Contemporary complex systems generally have multiple sensors embedded at various levels within their structure. Sensors are data gathering mechanisms that measure a systemic quantity (such as functionality or failure) providing the engineer with a multitude of reliability information. Data sets are said to be overlapping when drawn simultaneously from multiple sensors in a system. Current methodologies focus on system reliability analysis of non-overlapping data sets. We introduce a Bayesian methodology that allows analysis of overlapping data sets, exploiting their inherent inter-dependence to yield significant additional information.

Data gathered from any sub-system or component contextualizes data gathered from a sensor placed at the `top' of the system (i.e. systemic functionality) through dependence. A system that is functional in spite of a non-functional sub-system infers information about the reliability characteristics of the clearly functional remainder of the system. The same principle extends to any other sensor that has subordinate sensors upon which it is observationally dependent. We apply overlapping Bayesian analysis on several example systems to highlight the information inherent in overlapping data sets and compare these results to those obtained by constraining the data to be analysed as if it were non-overlapping.

The Bayesian methodology we introduce deals with on-demand and continuous life metric systems. The likelihood function for on-demand systems accommodates multiple degraded states and relies on an algorithm we introduce that rapidly generates combinations of disjoint cut-sets that imply the evidence. The likelihood function for continuous life-metric systems (such as those whose failure probability is time based) examines each sensor data when contextualised through all other data sets. We generalise these likelihood functions for uncertain data, allowing simplification through real-life measuring inaccuracies.

Finally, we use the methodologies developed above to assess probable information gain for various sensor placement permutations. We embed this process into a Bayesian experimental design framework to optimise sensor placement. This can then be fed into any multi-objective optimization framework, or used in isolation to allow informed sensor placement.