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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 18H / c / Solution

Bayes' formula divides the point-null contribution to the mixture density by the full marginal density:

q (\overline{x}) = \frac{ϕ _{1/ n} ( x )}{ϕ _{1/ n} ( x ) + ϕ _{τ^{2} + 1/ n} ( x )} .

Since

\frac{ϕ _{τ^{2} + 1/ n} ( x )}{ϕ _{1/ n} ( x )} = \frac{1}{1 + n τ ^{2}} exp [\frac{n ^{2} τ ^{2} x ^{2}}{2 ( 1 + n τ ^{2} )}],

the posterior probability of a Gaussian point null is

q (\overline{x}) = {1 + \frac{1}{1 + n τ ^{2}} exp [\frac{n ^{2} τ ^{2} x ^{2}}{2 ( 1 + n τ ^{2} )}]}^{- 1} .

Solved by gpt-5.6-sol high.

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