2.8 Exercises: Chapter 2
- Jeffreys-Lindley’s paradox
The Jeffreys-Lindley’s paradox ((Jeffreys 1961),(Dennis V. Lindley 1957)) is an apparent disagreement between the Bayesian and Frequentist frameworks to a hypothesis testing situation.
In particular, assume that in a city 49,581 boys and 48,870 girls have been born in 20 years. Assume that the male births is distributed Binomial with probability \(\theta\). We want to test the null hypothesis \(H_0. \ \theta=0.5\) versus \(H_1. \ \theta\neq 0.5\).
Show that the posterior model probability for the model under the null is approximately 0.95. Assume \(\pi(H_0)=\pi(H_1)=0.5\), and \(\pi(\theta)\) equal to \(\mathcal{U}(0,1)\) under \(H_1\).
Show that the p-value for this hypothesis test is equal to 0.0235 using the normal approximation, \(Y\sim \mathcal{N}(N\times \theta, N\times \theta \times (1-\theta))\).
- We want to test \(H_0. \ \mu=\mu_0\) vs \(H_1. \ \mu \neq \mu_0\) given \(y_i\stackrel{iid}{\sim}N(\mu,\sigma^2)\).
Assume \(\pi(H_0)=\pi(H_1)=0.5\), and \(\pi(\mu,\sigma)\propto 1/\sigma\) under the alternative hypothesis.
Show that
\(p(\mathbf{y}|\mathcal{M}_1)=\frac{\pi^{-N/2}}{2}\Gamma(N/2)2^{N/2}\left(\frac{1}{\alpha_n\hat{\sigma}^2}\right)^{N/2}\left(\frac{N}{\alpha_n\hat{\sigma}^2}\right)^{-1/2}\frac{\Gamma(1/2)\Gamma(\alpha_n/2)}{\Gamma((\alpha_n+1)/2)}\) and \(p(\mathbf{y}|\mathcal{M}_0)=(2\pi)^{-N/2}\left[\frac{2}{\Gamma(N/2)}\left(\frac{N}{2}\frac{\sum_{i=1}^N(y_i-\mu_0)^2}{N}\right)^{N/2}\right]^{-1}\). Then,
\[\begin{align*} PO_{01}&=\frac{p(\mathbf{y}|\mathcal{M}_0)}{p(\mathbf{y}|\mathcal{M}_1)}\\ & =\frac{\Gamma((\alpha_n+1)/2)}{\Gamma(1/2)\Gamma(\alpha_n/2)}(\alpha_n\hat{\sigma}^2/N)^{-1/2}\left[1+\frac{(\mu_0-\bar{y})^2}{\alpha_n\hat{\sigma}^2/N}\right]^{-\left(\frac{\alpha_n+1}{2}\right)}, \end{align*}\]
where \(\alpha_n=N-1\) and \(\hat{\sigma}^2=\frac{\sum_{i=1}^N (y_i-\bar{y})^2}{N-1}\).
Find the relationship between the posterior odds and the classical test statistic for the null hypothesis.
- Using the setting of the Example: Math test test \(H_0. \ \mu=\mu_0\) vs \(H_1. \ \mu \neq \mu_0\) where \(\mu_0=\left\{100, 100.5, 101, 101.5, 102 \right\}\).
What is the -value for these hypothesis tests?
Find the posterior model probability of the null model for each \(\mu_0\).