Point-null mixture prior

A point-null mixture prior assigns positive mass to one parameter value and distributes the remaining mass continuously over alternatives. Posterior point mass is obtained by dividing the null component of the marginal density by the full marginal density.

Table of contents
- Marginal likelihood for a Gaussian point-null mixture Point-null mixture prior
  - Posterior probability of a Gaussian point null Marginal likelihood for a Gaussian point-null mixture
    - Jeffreys-Lindley paradox for a Gaussian point null Posterior probability of a Gaussian point null

Marginal likelihood for a Gaussian point-null mixture

 0  0

For

\overline{X} ∣ μ \sim N (μ, 1/ n)

and an equal mixture of

μ = 0

and

μ \sim N (0, τ^{2})

m (x) = \frac{1}{2} ϕ_{1/ n} (x) + \frac{1}{2} ϕ_{τ^{2} + 1/ n} (x) .

(1)

Posterior probability of a Gaussian point null

 0  0

For the equal point-null mixture,

P (μ = 0 ∣ \overline{X} = x) = [1 + \frac{1}{1 + n τ ^{2}} exp (\frac{n ^{2} τ ^{2} x ^{2}}{2 ( 1 + n τ ^{2} )})]^{- 1} .

(1)

Jeffreys-Lindley paradox for a Gaussian point null

 0  0

With a fixed diffuse alternative prior, a frequentist p-value and the posterior probability of a point null can order evidence differently. For

n = 100

and

τ = 1

, the p-value exceeds the null posterior near zero, while sufficiently far in the tails the null posterior exceeds the p-value because it has a slightly slower Gaussian decay.