Bayesian Evaluation of Reliability Data

In the preceding sections the so-called classical methods for estimating reliability data from observed lifetimes and failure frequencies were treated. It was shown as well how the corresponding confidence intervals are determined. All procedures were based on the frequentist concept of statistics. Its underlying idea is the probability expressed as the limit of a relative frequency, i.e. 

If an experiment, in which the event denoted by E (e.g. the failure on demand of an electric shunt) can occur, is carried out n times with n tending towards infinity, the observed relative frequency for the occurrence of the event ns/n tends towards the unknown constant value u. This value is called probability of E (cf. Appendix C). 

The definition of probability as relative frequency implies that the event under investigation—here the failure of a component—must have occurred several times lest the confidence intervals be too large. If a component has rarely failed an evaluation using Bayes theorem is appropriate [25]. It is based on the so-called subjective notion of probability. 

In evaluating reliability data, as treated in detail in [25], Bayes theorem serves to combine lifetime observations for technical components coherently with the knowledge on the lifetime of this component type that existed prior to the lifetime observations. The formula is  

43) f(X) is the prior probability density function. It reflects the— subjective—assessment of component behaviour which the analyst had before the lifetime observations were carried out. L(EA.) is the likelihood function. It is the conditional probability describing the observed failures under the condition that f(5t) applies to the component under analysis. Eor failure rates L(E/X) is usually represented by a Poisson distribution of Eq. (9.30) and for unavaUabUities by the binomial distribution of Eq. (9.35). The denominator in Eq. (9.43) serves for normalizing so that the result lies in the domain of probabilities [0, 1] f(X/E) finally is the new probability density function, which is called posterior probability density function. It represents a synthesis of the notion of component failure behaviour before the observation and the observation itself. Thus it is the mathematical expression of a learning process. 

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