{ "background": "Process-control systems in industrial settings are critical for operational efficiency, yet diagnostic methods often fail to account for site-specific variability and uncertainty in performance data. This limits the accurate measurement of efficiency gains following system interventions.", "purpose and objectives": "This study develops and validates a novel Bayesian hierarchical model to quantify diagnostic efficiency gains within industrial process-control systems. The objective is to provide a robust framework that incorporates inherent data uncertainty and heterogeneity across different operational sites.", "methodology": "A Bayesian hierarchical model was formulated, specified as $y{ij} \\sim \\text{Normal}(\\alphaj + \\beta X{ij}, \\sigma^2)$, $\\alphaj \\sim \\text{Normal}(\\mu{\\alpha}, \\tau^2)$, where $y{ij}$ is the efficiency metric for system $i$ in plant $j$. The model was applied to diagnostic data from multiple industrial plants. Inference was based on posterior distributions with 95% credible intervals.", "findings": "The model estimated a central efficiency gain of 18.7% (95% CrI: 15.2% to 22.3%) following diagnostic implementation. Crucially, it revealed significant variation in baseline performance ($\\tau$) between plants, indicating that site-level factors substantially influence overall gains.", "conclusion": "The proposed model provides a statistically robust method for assessing efficiency improvements in process-control diagnostics, formally accounting for site heterogeneity and uncertainty in a single integrated framework.", "recommendations": "Industry practitioners should adopt hierarchical modelling approaches for plant-wide performance assessments to better inform targeted maintenance and capital upgrade decisions. Further research should integrate real-time sensor data into the model's structure.", "key words": "Bayesian inference, hierarchical modelling, industrial diagnostics, process control, efficiency measurement, uncertainty quantification", "contribution statement": "This paper presents a novel application of Bayesian hierarchical modelling to the field of industrial process-control diagnostics, providing a new method to disentangle plant-wide effects from system-specific efficiency