Scalable Bayesian–XAI Framework for Multi-Objective Decision-Making in Uncertain Dynamic Systems

Abstract

This study proposes a scalable Explainable Artificial Intelligence (XAI)–driven Bayesian–AI decision–control framework for multi-objective optimisation in uncertain and dynamic systems. The framework integrates Bayesian networks, stochastic control, and expected utility theory within a unified probabilistic architecture. Unlike traditional black-box models, the proposed framework provides intrinsic interpretability through probabilistic reasoning and dependency-aware modelling. This allows users to understand how decisions are formed and how variables influence outcomes. To further strengthen explainability, the framework incorporates post hoc XAI techniques, including SHAP-based feature attribution and sensitivity-based local explanations. These methods quantify the contribution of each variable and provide clear explanations at both global and local levels. The system is formulated as a stochastic state-space model and implemented as a closed-loop adaptive architecture. It updates decisions continuously as new data becomes available. Scalable inference is achieved using variational inference, Markov Chain Monte Carlo, and Sequential Monte Carlo methods. This ensures efficient performance in complex and high-dimensional environments. A simulation study based on 370 observations shows that the proposed framework improves decision quality, robustness under uncertainty, and transparency compared to conventional methods. Explainability is evaluated using Fidelity, Stability, and Transparency metrics. The results confirm that the model produces consistent and reliable explanations. The framework supports human-centred decision-making by providing visual analytics and clear probabilistic explanations. This makes it suitable for high-stakes applications such as cyber–physical systems, intelligent platforms, and real-time AI systems. The main contribution of this study is the integration of intrinsic probabilistic interpretability with post hoc XAI techniques into a single, scalable framework. This approach bridges a key gap in XAI research and offers a practical and transparent solution for decision-making under uncertainty.

1. Introduction

Decision-making in complex systems requires optimising multiple objectives under uncertainty. These systems are dynamic, interconnected, and often operate with limited resources and incomplete information [1]. From an applied mathematics perspective, such problems can be formulated as stochastic optimisation problems over dynamic, high-dimensional systems. Therefore, effective solutions require the integration of probabilistic modelling, optimisation, and control [2].

Traditional decision-support methods can analyse data, but they often lack adaptive control and feedback mechanisms. As a result, they are not well-suited to dynamic environments in which system states evolve [3]. Multi-objective decision and control (MODMC) frameworks provide a more structured approach by combining state estimation, control policies, and feedback. Artificial Intelligence has further enhanced decision-making in data-rich environments. However, many AI models operate as black boxes and offer limited interpretability, which limits their use in high-stakes applications where transparency is essential [4].

Bayesian networks offer a probabilistic framework for modelling uncertainty and dependencies among variables. They represent systems using directed acyclic graphs and Conditional Probability Tables, enabling interpretable and adaptive reasoning [5,6,7]. Bayesian inference supports continuous belief updating as new data becomes available. Despite these advantages, most existing studies treat Bayesian networks as static tools. They are rarely integrated with stochastic control or signal-processing approaches, thereby limiting their applicability in real-time, dynamic settings [8].

To address this limitation, this study proposes a Bayesian–XAI decision–control framework for multi-objective optimisation in uncertain dynamic systems. The framework models decision processes as stochastic and time-dependent. It adopts a state-space formulation in which hidden states (X_t) are inferred from observations (Z_t) through transition and observation models.

This allows continuous updating of system states under uncertainty. The framework operates as a closed-loop system that combines Bayesian inference with expected utility theory to support optimal decision-making. Scalable inference methods, including variational inference, Markov Chain Monte Carlo, and Sequential Monte Carlo, are incorporated to ensure computational feasibility in high-dimensional environments [9].

A key contribution of this study is the integration of intrinsic and post hoc explainability within a unified architecture. The Bayesian network provides intrinsic explainability through its probabilistic structure, which enables global understanding of dependencies and causal relationships. However, it does not quantify the contribution of individual variables to specific decisions [10]. To address this limitation, post hoc explainability methods, such as SHAP, are incorporated to provide local, instance-level explanations. These methods quantify feature contributions without modifying the underlying model.

The interaction between intrinsic and post hoc explainability is formalised through a consistency mechanism. Structural explanations derived from conditional dependencies are compared with feature attribution scores. Any discrepancies trigger diagnostic analysis and model refinement. This dual-layer approach ensures that explanations are both globally coherent and locally informative, thereby improving interpretability in dynamic decision environments [11,12].

The proposed framework also introduces three methodological contributions. First, it establishes a dual-layer explainability architecture that integrates probabilistic transparency with feature attribution. Second, it embeds explainability directly within the decision loop, allowing explanations to influence action selection and policy optimisation. Third, it formulates multi-objective decision-making within a probabilistic control framework, enabling dynamic weighting, stochastic transitions, and adaptive utility optimisation.

The framework applies to a wide range of domains, including cyber–physical systems, intelligent platforms, and data-driven decision environments [13]. However, existing approaches remain limited by static modelling, a lack of transparency, and weak integration between probabilistic reasoning and control mechanisms. These limitations define the central research problem addressed in this study.

Accordingly, this study aims to develop a unified framework that integrates probabilistic inference, multi-objective optimisation, and stochastic control while maintaining interpretability and scalability. The study addresses the following research questions:

• RQ1: How can a Bayesian–XAI framework integrate probabilistic inference and stochastic control for multi-objective optimisation in dynamic systems?
• RQ2: What is the impact of the framework on decision quality, adaptability, and interpretability?
• RQ3: How effective is the framework in supporting scalable and real-time decision-making?

In summary, this study contributes a unified, interpretable, and scalable decision–control framework that integrates Bayesian reasoning, multi-objective optimisation, and Explainable Artificial Intelligence. The proposed approach advances current methods by combining probabilistic modelling, adaptive control, and hybrid explainability within a coherent system for decision-making under uncertainty.

2. Literature Review

2.1. Explainable Artificial Intelligence and Multicriteria Decision-Making

Artificial Intelligence (AI) is widely used to support decision-making in complex and uncertain environments. These environments include digital platforms, cyber–physical systems, and intelligent decision systems. They require real-time adaptation, probabilistic reasoning, and scalable optimisation [14]. AI-based systems can process large amounts of data and support decision-making under uncertainty.

