Reliability engineering asks: how long will a system work before it fails? What is the probability of failure within a specified mission time? How should maintenance be scheduled to minimize downtime while controlling costs? These questions arise in aerospace, nuclear power, defense, medical devices, and any field where failures have serious consequences. Bayesian methods are naturally suited to reliability because prior information — from similar systems, accelerated testing, physics-of-failure models, and engineering judgment — is typically abundant and valuable, while failure data from the specific system of interest are often sparse.
Failure Rate Estimation
The most basic reliability problem is estimating the failure rate λ from observed failure and survival data. With a conjugate Gamma prior on λ and an exponential (constant failure rate) likelihood, the posterior is also Gamma, allowing closed-form updates. For more realistic failure models — Weibull, log-normal, or bathtub-shaped hazard functions — MCMC provides posterior samples for all model parameters.
Likelihood: L(data | λ) = λⁿ · exp(−λ · T)
Posterior: λ | data ~ Gamma(α₀ + n, β₀ + T)
where n = number of failures and T = total time on test.
Bayesian estimation naturally handles the common situation where some units are still operating at the end of a test (right-censored data), some were removed for reasons unrelated to failure (competing risks), and the test conditions may differ from field conditions (accelerated life testing).
System Reliability and Fault Trees
Complex systems have multiple components in series, parallel, or mixed configurations. Bayesian networks map naturally onto reliability block diagrams and fault trees, propagating uncertainty from component-level failure distributions to system-level reliability. When component failure data are sparse, hierarchical Bayesian models share information across similar components — for example, estimating failure rates for a class of valves while allowing individual variation.
Probabilistic risk assessment (PRA) for nuclear power plants is one of the most mature applications of Bayesian reliability. The NRC's NUREG databases provide prior distributions for component failure rates based on industry-wide operating experience. Plant-specific data update these priors, producing site-specific risk estimates. This Bayesian updating framework has been standard practice in nuclear safety since the 1980s and influences regulatory decisions about reactor operation and maintenance.
Degradation Modeling and Predictive Maintenance
Many failures are preceded by gradual degradation — crack growth, wear, corrosion, insulation breakdown. Bayesian degradation models track the evolution of a degradation indicator and predict the remaining useful life (RUL) of individual units. As new sensor measurements arrive, the posterior distribution over the degradation state and rate parameters is updated, producing increasingly precise RUL predictions. This enables condition-based maintenance that replaces components just before failure rather than on a fixed schedule.
Accelerated Life Testing
Testing products to failure under normal conditions may take years. Accelerated life testing (ALT) subjects units to elevated stress (temperature, voltage, vibration) and uses a physics-based model to extrapolate to normal conditions. Bayesian ALT analysis places priors on the acceleration model parameters, naturally quantifying the extrapolation uncertainty that is critical for predicting field reliability from laboratory tests.
"In reliability, the prior is not a nuisance — it is the engineering knowledge that makes analysis possible when you have two failures and need to make a decision." — William Q. Meeker and Luis A. Escobar, authors of Statistical Methods for Reliability Data
Current Frontiers
Digital twin reliability models combine physics-based simulations with Bayesian updating from real-time sensor data. Bayesian methods for software reliability estimate defect rates and predict failure-free operation periods. And the application of Bayesian reliability to machine learning systems — estimating the probability that an AI model will produce a safe output — is an emerging frontier at the intersection of reliability engineering and AI safety.