{ "background": "Process-control systems in industrial settings are critical for optimising yield, yet diagnostic tools for quantifying improvement in such systems, particularly in developing industrial contexts, are often limited to frequentist methods that do not fully account for multi-level variability and prior operational knowledge.", "purpose and objectives": "This study aimed to develop and evaluate a novel Bayesian hierarchical model to diagnose and quantify yield improvement in industrial process-control systems, providing a robust framework for inference under uncertainty.", "methodology": "A Bayesian hierarchical model was formulated, integrating plant-level and process-unit-level data. The core structure is given by $y{ij} \\sim \\text{Normal}(\\mu + \\alphai + \\betaj, \\sigma^2)$, with priors $\\alphai \\sim \\text{Normal}(0, \\tau\\alpha)$ and $\\betaj \\sim \\text{Normal}(0, \\tau\\beta)$, where $y{ij}$ is the yield, $\\alphai$ represents plant-specific effects, and $\\betaj$ represents unit-specific effects. The model was implemented using Hamiltonian Monte Carlo sampling and validated on operational data from multiple sites.", "findings": "The model successfully quantified yield improvements with credible intervals, identifying that approximately 78% of the posterior probability mass for the overall yield gain parameter lay above the threshold of practical significance. A key finding was the dominant contribution of unit-level control adjustments, which accounted for an estimated 65% of the total explained variance in yield improvement.", "conclusion": "The proposed Bayesian hierarchical model provides a statistically robust diagnostic tool that effectively quantifies yield improvements in multi-level process-control systems, offering advantages in uncertainty quantification over traditional methods.", "recommendations": "Adoption of this modelling framework is recommended for engineers and plant managers seeking to make data-informed decisions on process interventions. Further research should focus on integrating real-time data streams for dynamic updating of the model.", "key words": "Bayesian inference, hierarchical modelling, process control,