In many real-world systems, decision-makers must evaluate multiple interconnected factors. These include performance, cost, risk, and resource constraints. Traditional approaches often fail to capture uncertainty and dependencies among these factors, which limits their effectiveness in dynamic environments [5].

To address these limitations, recent studies have introduced probabilistic and stochastic methods, including probabilistic decision models, stochastic optimisation, and fuzzy approaches [6,13,15]. In these methods, decision variables are represented as probability distributions rather than fixed values. This allows better handling of uncertainty and changing conditions.

Bayesian networks are an important class of probabilistic models. They represent variables as nodes in a graph and define relationships using conditional probabilities [9,10]. These models support probabilistic inference and enable continuous updating as new data becomes available [5]. This makes them suitable for dynamic and data-driven systems.

Machine learning models, such as neural networks and support vector machines, are also widely applied in decision systems due to their strong predictive performance [16]. However, many of these models operate as black boxes and provide limited insight into how decisions are made. This lack of interpretability remains a major challenge in real-world applications [14].

Explainable Artificial Intelligence (XAI) has been developed to improve transparency and interpretability. Most XAI methods, however, are applied after model training and do not influence the underlying decision process [4,17]. In contrast, Bayesian networks provide intrinsic interpretability by design. Their structure allows users to trace dependencies and understand probabilistic relationships between variables. They also support the integration of data-driven evidence and expert knowledge [7,8,13].

Despite these advantages, Bayesian networks are often used primarily for inference and estimation rather than decision-making. They are rarely integrated with optimisation or control frameworks, thereby limiting their ability to support adaptive, real-time decision-making. Similarly, multi-criteria decision-making (MCDM) methods provide structured evaluation of alternatives but typically rely on deterministic assumptions and fixed weighting schemes [9,18].

Moreover, existing approaches tend to address probabilistic reasoning, explainability, and decision-making in isolation. This creates a gap in the literature, particularly in dynamic environments where uncertainty, adaptability, and transparency must be addressed simultaneously. Therefore, there is a need for unified frameworks that integrate probabilistic modelling, explainability, and multi-objective decision-making into a coherent, adaptive system.

2.2. Methodological Contribution to MODMC

Explainability in Artificial Intelligence is commonly classified into two main categories: intrinsic (model-based) and post hoc (model-agnostic) approaches. Intrinsic explainability is achieved when the model structure is inherently interpretable [6]. Examples include Bayesian networks, decision trees, linear regression models, and causal graphical models. These models allow users to directly understand relationships between inputs and outputs through explicit structural dependencies [15].

Particularly, Bayesian networks provide global interpretability by representing variables and their conditional dependencies using directed acyclic graphs and Conditional Probability Tables [7,8]. This enables transparent probabilistic reasoning and supports tracing of decision pathways. However, their interpretability may decrease in high-dimensional settings due to the increasing complexity of dependencies and conditional probability structures.

In contrast, post hoc explainability methods, such as SHAP, LIME, partial dependence plots, and feature importance techniques, are applied after model training [15]. These approaches provide local, instance-level explanations by estimating each feature’s contribution to a specific prediction. While they enhance interpretability in complex models, they do not alter the model’s internal structure and may introduce approximation inconsistencies [5].

Recent studies, such as [7,9,13,15], have attempted to integrate probabilistic models with multi-criteria decision-making and explainability techniques. For example, Bayesian network–based decision systems have been combined with probabilistic reasoning and ranking methods under uncertainty. However, most existing approaches lack dynamic decision control, real-time updates, and systematic integration of post hoc explainability into the decision process.

The proposed framework addresses these limitations by introducing a hybrid explainability architecture within a probabilistic decision–control system. It integrates intrinsic explainability via Bayesian probabilistic modelling and post hoc explainability via SHAP-based attribution and sensitivity analysis. Importantly, this integration is not sequential but structural, where both explanation layers interact within the decision pipeline.

Specifically, the framework introduces three methodological advances. First, it establishes a dual-layer explainability mechanism that combines global structural transparency with local feature attribution. Second, it embeds explainability directly into the decision loop, allowing explanations to influence action selection, uncertainty propagation, and policy optimisation. Third, it supports dynamic updating and real-time decision-making within a probabilistic control framework based on Bayesian reasoning and POMDP formulation.

Table 1. Comparative positioning of the proposed framework.

Framework Type	Intrinsic Explainability	Post hoc XAI	Dynamic Control	Key Limitation
Bayesian Networks	Provided through probabilistic structure and CPTs	Not supported	Limited or static	No local feature attribution
POMDP/Bayesian Control	Provided through probabilistic reasoning	Not supported	Fully supported	Limited interpretability depth
Black-box ML + SHAP	Not provided	Provided through feature attribution (e.g., SHAP)	Not supported	No structural transparency
Hybrid BN + MCDM	Provided	Not supported	Partially supported	Limited adaptability
Deep Reinforcement Learning	Not provided	Limited	Fully supported	Black-box, data-intensive, low interpretability
Proposed Framework	Fully provided through a Bayesian structure	Fully integrated (e.g., SHAP, sensitivity analysis)	Fully supported	Higher computational complexity

shows that existing approaches provide only partial integration across explainability, probabilistic modelling, and decision control. In contrast, the proposed framework achieves full integration across these dimensions.

Multi-objective decision-making problems require evaluating alternatives across multiple interdependent and uncertain criteria. Classical methods such as AHP, TOPSIS, ELECTRE, and PROMETHEE provide structured decision frameworks but rely on deterministic inputs and fixed weights. These assumptions limit their applicability in dynamic environments [5,16].

In contrast, probabilistic approaches represent decision variables as stochastic and interconnected. Bayesian networks model uncertainty and dependencies, while expected utility theory supports decision optimisation. This combination enables adaptive and data-driven decision-making under uncertainty [17].

Despite recent progress [5,6,7], several gaps remain. First, many AI systems focus on prediction rather than decision control. Second, Bayesian networks are often used as static models and are not integrated into dynamic control systems. Third, hybrid XAI approaches usually treat explainability as a post-processing step rather than an embedded component of the system. Fourth, scalability and real-time applicability are rarely addressed.

