CAS Exam MAS-II Spring 2019, Question 29
Study guide: Exam MAS-II
A random variable X follows a binomial distribution:
X \sim \text{Binomial}(n, \theta)The prior probability distribution of \theta is:
p(\theta) = 2\cos^2(4\pi\theta)An actuary uses the Metropolis algorithm to sample from the posterior distribution of \theta. At each iteration of the algorithm let \theta_{curr} and \theta_{prop} denote the current and proposed value of \theta.
Determine the acceptance probability of \theta_{prop}.
- A)
\min\left\{\dfrac{\theta_{curr}^{x}(1-\theta_{curr})^{n-x}}{\theta_{prop}^{x}(1-\theta_{prop})^{n-x}},\, 1\right\} - B)
\min\left\{\dfrac{\theta_{prop}^{x}(1-\theta_{prop})^{n-x}\cos^2(4\pi\theta_{prop})}{\theta_{curr}^{x}(1-\theta_{curr})^{n-x}\cos^2(4\pi\theta_{curr})},\, 1\right\} - C)
\min\left\{\dfrac{\theta_{curr}^{x}(1-\theta_{curr})^{n-x}\cos^2(4\pi\theta_{curr})}{\theta_{prop}^{x}(1-\theta_{prop})^{n-x}\cos^2(4\pi\theta_{prop})},\, 1\right\} - D)
\min\left\{\dfrac{\theta_{prop}^{x}(1-\theta_{prop})^{n-x}\cos^2(4\pi\theta_{curr})}{\theta_{curr}^{x}(1-\theta_{curr})^{n-x}\cos^2(4\pi\theta_{prop})},\, 1\right\} - E)
\min\left\{\dfrac{\theta_{curr}^{x}(1-\theta_{curr})^{n-x}\cos^2(4\pi\theta_{prop})}{\theta_{prop}^{x}(1-\theta_{prop})^{n-x}\cos^2(4\pi\theta_{curr})},\, 1\right\}
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