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For, since P(~h) = I - P(h). it says that P(h I e) =f{P(h) , pre I -h)) wherefis an in\ pre I h) creasing function of the prior probability P(h) of h and a decreasing function of the likelihood ratio pre 1 -h) . In other words, for pre 1 h) 22 CHAPTER 2 a given value of thc likelihood ratio, the posterior probability of h increases with its prior, while for a givcn value of the prior, the posterior probabi I ity of h is the greatcr, the less probable e is relative to ~h than to h. c Discussion Despite their scemingly abstract appearance, implicit in axioms (1)~(4) is some very interesting, significant and sometimes surprising information, and a good deal of this book will be taken up with making it explicit and explaining why it is significant.

The simplest continuous distribution, and one which we shall refer to many times in the following pages, is the so-called uniform distribution. A random variable X is uniformly distributed in a closed interval I if it has a constant positive probability density at every point in I and zero density outside that interval. 32 CHAPTER 2 Probability densities are of great importance in mathematical statistics-indeed, for many years the principal subject of research in that field was finding the forms of density functions of random variables obtained by transformations of other random variables.

In other words, as far as repeatable ... events are concerned probability is manifested in frequency (1994, p. 98) 46 CHAPTER 3 But not just frequency: fundamental to the identification of objective probability with relative frequency is the long run , for it is only in the long run, in general, that such frequencies behave as sufficiently stable features to be the counterparts of the postulated probability-distribution, and indeed to be objects of scientific enquiry at all. Moreover, there is a good deal of evidence that in suitable experimental contexts the relative frequency with which each of the various possible outcomes occurs settles down within a smaller and smaller characteristic interval as the number of observations increases.

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