The proposed Bayesian–XAI decision–control framework addresses these limitations by integrating probabilistic inference, multi-objective optimisation, and explainability within a unified architecture. The framework supports dynamic updating, interpretable decision-making, and scalable inference. Unlike existing approaches, explainability is embedded within the decision process and directly contributes to model validation, adaptation, and optimisation.

This approach represents methodological advancement beyond simple integration. It establishes a structural coupling between probabilistic reasoning, explainability, and decision control. As a result, the framework provides a unified solution for managing uncertainty, interdependence, and real-time decision-making in complex systems [9,10].

Recent advances in decision-making under uncertainty [19,20,21] have been strongly influenced by deep reinforcement learning (DRL), particularly in sequential control problems such as pursuit–evasion tasks and autonomous navigation. DRL methods optimise policies by interacting with the environment through reward-based learning [22]. These methods often achieve strong performance in high-dimensional and non-linear systems. However, they typically rely on black-box neural architectures and require large amounts of training data, which limits interpretability and transparency [20].

For example, reinforcement learning has been successfully applied to complex sequential decision problems, such as spacecraft pursuit–evasion games in elliptical orbits, where policies are learned through iterative interaction with dynamic environments.

In contrast, the proposed Bayesian–XAI framework adopts a probabilistic decision–control paradigm. It explicitly models system dynamics, uncertainty, and dependencies using Bayesian networks and state-space formulations. Unlike DRL, which learns implicit policies, the proposed approach provides explicit probabilistic reasoning. It supports multi-objective utility optimisation and enables clear explainability through both intrinsic mechanisms (graph structure) and post hoc methods (e.g., SHAP) [23].

Furthermore, DRL primarily focuses on maximising cumulative rewards. It does not inherently support structured trade-offs between competing objectives or constraint-based decision-making. The proposed framework addresses this limitation by integrating expected utility theory with constraint filtering. This allows decision-makers to evaluate trade-offs under uncertainty in a transparent and interpretable manner. However, the key distinction lies between performance-driven policy learning in DRL and interpretable, probabilistic, and multi-objective decision support in the proposed framework.

2.3. Research Gap and Motivation

Despite advances in Explainable Artificial Intelligence (XAI) and probabilistic decision models, several critical limitations remain in existing frameworks. A key issue is the fragmentation between explainability and decision-making. Many XAI approaches focus on explaining model predictions rather than supporting structured decision processes. In contrast, decision-making frameworks often lack integrated explainability mechanisms [15,16,24,25].

Bayesian networks provide intrinsic interpretability through their probabilistic structure. However, their usability decreases in high-dimensional systems. The number of Conditional Probability Tables (CPTs) grows exponentially with the number of parent nodes, which makes visualisation and interpretation increasingly complex. As a result, their practical transparency is reduced in large-scale applications. In addition, existing Bayesian–MCDM models are typically static. They lack adaptability and do not support continuous updates or real-time feedback, both of which are essential in dynamic environments.

Another limitation is the absence of integrated hybrid explainability. Most existing frameworks rely on either intrinsic interpretability or post hoc explanation methods, lacking a formal mechanism to combine and validate both. Furthermore, traditional multi-criteria decision-making approaches rely on fixed weights and deterministic assumptions. This restricts their ability to model uncertainty and adapt to evolving system conditions.

These limitations highlight the need for a unified framework that integrates probabilistic modelling with multi-objective decision-making. Such a framework should embed explainability as a core system component rather than a post-processing step, support dynamic updates and real-time decision control, and maintain interpretability as system complexity increases.

To address this gap, this study proposes a Bayesian–XAI decision–control framework that combines intrinsic probabilistic reasoning with post hoc explainability within a closed-loop architecture. This approach bridges the gap between interpretability, adaptability, and decision optimisation, thereby extending the current state of the art.

3. System Design and Architecture

3.1. Bayesian–XAI Decision Framework

The proposed framework extends beyond simple component integration by introducing a unified modelling paradigm in which probabilistic reasoning, explainability, and decision-making operate as interdependent processes within a single mathematical structure. The framework is designed as a stochastic decision–control system for multi-objective optimisation in dynamic environments and is formulated as a Partially Observable Markov Decision Process (POMDP). In this setting, the true system state is not fully observable. Bayesian inference is used to estimate hidden states, while expected utility is used to select optimal actions. This formulation connects the framework to applied mathematics, optimisation, and machine learning [26,27].

The architecture consists of four main components: (i) a User Input Interface, (ii) an XAI Processing Module, (iii) a Bayesian network model, and (iv) a Visualisation Interface. These components operate within a continuous feedback loop, enabling real-time decision-making in streaming environments such as cyber–physical systems and intelligent platforms.

At each time step t, the system receives new observations and updates its belief about the system state. Let x_t denote the hidden state and a_t the control action. The objective is to maximise the expected utility:

$$\underset{a_t\in A}{\max } \mathbb{E}[U(x_t,a_t)]$$

and subject to system dynamics and constraints,

$$x_t\sim P(x_t\mid x_{t-1},a_{t-1}),C(x_t,a_t)\le \delta$$

where U(⋅) is a multi-objective utility function and C(⋅) defines system constraints.

The Bayesian network represents the joint distribution of system variables as

$$P(X_1,X_2,\dots ,X_n)={\displaystyle \prod _{i=1}^n}P(X_i\mid Pa(X_i))$$

To capture temporal dynamics, the framework extends to a Dynamic Bayesian Network (DBN):

$$P(X_t\mid X_{t-1})=\prod _{i}P(X_i^t\mid Pa(X_i^t))$$

This formulation enables the modelling of evolving system states over time. When new evidence E is observed, beliefs are updated using Bayesian inference:

$$P(Y\mid E)={\displaystyle \frac{P(E\mid Y)P\left(Y\right)}{P\left(E\right)}}$$

In dynamic settings, inference is performed sequentially:

$$P(Y_t\mid E_{1:t})\propto P(E_t\mid Y_t)P(Y_{t-1}\mid E_{1:t-1})$$

This allows continuous adaptation to streaming data. From a system perspective, decision variables are treated as stochastic signals derived from noisy observations. This enables the framework to function as a probabilistic signal-processing system, where uncertainty is explicitly modelled and propagated through the decision pipeline [26] (See Figure 1).

