Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 60, February 2006

Reliability Basics

Bayesian Confidence Bounds

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.

Introduction

Bayesian confidence bounds are derived from Bayes's rule, which states that:

 

(1)

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 pdf of the parameters, which in turn is combined with the sample data in order to obtain the posterior pdf. Different forms of prior information exist, such as past data, expert opinion or non-informative. It can be seen from Eqn. (1) that we are now dealing with distributions of parameters rather than single value parameters. For example, consider a one-parameter distribution with a positive parameter θ1. Given a set of sample data and a prior distribution for θ1 , φ(θ1) Eqn. (1) can be written as:

(2)

In other words, we now have the distribution of θ1 and can make statistical inferences on this parameter, such as calculating probabilities. Specifically, the probability that θ1 is less than or equal to a value x, P( θ1 ≤ x ) can be obtained by integrating Eqn. (2), or:

(3)

Eqn. (3) essentially calculates a confidence bound on the parameter, where P( θ1 ≤ x ) is the confidence level and 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:

(4)

The only question at this point is what we should use as a prior distribution of θ1. For the confidence bounds calculation application, non-informative prior distributions are utilized. Non-informative prior distributions are distributions that have no population basis and play a minimal role in the posterior distribution. The idea behind the use of non-informative prior distributions is to make inferences that are not affected by external information, or when external information is not available. In the general case of calculating confidence bounds using Bayesian methods, the method should be independent of external information and it should rely on only the current data. Therefore, non-informative priors are used. Specifically, the uniform distribution is used as a prior distribution for the different parameters of the selected fitted distribution. For example, if the Weibull distribution is fitted to the data, the prior distributions for beta and eta are assumed to be uniform.

Eqn. (4) can be generalized for any distribution having a vector of parameters θ, yielding the general equation for calculating Bayesian confidence bounds:

(5)

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 T is given, then from Eqn. (5) and Ψ and for a given CL, the bounds on R are calculated. For more details, click here.

If R is given, then from Eqn. (5) and Ψ and for a given CL, the bounds on T are calculated. For more details, click here.

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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