Bayesian Methods in Reliability by Robert A. J. Badoux (auth.), P. Sander, R. Badoux (eds.) PDF
By Robert A. J. Badoux (auth.), P. Sander, R. Badoux (eds.)
When information is accumulated on failure or survival a listing of instances is acquired. the various instances are failure occasions and others are the days at which the topic left the test. those instances either supply information regarding the functionality of the procedure. the 2 varieties could be known as failure and censoring occasions (cf. Smith part 5). * A censoring time, t, supplies much less info than a failure time, for it really is * identified in simple terms that the object survived previous t and never while it failed. the knowledge is tn and of censoring therefore gathered as an inventory of failure occasions t , . . . , l * * * occasions t , t , . . . , t • 1 z m 2. 2. Classical tools The failure instances are assumed to keep on with a parametric distribution F(t;B) with and reliability R(t;B). There are numerous tools of estimating density f(t;B) the parameter B dependent basically at the information within the pattern with none past assumptions approximately B. the supply of strong pcs and software program applications has made the strategy of utmost probability the preferred. Descriptions of so much equipment are available within the booklet by way of Mann, Schafer and Singpurwalla (1974). as a rule the strategy of utmost likelihood is the main precious of the classical methods. the chance strategy relies on developing the joint chance distrilmtion or density for a sample.
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Bayesian procedures (and J others such as maximum likelihood) may be said to obey the likelihood principle; procedures like MVUE are said to violate the likelihood principle. 14. Assessment of Prior Densities The specification of the prior density component of Bayes' Theorem depends on individual experience and beliefs. This poses the very real problem of how to 48 convert such experiences and beliefs into the form of a probability density function p(9). This statisticians and problem psychologists, has been and many exhaustively theoretical by studied both and experimental papers have been published reporting the results of such studies.
Rather, we will usually have a probability model for x, g(xI8), depending on an unknown parameter 8, which itself appears in the model assumed for y. If p(8IY) is the posterior density for 8, given the previous data y, and if x and y are independent, given 8, we may obtain the predictive density for x from the formula 1I"(xIY) = f @ g(xI8)p(8IY) d8. If a 'single-figure' prediction is required, we simply choose a point estimate summary of the density 1I"(xly) using, for example, mode, median or mean.
What we then would do in a detailed analysis is to attempt to deduce from such assumptions the form that rational decision-making procedures should take. The answer that emerges is that rational decision-making requires us to (a) assume the existence of a utility function, (b) act so as to maximize expected utility, where 'rational' means conforming to the axioms laid down for preferences. In short, axiom systems of this kind (and there are many variations in the precise Bayesian list of axioms approach to chosen) lead to decision-making conformity with the axiom system.
Bayesian Methods in Reliability by Robert A. J. Badoux (auth.), P. Sander, R. Badoux (eds.)