CHAPTER VII - PROBABILITY INFERENCE. BAYES' METHOD

Abstract

This chapter presents a problem of which a sound mathematical basis for dealing with the problem of probability of hypotheses was suggested by Thomas Bayes. The aim is to make an inference on the unknown probability of deuce by means of the known outcome of 600 (= n) trials. The theory is known under various headings — probability of hypotheses, inverse probability, and probability of inference. The chapter also presents the idea leading to Bayes' solution of this problem and Bayes' solution. It also discusses analogies of Bayes' distribution to Bernoulli's distribution. The chapter presents the Bernoulli's distribution with the relative number of successes z = x/n, as the independent variable. The most significant analogy between the Bayes' distribution and the Bernoulli's distribution is the fact that the variance goes to zero as n increases indefinitely. This means that with increasing n, both distributions become more and more concentrated around the point whose abscissa is, at the same time, the mean value and the location of the maximum. The chapter also discusses the implications of this concentration in the case of the Bayes' problem for general p0 (x).