This is an expository paper dealing with Bayesian inference for three important mixture problems in quality and eliability. The traditional approach for estimation in these situations is the method of maximum likelihood. The corresponding inference based on large-sample theory can, however, be misleading in situations where the large-sample normal approximation is not adequate. The Bayesian approach, on the other hand, has been viewed as computationally intractable due to the complex nature of mixture models. Recent advances in Bayesian computational methods have alleviated this problem considerably. We illustrate the use of data augmentation methods for doing Bayesian inference in these applications. While the framework is formally Bayesian in nature, it can also be viewed as a computational device for calculating the likelihood function and doing likelihood-based inference. An additional advantage of data augmentation methods is that no further complications arise when failure time data are grouped or censored.
Keywords: Bayesian Methods, Likelihood, Manufacturing, Posterior Distributions, Quality Control, Reliability.
byVIJAYAN N. NAIR, University of Michigan, Ann
Arbor, MI 48109-1285
BOXIN TANG, University of Memphis, Memphis, TN 38152
LI-AN XU, Bristol-Meyer and Squibb, Princeton, NJ 08543-1285
Finite mixture models arise in a variety of industrial and scientific applications. In this paper, we consider Bayesian inference for three important classes of problems that arise in quality and reliability applications. The paper is intended to be expository, and it is aimed primarily at practitioners.
The first problem is concerned with the well-known "infant mortality" phenomenon in reliability where there are early failures due to manufacturing defects (see, for example, Meeker and Escobar (1998)). A common practice with such data is to fit a single distribution to the combined failure data - usually a distribution with a decreasing hazard rate. While this may be convenient, it does not adequately capture the fact that there are two different underlying causes of failure. A better approach is to view the failures as arising from a mixture population: one due to initial manufacturing defects and the other due to degradation-related reliability problems. This allows one to separately estimate the various parameters of interest including the mixing proportion and the usage-related reliability distribution.
A second type of mixture problem arises in high reliability applications, such as integrated circuits, where a significant proportion of the components do not fail (at least for a long time). Meeker (1987) considered a limited failure population model for such situations where the distribution is a mixture of a regular distribution function (corresponding to the failure time of defects) and a distribution with all of its mass at infinity (corresponding to units that do not fail).
The final application deals with count data on manufacturing defects. The Poisson distribution, which is often used to model such data, is not adequate in many applications. Lambert (1992) considered the analysis of data on soldering defects in printed wiring boards where there is an excessive number of zeros (no defects). She discussed the use of a zero-inflated Poisson model, a mixture of a Poisson distribution and a degenerate distribution at zero, for such data.
We will consider these three types of mixture problems in detail in this paper. Mixture models have also been used to explain spatial clustering of defects in integrated circuit manufacturing (Cunningham (1990); Friedman, et al. (1997)). Titterington, Smith and Makov (1985) provide an extensive list of other interesting applications where finite mixture models arise.
Many methods have been discussed in the literature for inference in finite mixture models. These include method of moments, maximum likelihood, and Bayesian inference. The method of moments lacks the attractive properties of maximum likelihood estimation and is mostly of historical interest, although it can be useful in providing initial estimates for computing maximum likelihood estimates (MLEs). The maximum likelihood method, in contrast, provides asymptotically efficient estimates. Yet inferences based on the large-sample theory will not be valid in situations where the limiting normal approximation is not adequate. An extra difficulty is that the likelihood can sometimes become unbounded for mixture models (Everitt and Hand (1981)). Nonetheless, likelihood estimation remains a commonly used approach (see, for example, Meeker (1987) and Lambert (1992)).
The Bayesian approach, on the other hand, has been viewed as computationally intractable due to the complex nature of mixture models. This is because the posterior for a mixture model generally does not have a simple form, and extensive computation is required to obtain the exact posterior. This difficulty motivated Titterington (1976) and Smith and Makov (1978) to consider the "quasi" Bayesian approach. Normal approximation to the posterior is also considered in the literature (see, for example, Tanner (1993)). However, recent advances in Bayesian computational methods have alleviated the computational problem considerably.
The goal of this paper is to introduce the advantages and flexibility of these methods for modeling three important mixture problems to quality and reliability practitioners.These computational techniques are being increasingly used within the statistical community, and it is important they become part of the tool kit of engineers as well. The data augmentation idea, which is the key component in this approach, is very natural in mixture problems, and it eases the computational burden in computing the posterior distributions a great deal. While it does not provide a universal solution to the problem, it is very useful in cases where knowledge of the category (or population) to which each observation belongs leads to a simple form for the posterior.
Although the approach is formally Bayesian in nature, when the prior distribution is taken to be "non-informative" the posterior distribution is essentially proportional to the likelihood. Thus, the approach can be viewed as a computational device for calculating likelihood functions and for doing likelihood-based inference. This approach allows one to compute the likelihood or posterior distribution, examine it to see if the standard large-sample normal approximations are valid, and, if not, to use the posterior distribution directly to make inference.
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