12.4 Exercises
Exercise 12.1 Using the dataset
wilson as in this section, do 10 additional iterations of the Metropolis-Hastings Algorithm by continuing the table. See if you can get the value of \(p_{1}\) to 2 decimal places of accuracy.
Exercise 12.2 An alternative model for the dog’s mass is the following differential equation:
\[\begin{equation} \frac{dW}{dt} = -k (W-p_{1}) \end{equation}\]
- Apply separation of variables and \(W(0)=5\) and the value of \(p_{1}\) from the previous problem to write down the solution to this equation. Your final answer will depend on \(k\).
- With the function
compute_likelihoodand the Metropolis-Hastings algorithm to estimate the value of \(k\) to three decimal places accuracy. The true value of \(k\) is between 0 and 1.
Exercise 12.3 Consider the linear model \(y=6.94+bx\) for the following dataset:
| x | y |
|---|---|
| -0.365 | 6.57 |
| -0.14 | 6.78 |
| -0.53 | 6.39 |
| -0.035 | 6.96 |
| 0.272 | 7.20 |
compute_likelihood apply 10 steps the Metropolis-Hastings algorithm to determine \(b\).
Exercise 12.4 For the
wilson dataset, repeat three steps of the parameter estimation to determine \(p_{1}\) as in this section, but this time use the log-likelihood (so you will compare the difference of log likelihoods). Which method do you think is easier?