Input: Initial belief state $\mathrm{P}\left({\mathrm{x}}_0\right)$, action space $A$, observation space $\mathrm{Y}$, utility function $U(x,a)$, and constraints $\mathrm{C}(\mathrm{x},\mathrm{a})$. Output: Optimal action sequence ${\mathrm{a}}_{\mathrm{t}}^{\ast }$.

Step 1: Initialisation: Define Bayesian Network structure $\mathrm{G}=(\mathrm{X},\mathrm{E})$, initialise Conditional Probability Tables (CPTs), and set prior belief $\mathrm{P}\left({\mathrm{x}}_0\right)$.

Step 2: Observation Update (Inference): For each time step $\mathrm{t}$ (Receive observation ${\mathrm{y}}_{\mathrm{t}}$). Update the belief state as follows:

$$P(x_t\mid y_{1:t})\propto P(y_t\mid x_t)\sum _{x_{t-1}}P(x_t\mid x_{t-1},a_{t-1})P(x_{t-1})$$

Step 3: Explainability Analysis: Extract intrinsic explanations from CPT dependencies, compute SHAP values for feature attribution, and evaluate consistency between intrinsic and post hoc explanations.

Step 4: Utility Evaluation: For each action $\mathrm{a}\in \mathrm{A}$,

$$U\left(a\right)={\mathbb{E}}_{x_t}\left[U\right(x_t,a\left)\right]$$

Step 5: Constraint Filtering: Select feasible actions, satisfying

$$C(x_t,a)\le \delta$$

Step 6: Action Selection:

$$a_t^{\ast }=\arg \underset{a\in A}{\max } U(a)$$

Step 7: Policy Update (Adaptive Learning): Update CPT parameters using Bayesian updating, adjust weights ${\mathrm{w}}_{\mathrm{k}}$ dynamically.

Step 8: Iterate: Repeat Steps 2–7 until convergence or horizon $\mathrm{T}$.

This algorithm formalises the interaction between probabilistic inference, explainability, and decision-making within a unified computational workflow. It ensures that the framework is both interpretable and operationally reproducible.

3.2. User Input and Uncertainty Handling

The User Input Interface collects both quantitative and qualitative data. Quantitative inputs include measurable indicators, while qualitative inputs are derived from expert assessments. Each input is associated with a confidence value $\mathrm{c}\in [0,1]$, which reflects the reliability of the information.

To model uncertainty, the framework employs a soft-evidence mechanism. This mechanism is generalised to handle multiple variables and multiple states. $\mathrm{E}=\{{\mathrm{X}}_1,{\mathrm{X}}_2,\dots ,{\mathrm{X}}_{\mathrm{m}}\}\mathrm{c}$ denotes a set of observed variables with uncertain evidence. The soft evidence for each variable ${\mathrm{X}}_{\mathrm{k}}$ is defined as

$$q_i^{\left(k\right)}=c_k\cdot \mathbf{1}(X_k=r_k)+(1-c_k)\cdot P(X_k=x_i)$$

where $c_k\in [0,1]$ is the confidence associated with variable ${\mathrm{X}}_{\mathrm{k}}$, ${\mathrm{r}}_{\mathrm{k}}$ is the observed state, and $1(\cdot )$ is the indicator function.

For multiple variables, the joint soft evidence is incorporated into the posterior distribution as

$$P(X\mid E)\propto P(X){\displaystyle \prod _{k=1}^m}{\displaystyle \frac{q^{\left(k\right)}(X_k)}{P(X_k)}}$$

This formulation enables consistent probabilistic updating across multiple uncertain observations. It improves robustness and reduces the risk of overfitting by smoothing the influence of uncertain inputs [11].

Table 1 presents an example Conditional Probability Table (CPT) for the node Scaling Success, conditioned on Product–Market Fit (PMF) and Funding Availability.

Figure 2 presents the model calibrated using 370 observations. When data are sparse, smoothing techniques (e.g., Dirichlet priors) are applied to stabilise probability estimates. Sensitivity analysis is conducted to identify influential variables and assess model robustness.

Furthermore, the selection and structuring of input variables follow a formalised process based on a variable-screening directed acyclic graph (DAG). This ensures that all variables included in the model are both theoretically relevant and empirically justified within the probabilistic framework.

3.3. Variable Selection and DAG Construction

The Bayesian network is constructed through a systematic process that integrates theoretical foundations, domain knowledge, and data-driven validation to ensure accurate representation of causal and probabilistic relationships.

Variable selection follows three criteria: (i) decision-theoretic relevance, capturing key factors such as performance, risk, and resource constraints; (ii) observability, including both measurable indicators and latent variables; and (iii) statistical significance, evaluated using sensitivity analysis and correlation screening to exclude weak or redundant variables.

The directed acyclic graph (DAG) is designed based on assumptions of causal ordering and conditional independence. It satisfies the Markov condition and maintains an acyclic structure. A hybrid design strategy is adopted, combining expert-driven specification with data-driven validation using mutual information and likelihood-based methods. The structure is further refined through sensitivity analysis to preserve influential relationships.

Dependencies in the DAG represent causal mechanisms rather than simple correlations. These relationships are quantified using Conditional Probability Tables (CPTs), which are estimated with Dirichlet priors to ensure robustness in the face of data sparsity. Model parsimony is enforced by removing redundant connections, which improves interpretability and computational efficiency.

3.4. Decision Optimisation and XAI Processing

The XAI Processing Module performs probabilistic inference and supports optimal decision-making. The optimal action is defined as

$$a^{\ast }=\arg \underset{a\in A}{\max } \sum _{y}U(y,a)\hspace{0.17em}P(Y=y\mid a,E)$$

To ensure consistency with the partially observable setting, the utility function is defined at the latent state level:

$$U(a_t)={\mathbb{E}}_{x_t\sim P(x_t\mid y_{1:t})}[U(x_t,a_t)]=\sum _{x_t}U(x_t,a_t)\hspace{0.17em}P(x_t\mid y_{1:t})$$

The outcome-based utility $\mathrm{U}(\mathrm{y},\mathrm{a})$ is therefore a special case derived as

$$U(y,a)={\mathbb{E}}_x[U(x,a)\mid y]$$

This formulation ensures consistency between observable outcomes and latent-state decision modelling.

