By H. S. Migon, D. Gamerman
Providing an built-in examine the 2 major faculties of statistical idea: the frequentist and the Bayesian, this book's major objective is to teach the parallels latest among the consequences given by means of either methodologies. the 2 colleges are in comparison, contrasted and tested within the mild of the latest advancements in theoretical and utilized information examine. This ebook is usually an replace at the box of data regularly, with discussions of recent study utilizing approximation and computationally-intensive equipment. From the graduate point, to these operating within the comparable components of Economics, arithmetic and Engineering this quantity is a far wanted consultant to an built-in view of data.
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Example text
P(x 1 0) = p(x I 0) = 0, then p(O It), for all priors p(O) p(x;t I0), if t = T(x) and 0, if I cp T(x). So, = p(x It, O)p(t I 0) = p(x I t)p(t 1 0), by the definition of sufficiency. But, by Bayes' theorem, p(O Ix) 6: p(x IO)p(O) Then p(O Ix) = = p(x I t)p(t I O)p(O)
So the doctor decides to ask John to undertake another test Y. 99. = . Note that the probability Of this new test yielding a positive result given that John doesn't have the illness is very smal� .. Alth_ough this test might be more expeHsive, its results are more efficient. The posterior distribution of e given X' P(B I X), will be the prior distribution for theY test. Before observing the result of testY, it is useful to ask ourselves what will the predictive distribution be, that is, what are the values of P(Y = y 1 X=1), for y =0, l.
5 Sufficiency and the exponential family As we have said before, one of the main goals of statistics is to summarize infor mation. e. functions of the observations X, that summarize all the information contained in X. Definition (classical). Let X be a random quantity with probability (density) I 0). Then, the statistic T T(X) is sufficient for the parameter 0 if function p(x = p(x II, 0) The definition states that given = p(x I 1). T, X does not bring any additional infonnation about 0. From the Bayesian point of view this means that X and 0 are independent conditionally on T.