2.8 Exercises

1. Jeffreys-Lindley’s Paradox

   The Jeffreys-Lindley’s paradox (H. 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, $Y \sim N(N \times \theta, N \times \theta \times (1 - \theta))$.

2. 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.

3. 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 $\mu_0$.

References

———. 1961. Theory of Probability. London: Oxford University Press.

Lindley, Dennis V. 1957. “A Statistical Paradox.” Biometrika 44 (1/2): 187–92.