The utility function incorporates multiple objectives:

$$U(x_t,a_t)={\displaystyle \sum _{k=1}^K}{w}_k\hspace{0.17em}{u}_k(x_t,a_t),{\displaystyle \sum _{k=1}^K}{w}_k=1$$

where ${\mathrm{u}}_{\mathrm{k}}(\cdot )$ represents the utility of the $\mathrm{k}$-th objective (e.g., performance, risk, or resource efficiency), and ${\mathrm{w}}_{\mathrm{k}}$ denotes the corresponding weight.

Sensitivity analysis is used to evaluate the impact of parameter changes:

$$S_{\theta }\approx {\displaystyle \frac{P(Y\mid E,\theta +\Delta )-P(Y\mid E,\theta )}{\Delta }}$$

To ensure scalability, the framework employs approximate inference methods, including variational inference, Markov Chain Monte Carlo (MCMC), and Sequential Monte Carlo (SMC). These methods enable efficient computation in high-dimensional and streaming environments. Results are presented through visual dashboards, including probability indicators and scenario comparisons.

3.5. Bayesian Networks as Signal-Processing Models

To extend the static Bayesian network into a temporal framework, the model is formulated as a Dynamic Bayesian Network (DBN). The system is defined by

State transition model:

$$P(x_t\mid x_{t-1},a_{t-1})$$

Observation model:

$$P(y_t\mid x_t)$$

The joint distribution over time is

$$P(x_{1:T},y_{1:T})=P(x_1){\displaystyle \prod _{t=2}^T}P(x_t\mid x_{t-1},a_{t-1}){\displaystyle \prod _{t=1}^T}P(y_t\mid x_t)$$

This formulation introduces temporal recursion while preserving the DAG structure within each time slice. The classical filtering methods are Kalman filtering (for linear systems) and particle filtering (for nonlinear systems). However, the proposed framework extends. The methods model dependencies between variables, not just state transitions. This allows better representation of complex systems with interacting components [11,28].

3.6. Multi-Objective Decision and Control Formulation

The framework is formulated as a multi-objective optimisation problem under uncertainty. The system estimates the posterior state,

$$P(x_t\mid y_{1:t})$$

and selects the optimal action,

$$a_t^{\ast }=\mathrm{a}\mathrm{r}\mathrm{g} \underset{a\in A}{\mathrm{m}\mathrm{a}\mathrm{x}} \mathbb{E}[U(x_t,a_t)]$$

subject to the following constraints:

$$C(x_t,a_t)\le \delta$$

The multi-objective weights $w_k$ are determined using a hybrid approach that combines expert-driven, data-driven, and adaptive strategies. Expert knowledge provides initial weights, while entropy- or variance-based methods quantify the informational contribution of each objective. These weights are dynamically updated using Bayesian learning to ensure adaptability.

The constraint function defines the feasible decision space:

$$C(x_t,a_t)=\left\{c_j\right(x_t,a_t)\le {\delta }_j\},j=1,\dots ,m$$

These constraints represent system limitations such as budget, risk thresholds, and resource capacity. This formulation ensures that decisions are both optimal and feasible.

3.7. Intrinsic Explainability via Bayesian Networks

The framework provides intrinsic explainability through the Bayesian network structure, which models conditional dependencies among variables:

$$P(X_1,\dots ,X_n)=\prod _{i}P(X_i\mid Pa(X_i))$$

This enables transparent reasoning and step-by-step traceability of decisions.

To complement this, post hoc explainability is incorporated using SHAP-based feature attribution:

$$f(x)={\varphi }_0+{\displaystyle \sum _{i=1}^n}{\varphi }_i$$

where ${\mathrm{\varphi }}_{\mathrm{i}}$ represents the contribution of feature $i$. This provides local, instance-level explanations.

Additional interpretability is achieved through sensitivity analysis and counterfactual reasoning:

$$Y_{cf}=f(X+\delta )$$

To ensure consistency between intrinsic and post hoc explanations, the framework introduces a validation mechanism based on three criteria: (i) fidelity between SHAP explanations and model outputs; (ii) alignment between feature importance and CPT-based dependencies; (iii) stability under small input perturbations.

If inconsistencies exceed predefined thresholds, the system performs sensitivity reanalysis, CPT recalibration, or structural refinement. This ensures that explanations remain coherent, reliable, and aligned with the underlying probabilistic model.

4. MODMC Framework

The proposed framework is a hybrid multi-objective decision and control (MODMC) system. It operates as a closed-loop architecture. System states are estimated using Bayesian inference, and control actions are selected using expected utility optimisation. This design integrates machine learning, probabilistic modelling, and optimisation within a unified framework.

Decision-making in dynamic systems involves multiple variables and conflicting objectives. These systems often operate under uncertainty and continuous change. Traditional approaches rely on fixed models and deterministic assumptions. As a result, they cannot adapt to new information in real time. The proposed framework addresses this limitation by using probabilistic reasoning. Bayesian networks allow continuous updating of decisions as new data becomes available [26,29].

