Homework exercise on MCMC - due Wed. Oct. 24

The purpose of this homework is to make sure you can run MCMC (so more
or less like writing ml code that we did in week 3 of ml) plus make
sure you have some understanding of the ir"t" model

Given that we (you) will be using Simon Jackman's Winbugs code, this
may be easier for you to do in Winbugs directly (at the cost that you
have to be more careful about picking up the output).

You will need to change the updater method (don't worry!), taking the
new updater from Week 5 (Neal's updater file). On my box it goes in 
C:\Program Files\WinBUGS14\Updater\Rsrc, on your boxes, either mutatis
mutandis or ymmv depending on your generation. (Alex will make sure
this works in the lab.)

The exercise is first done with Jackman's legislative scaling, as per
the APSR article (winbugs odc file, just open it in winbugs and click
on some arrows to see stuff) and then similar data and methods for
Supreme Court judges (as data for today's exercise). The second
exercise is to allow you a bit more freedom.

Working in groups is fine, but, like typing, winbugs is not a group
skill.

1. Legislatures

a. Run and replicate Jackman's results. Do with 10000 iterations plus 1000 iterations 
   burn-in,  thinning by saving every 25th obs, giving 400
   samples. This should take something over an hour in the lab.)

b. Monitor a few interesting nodes (that is, a few members of
   Congress). What is happening over time.


2. Judges. 

   a. Run Jackman's model as a baseline. Do with 40,000 iterations plus
      10,000 iterations burn-in. Thin the output such that only every
      100th iteration is stored.

   b. Monitor a few interesting nodes. What is happening over time.

   c. Suppose you forget to drop the burn-in period. What happens when
      you take all obs.

   d. Run 4 chains instead of 1. Monitor convergence. Do the results
      differ? Remember you need 4 different starting values (look at
      examples in GH).

   e. Going back to a, run model with 1000 iterations plus 100 iterations burn-in 
      and then with 10000 iterations plus 1000 iterations burn-in. Do the results
      differ?
      
   f. Going back to a, change the priors to be a lot less
      gentle. Do this by increasing precision by a factor of 100 and
      then perhaps even 10,000. What happens? What if you decrease prior
      precision by a factor of 100?

   g. Going back to a, change the two judges who are fixed. What
      happens? Interpret.

   h. Same question as g, but just flip Jackman's -1 and 1 assignment for the
      lefty and righty judges.

   i. Try running without  assigning two judges to -1 and 1. What happens?

   j. Interpret the results from a and write a page or two  (not a paragraph, 
      not five pages) on what you learned about the Supreme Court, akin to the writeup 
      by Jackman, et al in the APSR.