BDA3 Chapter 3 Exercise 10

Here’s my solution to exercise 10, chapter 3, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.

\(\DeclareMathOperator{\dbinomial}{Binomial} \DeclareMathOperator{\dbern}{Bernoulli} \DeclareMathOperator{\dpois}{Poisson} \DeclareMathOperator{\dnorm}{Normal} \DeclareMathOperator{\dt}{t} \DeclareMathOperator{\dcauchy}{Cauchy} \DeclareMathOperator{\dexponential}{Exp} \DeclareMathOperator{\duniform}{Uniform} \DeclareMathOperator{\dgamma}{Gamma} \DeclareMathOperator{\dinvgamma}{InvGamma} \DeclareMathOperator{\invlogit}{InvLogit} \DeclareMathOperator{\dinvchi}{InvChi2} \DeclareMathOperator{\dsinvchi}{SInvChi2} \DeclareMathOperator{\dchi}{Chi2} \DeclareMathOperator{\dnorminvchi}{NormInvChi2} \DeclareMathOperator{\logit}{Logit} \DeclareMathOperator{\ddirichlet}{Dirichlet} \DeclareMathOperator{\dbeta}{Beta}\)

For \(j = 1, 2\), let

\[ \begin{align} y_j \mid \mu_j \sigma_j^2 &\sim \dnorm(\mu_j, \sigma_j^2) \\ p(\mu_j, \log \sigma_j^2) &\propto 1. \end{align} \]

We show that

\[ \frac{s_1^2 \sigma_2^2}{s_2^2 \sigma_1^2} \sim F(n_1 - 1, n_2 - 1) . \]

Equation 3.5 in the book shows that \(\sigma_j^2 \mid y \sim \dinvchi(n_j - 1, s_j^2)\). It follows that \(\frac{\sigma_j^2}{(n_j - 1) s_j^2} \sim \dinvChi(n_j - 1)\). Thus, \(\frac{(n_j - 1) s_j^2}{\sigma_j^2} \sim \dchi(n_j - 1)\). The result follows from the fact that the ratio of two \(\chi^2\) random variables (divided by the ratio of their degrees of freedom) has an \(F\)-distribution.