# Bayesian Inference in Statistical Analysis (Wiley Classics by George E. P. Box, George C. Tiao

By George E. P. Box, George C. Tiao

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Additional info for Bayesian Inference in Statistical Analysis (Wiley Classics Library)

Example text

33) As an example, consider again the binomial mean n. L(n I y) = log I(n I y) = const + y log n + The log likelihood is (n - y) log (1 - n). 24). 37b) is precisely the metric employed in plotting the nearly data translated likelihood curves in Fig. 6. We recognize the sin -l-Jrr transformation as the weJlknown asymptotic variance stabilizing transformation for the binomial, originaJly proposed by Fisher. [See, for example, Bartlett (1937) and Anscom be (1948a)]. In the above we have specifically supposed that the quantity (- -;;1 aa8L) 2 2 iJ e is a function of only.

3) pee I 0') cc C. e e This noninformative pnor distribution is shown line , Since p(K I 0') == pee I 0') In Fig. 4) the corresponding noninformative prior for K is not uniform but is locally proportional to 2 , that is, to K - 2 In general, if the noninformative prior is locally e Nature of Bayesian Inference 28 uniform in 4>(8), then the corresponding noninformative prior for 8 is locally proportional to Id4>/d81, assuming the transformation is one to one. It is to be noted that we regard this argument only as indicating in what metric (transformation) the local behaviour of the prior should be uniform.

Thus "'0 1 (e-900)2J P (8) = - -1 - exp l' - - - . A 2IT 20 . 10a) According to A, a priori 0 ~ N(900, 20 2 ) where the notation means that 0 is distributed Normally with "mean 900 and variance 20 2 " This would imply, in particular, that to A the chance that the value of 8 could differ from 900 by more than 40 was only about one in twenty. By contrast, we suppose that B has had little previous experience in this area ,a nd his rather vague prior beliefs are represented by the Normal distribution PB(8) = I ,-,,2IT HO exp [I (e -2 -80800 )2] .