A reminder that there are two main parts to the mock data challenge:

Part 1

Step 1: Use ordinary least squares to fit the linear model y = a + bx to the Part 1 mock data
(a) compute LS estimators of
a and b,
(b) estimate the variance of the (assumed Gaussian) noise which has been added to the mock
y values
(c) estimate errors on your least squares estimates of
a and b, and their covariance
Step 2: By casting the data analysis challenge not as a least squares problem, but as a maximum likelihood
problem, form an appropriate likelihood function for the mock data, which depends on the parameters (a,b).
Then, by computing the log likelihood on a rectangular grid of values of a and b (you need to think carefully about the range of a and b values you should consider, and the spacing between them), and in turn computing the value of chi-squared for each (a,b) pair on your grid, you should find the minimum value of chi-squared. You then should turn your grid of values into a rectangular array of Delta chi-squared values. Finally, using the information in the table in Section 6, you should compute and plot Bayesian credible regions for the parameters at e.g. 68.3%, 95.4%, 99.73%.
Step 3: Finally, using the Metropolis algorithm, and assuming a Gaussian likelihood function for the model parameters a and b, write an MCMC code to generate a sample from the likelihood function - thinking carefully about your choices of proposal density and prior range for a and b. Use this sample to estimate the mean values, errors and covariance of the parameters a and b from their sampled marginal distributions. Devise a method for estimating and Bayesian credible regions for the parameters, using your MCMC sample.

Part 2

Step 1: similar to Step 3 of Part 1 – fit a quadratic model, of the form y = a + bx + cx2, to the Part 2 data, using e.g. the Metropolis algorithm to sample the posterior distributions of the parameters a, b and c and generating plots of the marginal posterior for each pair of parameters – thinking carefully about how to estimate the variance of the (assumed Gaussian) noise that has been added to the mock y values.

Writing it all up

Your Mock Data Analysis submission should take the form of a short report, similar to what you would write for a laboratory experiment, or indeed if you were writing this up as a short scientific paper. You should, therefore, include in your report:
  • a short abstract summarising your overall results
  • a description of the data you have been provided with
  • a short section summarising the method(s) used to solve each step of the challenge, including a discussion of your choices of parameters (e.g. grid spacing of your likelihood search for part 1 step 2, proposal density widths for your MCMC code for part 1 step 3, approach to quadratic model calculations in part 2, etc).
  • some graphs presenting your results.
You should also include (in an appendix) a copy of the codes that you used.
If you have any questions about the format of your report, and what you should – and need not – include, please don’t hesitate to ask me.