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By D V Lindley

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The cost function is en (so c0 = 0, cl = c) and the loss functions have av — a2, b1 = b2 = b, say, so we may write19 L(d, 9) = b\9 — 90\. The obvious application is to industrial sampling inspection for attributes 20 where 90 is the break-even value. Hald shows that as c -»0 (so that the optimum sample size becomes large) the latter is given by with optimum expected loss (200(1 — 00)p(00)bc}1/2 + O(l). These ideas have also been extended to several stages by Hald and Keiding (1969). For two stages of sizes nl and n 2 the asymptotic results are where N = c~l.

Typically p(s|0) does not depend on 0 but sometimes it does as when sampling fibres, the chance of a fibre being included in s depending on its unknown length. Notice that the model supposes that a single sample is taken; it can be extended to sequential sampling. The sample having been taken, measurements x = ( x 1 , xz, • • • , xn) are made on the individuals in the sample. Here Xj is the measurement on the jth sample member. , that is, the measurement is exactly the description of the individual.

LINDLEY under H, independent prior beta distributions with parameters at, bh the posterior distributions will also be beta. The above result can be applied with 22 £ = 0j — 6 rj = |(0 j + 0 2 ). The calculations of p(£0\H) and p(£ 0 |x,H) with £0 = 0 follow easily from properties of the beta distributions. 2)) suppose The expected utilities are then and similarly, and the posterior odds are directly relevant to the solution of the decision problem. If R is the ratio of the two integrals, then H is accepted if RLO exceeds unity, where 0 is the prior odds.

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Bayesian statistics, a review by D V Lindley

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