13.5 Exercises
phosphorous dataset for 1 iteration. Then time the MCMC parameter estimate for 10, 100, 1000, and 10000 iterations, recording the times for each one. Make a scatterplot with the number of iterations on the horizontal axis and time on the vertical axis. How would you characterize the relationship between the number of iterations and the time it takes to run the code?
Exercise 13.3 For the parks data (Equation (13.2)) studied in this section, compare the 1:1 and the posterior parameter plots. Summarize the following:
- The posterior parameter estimates, with 95% confidence interval.
- The posterior parameter histograms.
Apply your knowledge of equifinality and other observations to determine by how much you have estimated the parameters \(a\) and \(b\) fom the data.
Exercise 13.4 Run an MCMC parameter estimation on the dataset yeast from Gause (1932), where the equation for the volume of yeast \(V\) over time is given by the following equation for an yeast growing in isolation is:
\[\begin{equation}
V = \frac{K}{1+e^{a-bt}},
\end{equation}\]
where \(K\) is the carrying capacity, \(a\) and \(b\) respective rate constants. Apply the data for Sacchromyces to do an MCMC estimate for this equation. You may assume the following prior values on your parameters:
- \(K:\) 0 to 20
- \(b\): 0 to 1
- \(a\): automatically equals to \(a = \ln(K/0.45-1)\)
Exercise 13.6 Run an MCMC parameter estimation on the dataset wilson according to the following differential equation:
\[\begin{equation} \frac{dP}{dt} = b(N-P) \end{equation}\]
Be sure to report all outputs from the MCMC estimation (this includes parameter estimates, confidence intervals, log likelihood values, and any graphs).