CAS Exam MAS-II Fall 2018, Question 16
Study guide: Exam MAS-II
The Metropolis algorithm is chosen to sample the posterior distribution of a parameter \theta. Assume the following:
- An unnormalized posterior defined by the function
f(\theta) = \begin{cases} 0, & \text{for } \theta < 0 \\ e^{-\frac{\theta}{2}}, & \text{for } \theta \ge 0 \end{cases} - A sampler initialization point of 3.0 before Iteration 1
- Each proposal in the Metropolis algorithm is the sum of the current position of the sampler and the step within a given iteration
- A proposal is accepted if the acceptance ratio,
\dfrac{f(\theta_{prop})}{f(\theta_{curr})}, is greater than the random uniform number
The first four iterations for the sampler are provided in the table below.
| Iteration | Step | Random Uniform Number |
|---|---|---|
| 1 | 1.0 | 0.820 |
| 2 | -1.4 | 0.939 |
| 3 | 2.2 | 0.233 |
| 4 | -3.7 | 0.468 |
Calculate the position of the sampler after iteration 4.
- A) Less than 1.0
- B) At least 1.0, but less than 2.0
- C) At least 2.0, but less than 3.0
- D) At least 3.0, but less than 4.0
- E) At least 4.0
Answer this question and see the worked solution