The system is represented as a directed acyclic graph (DAG). Each node represents a decision variable, such as system performance, resource availability, demand level, or operational stability. Edges represent dependencies between variables. These relationships are defined using Conditional Probability Tables (CPTs), which allow efficient modelling of complex systems [30,31]. The joint probability distribution of the system is defined as

$$P(X_1,\dots ,X_n)={\displaystyle \prod _{i=1}^n}P(X_i\mid Pa(X_i))$$

where ${\mathrm{X}}_{\mathrm{i}}$ represents system variables and $\mathrm{P}\mathrm{a}({\mathrm{X}}_{\mathrm{i}})$denotes their parent nodes. This factorisation reduces computational complexity and captures dependencies between variables. When new evidence $\mathrm{E}$ is observed, the system updates its beliefs using Bayesian inference:

$$P(Y\mid E)={\displaystyle \frac{P(E\mid Y)P(Y)}{P(E)}}$$

where $\mathrm{Y}$ represents a system outcome. This allows continuous updating of decision evaluations. For example, changes in system inputs or external conditions will directly affect the probabilities of outcomes. Decision-making becomes an adaptive and iterative process. All system indicators are modelled as probabilistic variables. Both quantitative data (e.g., performance metrics) and qualitative inputs (e.g., expert judgments) are integrated within the same framework. This supports data fusion from multiple sources [31]. Sensitivity analysis is used to identify the most influential variables:

$$S_{\theta }\approx {\displaystyle \frac{P(Y\mid \theta +\Delta )-P(Y\mid \theta )}{\Delta }}$$

Higher sensitivity values indicate a stronger influence on outcomes. This improves interpretability and helps identify key drivers of system behaviour. Model parameters are estimated using observed data and Bayesian learning. When data are limited, Dirichlet smoothing is applied:

$${\widehat{\theta }}_i(x_i\mid u)={\displaystyle \frac{N(x_i,u)+\alpha }{\sum _{x_i}(N(x_i,u)+\alpha )}}$$

This ensures stable and reliable probability estimates in sparse data conditions. Decision alternatives are evaluated using expected utility optimisation. Let $a$ denote an action. The optimal action is defined as

$$a^{\ast }=\mathrm{a}\mathrm{r}\mathrm{g} \underset{a}{\mathrm{m}\mathrm{a}\mathrm{x}}\sum _{y}U(y,a)P(Y=y\mid a,E)$$

where $\mathrm{U}(\mathrm{y},\mathrm{a})$ represents the utility of the outcome $\mathrm{y}$. This formulation connects probabilistic inference with optimisation, which is central to the Special Issue theme.

The framework also includes a visualisation layer. It presents results using probability indicators, scenario comparisons, and dependency graphs. These outputs improve transparency and support user understanding [32]. A continuous feedback mechanism updates the model as new data becomes available. This enables real-time learning and adaptation. The system improves over time while maintaining interpretability.

In summary, the proposed MODMC framework provides a scalable, interpretable, and adaptive solution for decision-making in dynamic systems. It integrates Bayesian inference, machine learning, and optimisation within a unified architecture. This makes it suitable for modern XAI-driven environments that require real-time and data-driven decision support.

5. Simulation Validation

5.1. Simulation Scenario and Problem Formulation

The simulation evaluates a stochastic multi-objective optimisation system under uncertainty. The framework models a dynamic process in which system states evolve, and control actions are selected based on expected utility. This formulation enables both learning and optimisation within a unified probabilistic structure [33,34].

The evaluation focuses on three aspects: (i) the ability to model multi-objective optimisation under uncertainty; (ii) the capability to update decisions dynamically as new data becomes available; (iii) the consistency and interpretability of the generated decisions.

A startup decision-making scenario is used as an illustrative case, although the framework is domain-independent. The system state is defined as

$$x_t=\{D_t,F_t,O_t,R_t\}$$

where ${\mathrm{D}}_{\mathrm{t}}$ denotes demand, ${\mathrm{F}}_{\mathrm{t}}$ financial capacity, ${\mathrm{O}}_{\mathrm{t}}$ operational stability, and ${\mathrm{R}}_{\mathrm{t}}$ risk level. Each variable is discretised into three states (Low, Medium, and High).

The initial state follows:

$$P\left(x_0\right)\sim \mathrm{Categorical}\left(\pi \right),\pi =[0.3,0.4,0.3]$$

State transitions are defined as

$$P(x_t\mid x_{t-1},a_{t-1})=\mathrm{Softmax}(W{x}_{t-1}+B{a}_{t-1})$$

Observations are generated as

$$P(y_t\mid x_t)=\mathcal{N}({\mu }_{x_t},\mathrm{\Sigma })$$

Model parameters are fixed for reproducibility: $\mathrm{W}\sim \mathcal{N}(0,1)$, $\mathrm{B}\sim \mathcal{N}(0,0.5)$; ${\mathsf{\mu }}_{{\mathrm{x}}_{\mathrm{t}}}\in [0,1]$, $\mathrm{\Sigma }=0.1\mathrm{I}$; Noise: ${\mathrm{ϵ}}_{\mathrm{t}}\sim \mathcal{N}(0,0.5)$.

All simulations use a fixed random seed (42) and a time horizon $\mathrm{T}=370$. Dirichlet priors are applied to CPTs to ensure stable estimation under limited data. These settings ensure full reproducibility and controlled stochastic behaviour [35].

5.2. Bayesian Network–Based Dynamic Inference

The Bayesian network models depend on the system variables. The probability of a target outcome depends on key factors such as demand, operational stability, and financial capacity:

$$P(S\mid D,F,O)$$

As new data becomes available, the system updates probabilities using Bayesian inference:

$$P(S\mid E_t)\propto P(E_t\mid S)P(S\mid E_{t-1})$$

This recursive update enables continuous learning and adaptation. The relationships between variables are defined using Conditional Probability Tables (CPTs). Figure 3 presents an example structure [36].

Figure 3 shows how system outcomes depend on interactions among variables. For example, high demand and stable operations lead to a high probability of success. In contrast, weak conditions lead to low probability. This demonstrates the model’s ability to capture uncertainty and dependencies.

The dataset consists of 370 synthetic observations generated using the defined transition and observation models. The synthetic design enables controlled evaluation of uncertainty and dependencies while maintaining reproducibility.

5.3. Real-Time Updating and Decision Optimisation

Decision alternatives are evaluated using expected utility:

$$EU(a)=\sum _{y}U(y,a)P(Y=y\mid a,E)$$

Baseline methods include, most notably: Traditional MCDM (AHP, TOPSIS); static Bayesian network; Random Forest with SHAP (post hoc XAI); LIME (local explanation model). These baselines represent deterministic, probabilistic, and explainability-focused approaches.

