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 Incoming links: Marginal likelihood for a Gaussian point-null mixture

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 18H / b / Solution Created 2026-09-24 Updated 2026-09-25

Conditionally on

μ

,

\overline{X} ∣ μ \sim N (μ, \frac{1}{n}),

so

f (\overline{x} ∣ μ) = \frac{n}{2 π} exp [- \frac{n}{2} (\overline{x} - μ)^{2}] .

Under the continuous half of the prior, write

\overline{X} = μ + Z

with independent

μ \sim N (0, τ^{2}), Z \sim N (0, 1/ n) .

Thus

\overline{X} \sim N (0, τ^{2} + 1/ n)

there. Mixing this density with the point-null density gives the marginal likelihood for a Gaussian point-null mixture

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

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