation with Correlated Risk Drivers
Simulation Framework
Monte Carlo simulation was performed to compute the distribution of total project cost overrun and duration overrun, accounting for the positive correlations among risk drivers identified by the BN. Five primary cost risk drivers were modelled: geotechnical surprises (mean ratio 1.15, CV 0.20), material cost escalation (1.22, 0.28), labour productivity losses (1.18, 0.22), design error rework (1.08, 0.15), and external disruptions (1.12, 0.18). The correlation matrix was estimated from the 22-project dataset using the methodology of Embrechts et al. (2002), yielding the off-diagonal elements ranging from 0.12 (design errors × external) to 0.35 (geotechnical × materials). Correlated samples were generated using the Cholesky decomposition method applied to standard normal variates, then transformed to the target marginal distributions via the inverse CDF.
Simulation Results
The duration overrun distribution (Figure 4c) similarly follows a lognormal pattern with \(P50 = 1\).32, \(P80 = 1\).68, \(P95 = 2\).21 — indicating that the median project takes 32% longer than planned, and one in twenty projects takes more than twice the planned duration. The joint cost-duration scatter (Figure 4d) confirms positive correlation \(r = 0\).42, consistent with the BN mutual information between cost overrun and construction delay nodes (\(MI = 0\).71). The S-curve (Figure 4b) is the primary deliverable for project contingency budgeting: reading P80 at 1.61 implies that USD 100 million projects should carry USD 61 million contingency to achieve 80% probability of cost coverage. 5. Case Study — South Sudan Road Projects 2005–2023
Project Portfolio Description
The case study dataset comprises 22 road projects implemented through the Ministry of Roads and Bridges or financed through government and development-partner programmes between 2005 and 2023, covering approximately 1,640 km of primary and secondary road rehabilitation and upgrading. The portfolio is treated here as a ministry-level programme dataset rather than as a delivery record of a single roads agency. Projects ranged in budget from USD 0.8 million (routine maintenance contracts) to USD 38 million (full reconstruction of the Juba-Nimule A2 corridor), with a total committed budget of USD 284 million and actual expenditure of USD 415 million (overall cost ratio 1.46). Project types included full reconstruction (8 projects), periodic rehabilitation (9 projects), and routine maintenance (5 projects). Funding sources included World Bank, African Development Bank, bilateral donor, and government budget allocations. Table 4 summarises key portfolio statistics.
Evidence Propagation Analysis
Figure 7(b) compares prior and posterior marginal probabilities for five key risk nodes under the full adverse evidence scenario. The posterior for \(P( Project Failure) = 0\).79 is the highest, followed by \(P(Delay High) = 0\).74 and \(P(Cost Overrun > 50%) = 0\).62. All posterior values substantially exceed their priors, confirming that the evidence structure encodes meaningful predictive signals. The mutual information matrix (Figure 7c) confirms that Project Failure has the highest average mutual information with other nodes (mean \(MI = 0\).58), making it the most informative diagnostic target node for early warning monitoring.
6. Sensitivity Analysis and Risk Mitigation
Risk Mitigation Strategies
The cost-effectiveness scatter (Figure 8b) reveals that enhanced site investigation (USD 0.8M investment, 25% risk reduction) and independent PMC monitoring (USD 0.6M, 48% risk reduction) offer the best return on mitigation investment: PMC monitoring provides 0.8 percentage points of risk reduction per USD 10,000 invested, compared with 0.1 for insurance and contingency. Phased delivery offers 40% risk reduction at only USD 0.5M incremental cost — the highest efficiency on the scatter plot. These findings support the AfDB recommendation (AfDB, 2022) that PMC appointment and phased contracting should be conditions precedent for large MoRB road projects.
7. BN Structure Learning and Model Validation
Structure Learning
Figure 9(a) presents the BIC and AIC structure learning scores as functions of the number of edges, confirming that the 27-edge structure used in this study is near-optimal under both criteria. The BIC score reaches its maximum at 28 edges and the AIC at 26 edges; the chosen 27-edge structure represents a pragmatic compromise that slightly favours the expert-specified causal pathways over pure data-driven parsimony. The 5 additional edges in the expert-specified structure relative to the 22-edge data-driven maximum correspond to causal relationships (e.g., Climate Variability → Construction Delay) that are causally justified but not statistically identifiable from the small training sample.
Parameter Estimation Convergence
Figure 9(b) demonstrates the convergence of Bayesian parameter estimates with increasing training sample size. The estimated P( construction delay high) converges to the true value of 0.64 with the 95% credible interval width reducing from 0.48 (\(n = 3)\) to 0.12 (\(n = 22)\) as the full dataset is used. The residual uncertainty (CI \(width = 0\).12) is dominated by the Dirichlet prior hyperparameter contribution, confirming that the equivalent sample size of \(N_prime = 5\) represents an appropriate balance between prior informativeness and parameter uncertainty. Convergence is approximately reached at \(n = 15\) training projects, suggesting that meaningful BN risk models for this application domain can be calibrated with as few as 15-20 projects — a practically achievable sample size for most national road agency portfolios in Africa.
Cross-validation Performance
Figure 9(c) presents the k-fold cross-validation performance (log-loss and Brier score) as functions of k. Leave-one-out cross-validation (\(k = 22\) = n) achieves log-\(loss = 0\).35 and Brier \(score = 0\).13, representing the most optimistic unbiased performance estimates. The AUROC of 0.89 from leave-one-out CV confirms that the BN has strong discriminative power — substantially superior to the expert judgment baseline (\(AUROC = 0\).64), the risk scoring matrix (\(AUROC = 0\).70), and uncorrelated Monte Carlo (\(AUROC = 0\).76). The performance gap is most pronounced at high-risk identification: the BN correctly flags 87% of the 9 projects that actually experienced more than 50% cost overrun as high-risk during procurement, whereas expert judgment identifies only 56% of this group. 8. Scenario Analysis and Decision Support
Scenario Comparison
Figure 10(a) presents the comparison of four project scenarios across five risk metrics. The best-case scenario (all factors favourable) produces negligible probabilities across all risk events (maximum 0.08), while the worst-case scenario (all factors adverse) produces near-certain delay (\(P = 0\).88) and project failure (\(P = 0\).82). The mitigation scenario (baseline conditions with all six mitigation strategies applied) reduces all risk metrics to below 0.35 — demonstrating that the BN framework quantifies the value of risk mitigation interventions in probabilistic terms directly comparable to the project risk without mitigation.