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By Jiruse M., Machek J.

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FAMILIES OF EXCHANGEABLE EVENTS 11 A trivial but fundamental consequence of the de nition of exchangeability is illustrated in the following. 10. Let E fE1 ::: Eng be an exchangeable family. 11. In a lot of n units fU1 ::: Ung, let Sm be the number of defectives out of the sample fU1 ::: Umg. 10). If P fSn = kg = 1, for some k n, then the distribution of Sm is obviously hypergeometric. 10) shows that the distribution of Sm is a mixture of hypergeometric distributions. If Sn is binomial b(n p) then Sm is binomial b(m p).

6, respectively. In the former case the aim of statistical analysis is typically the estimation of the quantity since we draw observations from a real nite population and has a clear physical meaning of its own. In the latter case the object of statistical interest is rather the prediction problem. However it can be natural, as already mentioned, to assume in nite extendibility and then to de ne the variable , also for the present case. has the meaning of being the limit of P fXN +1 = 1jX1 ::: XN g for N !

Given ZN (X1 X2 ::: XN ), when N ! 1. In this respect, the following fact is relevant: consider the variables Z1 (X1 ) ::: ZN ;1 (X1 ::: XN ;1 ) for exchangeable quantities X1 ::: XN . 9). More precisely, it is only a technical matter to show the following result. 55. d. with respect to . 56. 35), by assigning the sequence of su cient statistics. Note also that, in such cases, the \parameter" , being the limit of the sequence fZN (X1 ::: XN )g, is given a meaning as a function of the whole sequence of the observable quantities X1 X2 :::(see also the discussion in Dawid, 1986).

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A Bayesian estimate of the risk of Tick-Borne deseases by Jiruse M., Machek J.


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