2.8 Exercises
Jeffreys-Lindley’s Paradox
The Jeffreys-Lindley’s paradox (Jeffreys 1961; Dennis V. Lindley 1957) represents an apparent disagreement between the Bayesian and Frequentist frameworks in a hypothesis testing scenario.
In particular, assume that in a city, 49,581 boys and 48,870 girls have been born over 20 years. Assume that the male births follow a Binomial distribution with probability \(\theta\). We wish to test the null hypothesis \(H_0: \ \theta = 0.5\) versus the alternative hypothesis \(H_1: \ \theta \neq 0.5\).
- Show that the posterior model probability for the null model is approximately 0.95. Assume \(\pi(H_0) = \pi(H_1) = 0.5\), and that \(\pi(\theta)\) follows a uniform distribution, i.e., \({U}(0,1)\), under \(H_1\).
- Show that the p-value for this hypothesis test is 0.0235 using the normal approximation, \(Y \sim N(N \times \theta, N \times \theta \times (1 - \theta))\).
We want to test \(H_0: \ \mu = \mu_0\) versus \(H_1: \ \mu \neq \mu_0\) given \(Y_i \stackrel{iid}{\sim} N(\mu, \sigma^2)\).
Assume \(\pi(H_0) = \pi(H_1) = 0.5\), and that \(\pi(\mu, \sigma) \propto 1/\sigma\) 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, the posterior odds ratio is:
\[ PO_{01} = \frac{p(\mathbf{y}|\mathcal{M}_0)}{p(\mathbf{y}|\mathcal{M}_1)} = \frac{\Gamma((\alpha_n+1)/2)}{\Gamma(1/2)\Gamma(\alpha_n/2)} (\alpha_n \hat{\sigma}^2 / N)^{-1/2} \left[ 1 + \frac{(\mu_0 - \bar{y})^2}{\alpha_n \hat{\sigma}^2 / N} \right]^{-\left(\frac{\alpha_n + 1}{2}\right)}, \]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 ratio and the classical test statistic for the null hypothesis.
Math Test Continues
Using the setting of the Example: Math Test in Subsection 2.6, test \(H_0: \ \mu = \mu_0\) versus \(H_1: \ \mu \neq \mu_0\) where \(\mu_0 = \{ 100, 100.5, 101, 101.5, 102 \}\).
- What is the p-value for these hypothesis tests?
- Find the posterior model probability of the null model for each \(\mu_0\).