CAS Exam MAS-II Spring 2019, Question 28
Study guide: Exam MAS-II
You have two random variables \phi and \theta where
\begin{align*}
\phi \mid \theta &\sim \text{Binomial}(n, \theta) \\
\theta &\sim \text{Beta}(a, b)
\end{align*}and n is a fixed positive integer. Thus, the prior and the conditional distributions are given by
\begin{align*}
\pi(\theta) &= \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\theta^{a-1}(1-\theta)^{b-1} \\
f(\phi \mid \theta) &= \binom{n}{\phi}\theta^{\phi}(1-\theta)^{n-\phi}
\end{align*}where \Gamma(\cdot) denotes the Gamma function and \int \theta^{a-1}(1-\theta)^{b-1}d\theta = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.
An actuary wants to use a Gibbs sampler to sample from the joint distribution of \theta and \phi. She sets an initial value for \phi and \theta denoted by (\phi_0, \theta_0) and subsequently in the i^{th} step samples \phi_i \sim f(\phi \mid \theta_{i-1}) and \theta_i \sim f(\theta \mid \phi_i).
Determine which conditional distribution for \theta \mid \phi_i correctly implements the Gibbs sampler.
- A)
\text{Beta}(\phi_i,\, n - \phi_i) - B)
\text{Beta}(b + \phi_i,\, a + n - \phi_i) - C)
\text{Beta}(a + b + \phi_i,\, n - \phi_i) - D)
\text{Beta}(a + \phi_i,\, b + n - \phi_i) - E) None of the above
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