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Reliability Basics | |||||||||||

One of the many new additions to Weibull++ 7 is the method of estimating confidence bounds based on the Bayes theorem. This type of confidence bounds relies on a different school of thought in statistical analysis, where prior information is combined with sample data in order to make inferences about model parameters and their functions. An introduction to Bayesian methods is given in this article.
Bayesian confidence bounds are derived from Bayes's rule, which states that:
where: *f(θ|Data)*is the posterior*pdf*of*θ.**θ*is the parameter vector of the chosen distribution (*i.e.*Weibull, lognormal, etc.)*L(.)*is the likelihood function.*φ*(*θ)*is the prior*pdf*of the parameter vector.*ς*is the range of*θ.*
In other words, the prior knowledge is
provided in the form of the prior
θ
_{1 },_{ }φ(θEqn. (1) can be written as:_{1})
In other words, we now have the
distribution of θis less than or equal to a
value _{1
}x, P( θcan be obtained by integrating
Eqn. (2), or:_{1 }≤ x
)
Eqn. (3) essentially calculates a
confidence bound on the parameter, where x is the
confidence bound. (Note: In Bayesian statistics, the term confidence bounds
is not correct. Credible bounds is the correct term. However, since
from an application perspective the result has the same interpretation, we
will use the term confidence bounds to avoid confusion.) Substituting Eqn.
(2) into Eqn. (3) yields:
The only question at this point is what we
should use as a prior distribution of beta and eta are assumed
to be uniform.Eqn. (4) can be generalized for any
distribution having a vector of parameters
where: *CL*is confidence level.*θ*is the parameter vector._{ }*L(.)*is the likelihood function.*φ*(*θ)*is the prior*pdf*of the parameter vector.- ς is the range of
*θ.* - ξ is the range in which
*θ*changes from*Ψ(T,R)*till*θ*maximum value or from*θ's*minimum value till*Ψ(T,R). Ψ(T,R)*is the function such that if*T*is given then the bounds are calculated for*R*and if*R*is given, then the bounds are calculated for*T.*
If If When the data set being analyzed is of small size, the Bayesian bounds method is usually preferred over the Fisher Matrix, the Likelihood Ratio and the Beta Binomial methods. The advantage of the Bayesian bounds method lies in the fact that it makes the fewest assumptions about the distribution of the parameters. The Fisher Matrix method relies on a normality assumption. The Likelihood Ratio method relies on the assumption that
follows a Chi-Square distribution. The Beta Binomial method is a non-parametric method, which discourages making predictions outside the range of data. (Note also that in Weibull++ 7 the Beta Binomial method is only available for the Mixed Weibull distribution.) The Bayesian confidence method is free of all of these assumptions since the posterior distribution is calculated directly. | |||||||||||

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