Introduction to the Weibull-Bayesian Distribution

Reliability HotWire

Issue 55, September 2005

Reliability Basics

Introduction to the Bayesian-Weibull Distribution

Introduction

Until the release of Version 7, Weibull++ has dealt exclusively with what is commonly referred to as classical statistics. In this article, another school of thought in statistical analysis will be covered, namely Bayesian statistics. The premise of Bayesian statistics is to incorporate prior knowledge along with a given set of current observations in order to make statistical inferences. The prior information could come from operational or observational data, from previous comparable experiments or from engineering knowledge.

This type of analysis is particularly useful if there is a lack of current test data and when there is a strong prior understanding about the parameter of the assumed life model and a distribution can be used to model the parameter. By incorporating prior information about parameter(s), a posterior distribution for the parameter(s) can be obtained and inferences on the model parameters and their functions can be made.

This article describes the Bayesian concepts, their applications in life data analysis and the Bayesian-Weibull distribution in Weibull++.

Bayes's Rule
Bayes's rule provides the framework for combining prior information about the assumed distribution's parameter(s) θ with sample data to make inferences about the model. The prior knowledge about the parameter(s) is expressed in terms of a pdf φ(θ). L(Data|θ) is the likelihood function for the available data. The conditional distribution of θ given the data set, using Bayes's rule, provides the updated information about the parameters θ. This is expressed with the following posterior pdf:

(1)

where:

θ is a vector of the parameters of the chosen distribution
ς is the range of θ
L(Data|θ) is the likelihood function based on the chosen distribution and data
φ(θ) is the prior distribution for each of the parameters

The integral in Eqn. (1) is often referred to as the marginal probability (which is a constant) and can be interpreted as the probability of obtaining the sample data given a prior distribution. Generally, the integral in Eqn. (1) does not have a closed-form solution and numerical methods are needed for its solution.

It can be seen from Eqn. (1) that there is a significant difference between classical and Bayesian statistics. First, the idea of prior information does not exist in classical statistics. All inferences in classical statistics are based on the sample data. On the other hand, in the Bayesian framework, prior information constitutes the basis of the theory. Another difference is in the overall approach of making inferences and in their interpretation. For example, in Bayesian analysis the parameters of the distribution to be ''fitted'' are the random variables. In reality, there is no distribution fitted to the data in the Bayesian case. For instance, consider the case where a data set is obtained from a reliability test. Based on prior experience on a similar product, the analyst believes that the shape parameter of the Weibull distribution has a value between β₁ and β₂, and wants to utilize this information. This can be achieved by using the Bayes theorem. At this point, the analyst is automatically forcing the Weibull distribution as a model for the data with a shape parameter between β₁ and β₂. In this example, the range of values for the shape parameter is the prior distribution, which in this case is Uniform. By applying Eqn. (1), the posterior distribution of the shape parameter will be obtained. So the analyst will end up with a distribution for the parameter rather than an estimate of the parameter as in classical statistics.

Bayesian-Weibull Analysis
Applying Bayes's rule, Eqn. (1), on the Weibull distribution, we obtain the following joint pdf distribution of β and η posterior:

The posterior distribution of failure time T is:

where:

is the pdf function of the two-parameter Weibull distribution.

This model considers prior knowledge on the beta parameter of the Weibull distribution when it is chosen to be fitted on a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the one-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes'' model, unlike the one-parameter Weibull that is commonly referred to as such) offers an alternative to the one-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Initial studies performed by the ReliaSoft R&D group show some very promising results using this model.

In this model, η is assumed to follow a non-informative prior distribution with the density function φ(η)=1/η. The prior distribution of β, denoted as φ(β), can be selected from the following distributions: normal, lognormal, exponential and uniform.

Prior Distributions
Prior distributions play a very important role in Bayesian statistics. They are essentially the basis of Bayesian analysis. Different types of prior distributions exist, namely, informative and non-informative. Non-informative prior distributions (a.k.a. vague, flat and diffuse) 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 greatly affected by external information, or when external information is not available. The uniform distribution is frequently used as a non-informative prior distribution.

On the other hand, informative priors have a stronger influence on the posterior distribution. The influence of the prior distribution on the posterior is related to the sample size of the data and the form of the prior. Generally speaking, large sample sizes are required to modify strong priors, where weak priors are overwhelmed by even relatively small sample sizes. Informative priors are typically obtained from past data.

The Posterior R(T) and λ(t)

The expected reliability at time T is:

where:

The expected failure rate at time T is:

or:

where:

is the two-parameter Weibull failure rate function.

Confidence Bounds

Confidence bounds estimations will be discussed in future issues of the Reliability HotWire.