By R.F. Wrighton
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Extra info for Elementary Principles of Probability and Information
73-74 below. Let us consider a source which, at each stage emits l ’s and 0’s with prob abilities p and 1 —p respectively. We refer to this as a binary stochastic source with parameter p. We adopt now the second of the two viewpoints mentioned above. In a very long sequence of cells the number of l ’s will almost certainly be very nearly equal to p. We infer, therefore, that it is possible, by coding over sufficiently long messages, to compress the output of the source in the ratio H(p): 1 with negligible loss.
As an application of the same argument, we reconsider the Bemoulli-Bayes problem. We have an urn containing a very large number of balls known to be either black or white, 44 PROBABILITY AND INFORMATION and denote the unknown proportion of white balls by p . We are allowed to acquire information about p by sampling. Since the number of balls is very large, it does not matter whether we do so with or without replacement. Suppose we take an «-fold sample, observe r white balls, and assert that This is just the same as asserting that We can now easily work out the probability of false assertion relative to all possible values of p, using the numerical form of Bernoulli’s theorem (LaplaceDeMoivre approximation).
4 . INVERSE PROBABILITY 37 diffuse, and certainly by no means exclusively, empirical grounds. In general, we may say that a formalisable problem arises if a choice has to be made between different summetrical set-ups on the basis of a finite number of random trials. Alternative Formulations o f the Urn-sampling Problem The simplest characteristic situation of the type to which we have drawn attention concerns the assessment of the contents of an urn containing balls of different labellings by means of random sampling.