Inverse Inference of Component Reliability for k-Out-of-n Systems Based on Maximum Entropy and Hazard-Rate Matrix Representation

This study presents a rational inverse inference framework for k-out-of-n systems that derives component-level reliability characteristics from system-level failure and monitoring data. The framework employs a hazard-rate matrix to represent the degradation hierarchy and applies the principle of maximum entropy to allocate system-level probabilities to latent component-state configurations without bias, yielding analytical solutions for component hazard rates. The key innovation lies in combining maximum entropy with the hazard-rate matrix, which overcomes the ill-posed nature of the inverse problem and enables systematic integration of heterogeneous auxiliary information within a unified formulation, including system-level multi-state observations, component-wise moment constraints, sub-component data, and inter-component dependencies. This flexibility addresses a major limitation of existing inverse methods, such as Bayesian approaches, which are typically restricted to a single data type and often require strong prior assumptions or extensive failure datasets. The practical applicability of the framework is demonstrated through a case study of a west-to-east gas pipeline pumping system, highlighting its effectiveness in processing multiple information types and delivering actionable component-level reliability assessments for maintenance decision support. To the best of our knowledge, this is the first study to formulate and solve the inverse inference problem for k-out-of-n systems in a theoretically grounded and information-theoretically optimal manner.