ラプラスの統計的推測(蕗谷硯児教授退任記念号)

概要

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The first person to attempt an answer to the question of how to determine probability from observed frequencies was T. Bayes. However, Bayes' formulation of this problem is ambiguous. Further progress was made some years later by P. S. de Laplace, using by what is called the Bayes' Principle. He derived the true answer of this problem. An approch of this question by Laplace was analytical. He made an exhaustive investigation of the Beta probability integral. To go further, Laplace had to prove the law of large numbers and to evaluate the integral [numerical formula] for arbitrary T. He then applied those results to a problem in mathematical statistics. During the 26 year period from 1745 to 1770, 251, 527 boys and 241,945 girls had been born in Paris. Setting x the probability of a male birth, Laplace made a calculation using his analysis, and demonstrated that the probability that x≦1/2 was 1.15×10^<-42>. He therefore concluded that it was "morally certain" that x>1/2. This is a kind of the hypothesis testing. In the 1783 paper, Laplace intended to analyzing the data of vital statistics of Paris from 1771 to 1784 and to estimate for the population of France over two-year period 1781-1782. In modern terminology, Laplace concluded that a confidence interval for population of France with probability 0.999999 is 25,299,417±500,000. For this reason, I think that the mathemathical statistics orignated with Laplace.