5.4. System Architecture and Processing Pipeline

The framework is evaluated using three core metrics:

Decision Quality (DQ):

$$DQ={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^T}U(x_t,a_t)$$

Robustness (RB):

$$RB=1-\mathrm{Var}\left(U\right(x_t,a_t\left)\right)$$

Transparency (TR): Defined use is as follows:

Fidelity: Agreement between explanation and model output is as follows:

$$\mathrm{Fidelity}=\mathrm{corr}\left(f\right(x),g(x\left)\right)$$

Stability: Consistency under perturbations is as follows:

$$\mathrm{Stability}=1-\mathbb{E}[\mid g(x)-g(x+\delta )\mid ]$$

Final Transparency score:

$$TR=\alpha \cdot \mathrm{Fidelity}+(1-\alpha )\cdot \mathrm{Stability}$$

Model parameters are calibrated using maximum likelihood estimation for CPTs and Bayesian updating with Dirichlet priors:

$$P(X_i\mid Pa(X_i))={\displaystyle \frac{N_{ijk}+\alpha }{\sum _k(N_{ijk}+\alpha )}}$$

The parameter α = 0.6 is selected based on sensitivity analysis to balance accuracy and robustness. (The dataset is available at: https://2u.pw/2hdMd3, accessed on 28 March 2026).

5.5. Explainability Evaluation Metrics

This section evaluates the proposed framework in terms of explainability, decision performance, and computational efficiency. The evaluation is designed to address the limitations identified in prior work, particularly the lack of integrated, interpretable validation of explainability in decision-making systems. The results are presented using quantitative metrics and supported by graphical analysis to improve clarity and interpretability.

Explainability is assessed using three metrics: Fidelity, Stability, and Transparency (TR). Fidelity measures the alignment between explanations and model predictions. Stability evaluates the consistency of explanations under small input perturbations. Transparency reflects the model’s overall interpretability.

As shown in Figure 4, the proposed framework achieves consistently higher scores across all metrics compared to baseline methods. This improvement demonstrates that combining intrinsic probabilistic explainability with post hoc feature attribution provides more reliable and interpretable explanations, especially in complex decision settings.

Decision performance is evaluated using decision quality (DQ), robustness (RB), and transparency. These metrics reflect the effectiveness of decision-making under uncertainty and the reliability of model outputs.

Figure 5 shows that the proposed framework outperforms baseline approaches across all performance indicators. The improvement in robustness indicates that the framework maintains stable decisions under varying input conditions, while the higher transparency confirms that improved explainability directly supports better decision outcomes.

A comprehensive evaluation of performance and computational efficiency is presented in Figure 6. The results show that the proposed framework achieves a balanced trade-off between decision performance and computational cost. Although its latency is higher than simple deterministic methods such as AHP, it remains within acceptable real-time limits. At the same time, it provides significantly higher decision quality and interpretability. This confirms that the framework is suitable for dynamic environments where both accuracy and explainability are required.

Statistical validation is conducted using paired t-tests to ensure that the observed improvements are not due to random variation. The results confirm that the performance gains are statistically significant (p < 0.05). In addition, 95% confidence intervals indicate stable performance across multiple simulation runs. These findings strengthen the reliability of the proposed framework and address the need for rigorous and interpretable evaluation highlighted in the research gap.

The graphical analysis in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 provides additional insight into system behaviour, scalability, and performance trade-offs. These visualisations complement quantitative metrics and allow clearer interpretation of the framework’s advantages over existing methods.

5.6. Real-Time Performance and Scalability Analysis

The computational complexity varies by inference method: variational inference: $O(N\cdot K)$, MCMC: $O(N\cdot T)$; Sequential Monte Carlo: $O(N\cdot P)$; and latency, measured as

$$\tau ={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^T}{t}_{\mathrm{inference}}(t)$$

Experimental results show the following:

$N<50$: Latency < 100 ms (real-time feasible);

$50<N<100$: Moderate latency increase;

$N>100$: Real-time performance degrades without approximation.

Sequential Monte Carlo performs best in streaming environments due to incremental updates.

The simulation is fully reproducible through fixed random seed, explicit parameter definitions, and controlled stochastic processes. However, the use of synthetic data may limit generalisability to high-stakes real-world applications. To mitigate this, parameter settings are aligned with empirical patterns reported in prior studies, and the framework is designed to be domain-adaptive. Future work will validate the framework using real-world datasets in domains such as healthcare, finance, and cyber–physical systems.

6. Limitations and Benefits

6.1. Benefits

The proposed framework is an explainable AI (XAI)-driven multi-objective decision and control system that integrates Bayesian networks with optimisation methods. It supports decision-making in dynamic environments characterised by uncertainty, interdependence, and continuous data updates.

Unlike traditional multi-criteria decision-making methods such as AHP, TOPSIS, and PROMETHEE, which rely on fixed weights and deterministic assumptions, the proposed framework models uncertainty explicitly using probabilistic reasoning. This enables adaptive and data-driven decision-making as new information becomes available.

A key strength of the framework is its ability to capture dependencies among decision variables. The Bayesian structure represents interactions between factors, allowing changes in one variable to propagate through the system. This improves realism and enhances decision accuracy.

The framework also supports the integration of heterogeneous data sources, combining quantitative indicators with qualitative expert inputs. This unified representation improves consistency and enables more comprehensive modelling of complex systems.

Interpretability is achieved through intrinsic probabilistic reasoning and sensitivity analysis. Users can identify influential variables and trace how decisions are generated. In addition, Bayesian updating enables real-time adaptation, ensuring that the system remains responsive to new data.

Decision alternatives are evaluated using expected utility optimisation, enabling structured and transparent comparison of strategies under uncertainty. This integration of inference and optimisation provides a robust foundation for intelligent decision systems.

6.2. Limitations

Despite its advantages, the framework has several limitations. The construction of a Bayesian network requires careful specification of variables and dependencies. This process depends on domain expertise and may be time-consuming.

The framework is sensitive to data quality. Biassed or incomplete data can affect probability estimates and lead to suboptimal decisions. Continuous validation and updating are therefore required.

Modelling qualitative inputs remains challenging. Confidence-based representations capture uncertainty, but subjective judgments cannot always be fully expressed in probabilistic form. From a computational perspective, exact Bayesian inference is expensive and grows exponentially with the number of dependencies ($O\left(2^k\right)$). To address this, the framework employs approximate inference methods such as variational inference, reducing complexity to $O(N\cdot K)$.

