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Article Excerpt
1. Background
Some years ago Richard E. Barlow wrote the book Reliability Engineering (Barlow, 1998), where he took a subjectivist approach based on basic engineering judgements, from which engineering decisions, based on Bayesian belief nets, could be built. He also included in the book an interesting section on the Total Time on Test (TTT) plotting technique, suggested by Barlow and Campo (1975). However, the inclusion of the TTT plotting technique was not very well argued for within the subjectivist framework.
In this paper I will argue for an extension to the usual Bayesian or subjectivist framework naturally including techniques, such as the TTT plotting above, challenging the judgements whereupon statistical inference and subsequent decisions are built. The starting point will be in the knowledge theory developed by the Harvard philosopher Clarence I. Lewis. Lewis called himself a conceptualistic pragmatist to differentiate himself from the earlier pragmatists such as C. S. Peirce, William James and John Dewey. He emphasized that when interpreting the sensuously given we always have to use our prior conceptions to build our experience. Lewis' ideas were an important source of inspiration to the originator of the modern approach to quality improvement, Walter A. She-whart (see Shewhart (1931, 1939) and subsequently to W. Edwards Deming (see Deming (1986, 1993)) and to the learning approach, symbolized by the so-called PDSA cycle, which, partly via a Japanese influence, permeates the current quality movement. These ideas seem quite similar to the Bayesian thoughts. However, Lewis challenged us to always reflect on our prior understanding, and, if felt necessary, make changes. In Lewis' theory, learning is not only incremental as in the Bayesian theory, but also dramatic shifts might occur. This will be the starting point of the current paper, the purpose of which is to discuss the relations between the two approaches.
We live in a world of variation. Everywhere in our world we find variation, be it in the physical world, in the biological world, or in the man-made artificial world (Bergman, 2003). The possibility to learn and act in spite of this fact is well known to us. At least partly, our worlds are predictable, at least within limits, and in spite of uncertainties and variation, we can learn. Statistics is an important discipline for handling variation and on which to base predictions in spite of all variation.
Two important themes in the contemporary statistics discourse are the subjectivist approach to statistical inference and the usage of statistics in the quality movement. In both themes prediction and learning play important roles. In the subjectivist approach to statistics the update of a priori knowledge is emphasized and, similarly, the PDSA cycle of learning, to be discussed further below, is emphasized in the quality movement. In both approaches learning is seen as important in order to improve our capability for decision making and action in the future.
The origins of the subjectivist approach to probability are quite well known with important milestones provided by Thomas Bayes, Pierre Laplace, Frank Ramsey, Bruno de Finetti and Leonard Savage. Interesting applications to reliability engineering are discussed by Barlow (1998). The corresponding origins of the quality movement, with the seminal works of Walter A. Shewhart (Shewhart, 1931, 1939), and his approach to learning with an origin in the works of the philosopher C. I. Lewis (Lewis, 1929, 1934), seems less well known, even within the quality movement. In the next sections we will give brief discussions on the subjectivist approach to statistics and on the search for statistical control, predictability and learning in the quality movement. Some problematic issues will be highlighted.
In Section 5, we will make a first attempt to combine the two approaches to learning. Finally, we will conclude the paper with a short discussion, some remarks and a summary.
2. The a priori in the subjectivist approach to probability
Bayesian analysis provides an excellent framework for evolutionary learning. Based in our pre-understanding of the world--judgements of patterns of variation in the world around us and our prior understanding formalized in probability distributions--we gather data and, in order to make decisions, we update our prior understanding. Decisions are assumed to be followed by action. Knowledge is for action in the subjectivist's world.
The Bayesian paradigm takes its fuel from Bayes' theorem--from an a priori probability distribution in combination with a likelihood function a posterior probability distribution is obtained. The general form of the likelihood function is derived from a judgement on the nature of the variation pattern of the phenomenon under study and its actual shape is determined by the observations made: i.e., conditionally on some unobserved quantity [theta], a probability distribution, or, more generally, a stochastic belief model, is assumed to describe the general variation pattern of the phenomenon under study and its potential observations. The likelihood function is a weighting function derived from the belief model concerning the variation pattern and, essentially, from the corresponding conditional probability, given the value of [theta], of observing what has actually been observed.
The value of the parameter or quantity [theta], is considered uncertain and it is assumed that, from the domain knowledge of the decision maker, this uncertainty is describable utilizing a probability distribution, the a priori distribution.
The posterior distribution can be used to make predictive inference (the only interesting feature from an action point of view) about the future, and suitable decisions and subsequent actions can be taken to "optimize" the consequences of the actions. (Of course, there is a hidden assumption in this--actions are taken as determined by the decisions but that is perhaps not always...
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