Credibility and Confidence Intervals by Laplace and Gauss

Abstract

It follows from Laplace’s 1774 and 1785 papers that the large-sample inverse probability limits for θ are given by the relation

$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)} /n|h} \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right), $$

for u > 0. In 1812 ([159], II, §16) he uses the normal approximation to the binomial to find large-sample direct probability limits for the relative frequency as

$$ P\left( {\theta - u\sqrt {\theta \left( {1 - \theta } \right)/n} < h < \theta + u\sqrt {\theta \left( {1 - \theta /n} \right)} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u} \right). $$

Noting that θ = h + O(n^−1/2) so that

$$ \sqrt {\theta \left( {1 - \theta \,} \right)/n} = \sqrt {h\left( {1 - h/n} \right)} + O\left( {n^{ - 1} } \right) $$

and neglecting terms of the order of n⁻¹ as in the two formulas above he solves the inequality (8.2) with respect to θ and obtains for u > 0

$$ P\left( {h - u\sqrt {h\left( {1 - h/n} \right)} < \theta < h + u\sqrt {h\left( {1 - h} \right)/n} |\theta } \right) \cong \Phi \left( u \right) - \Phi \left( { - u.} \right) $$