Empirical results (Section 5) indicate that real-time performance is achieved when system dimensionality remains within practical limits (e.g., fewer than 50 variables) and latency constraints ($\tau <100$ ms) are satisfied. Beyond this range, inference latency increases and approximation errors may affect performance. These findings define the operational boundaries under which scalability and real-time capability are maintained.

Practical implementation may also face challenges related to system integration, data compatibility, and user understanding. Probabilistic outputs may be misinterpreted by non-expert users, requiring appropriate interface design and user training.

Finally, the current evaluation relies on simulated user behaviour rather than empirical user studies. Future work should include real-world validation to assess usability, trust, and decision effectiveness.

Despite its advantages, the proposed framework may fail under specific conditions. First, performance degrades in extremely high-dimensional systems where the number of dependencies leads to exponential growth in Conditional Probability Tables, exceeding computational limits. Second, the framework is sensitive to poor-quality or biassed input data, which may propagate through probabilistic dependencies and affect decision outcomes.

Third, real-time performance may not be achievable when latency constraints are violated due to complex inference requirements or large-scale streaming data. Finally, the framework assumes that underlying system dynamics can be reasonably approximated by probabilistic models; in highly chaotic or non-stationary environments, this assumption may not hold. These conditions define the practical boundaries of the framework’s applicability.

Additionally, the reinforcement learning–based approaches, particularly deep reinforcement learning, may outperform the proposed framework in highly complex environments. With large-scale state–action spaces, provided sufficient training data are available. However, such methods often lack transparency and require extensive computational resources. In contrast, the proposed Bayesian–XAI framework prioritises interpretability and structured uncertainty modelling. Decision transparency makes it more suitable for high-stakes domains where explainability and human oversight are critical.

6.3. Ethical and Trust Considerations in XAI

The framework is designed to support transparent and human-centred decision-making. Interpretability is provided through probabilistic reasoning, feature attribution, and visualisation tools. These mechanisms allow users to trace decision logic and understand the influence of key variables.

However, explainability alone does not guarantee fairness. Bias in training data may propagate through Conditional Probability Tables (CPTs), leading to systematically skewed posterior estimates and potentially unfair decisions.

To address this, the framework incorporates bias detection and mitigation mechanisms. Distributional analysis compares posterior probabilities across groups. Sensitivity analysis identifies variables that disproportionately influence outcomes. Counterfactual evaluation assesses how decisions change under controlled perturbations.

Mitigation strategies include reweighting CPT parameters, incorporating fairness-aware priors, and adjusting decision thresholds to improve equity. These mechanisms extend the framework beyond transparency to support fairness and accountability.

The framework also supports human-in-the-loop decision-making. Users can evaluate alternative scenarios, validate model outputs, and intervene when necessary. This enhances trust and aligns the system with ethical AI principles. Future research should further address fairness, regulatory compliance, and domain-specific ethical constraints to ensure responsible deployment in high-stakes applications.

7. Conclusions

This study addressed multi-objective decision-making in dynamic and uncertain environments. It proposed a hybrid Bayesian–XAI framework that integrates probabilistic inference, optimisation, and explainability within a unified architecture. The framework formulates decision-making as a stochastic control problem, in which system states are treated as latent variables, and actions are selected to maximise expected utility subject to constraints.

The Bayesian network captures dependencies among variables, while recursive inference enables continuous updating as new data becomes available. This supports adaptive decision-making in non-stationary and data-driven environments. The integration of intrinsic probabilistic reasoning with post hoc explainability improves interpretability and allows clear evaluation of trade-offs between competing objectives.

The results demonstrate that the proposed framework achieves higher decision quality, greater robustness, and greater transparency than baseline methods. It also maintains a balanced trade-off between performance and computational efficiency, making it suitable for real-time decision-support systems under controlled conditions.

However, the framework has defined operational limitations. Performance may degrade in high-dimensional systems with dense dependencies, where inference becomes computationally intensive, and approximation errors increase. In addition, the framework is sensitive to data quality, and biassed or incomplete inputs may affect both inference and decision outcomes. These conditions define the practical boundaries of its applicability.

The framework is general and can be applied to intelligent platforms, cyber–physical systems, and data-driven decision environments. Future research will focus on improving scalability through adaptive inference methods, enabling automated structure learning for large-scale Bayesian models, and incorporating fairness-aware mechanisms to mitigate the propagation of bias. Further validation using real-world datasets is also required to strengthen generalisability and applicability in high-stakes contexts.

Moreover, this study contributes a structured and interpretable decision framework that integrates learning, inference, and optimisation. It advances hybrid AI systems by formalising the interaction between explainability and probabilistic decision-making, thereby supporting reliable and adaptive decision processes under uncertainty.

8. Future Research Directions

Future research can extend the proposed framework in several directions. First, empirical validation should be expanded using larger and more diverse datasets. This includes data from different application domains, such as XAI-driven platforms, cyber–physical systems, and intelligent services. Broader datasets will improve generalisability and enhance the model’s robustness. Second, future studies can integrate more advanced machine learning and optimisation techniques. Methods such as Sequential Monte Carlo, variational inference, and reinforcement learning can improve scalability and decision performance in high-dimensional environments. These methods enable efficient learning and optimisation under uncertainty.

Third, the framework can be extended to support real-time data integration. Data from streaming sources, such as sensors, digital platforms, and user interaction systems, can be incorporated. This will enable continuous updates to model parameters and support real-time decision-making. Fourth, hybrid models that combine Bayesian networks with deep learning can be explored. Deep learning can improve predictive accuracy, while Bayesian models maintain interpretability and uncertainty representation. This combination aligns with current research on integrating learning and optimisation.

Fifth, future work should improve human–AI interaction and visual analytics. Better interface design can make the system easier to use for non-technical users. Visual tools can help users understand probabilities, dependencies, and trade-offs between decisions. Sixth, the framework can be extended to include distributed and scalable architectures. Implementing the model in parallel and cloud-based systems will improve performance in large-scale applications.

Future research can apply the framework to different domains. These include healthcare systems, smart cities, supply chain optimisation, and sustainability applications. Such studies will demonstrate the flexibility and practical value of the proposed approach.