Assignment on Bayesian Inference

Part 1: Spot the Difference

Before attempting this exercise, you should familiarise yourself with the Gaussian Parameter Inference example from Lab Sheet 1.

A situation that arises in many circumstances is that we have two data sets and we want to know if they are likely to have been generated by different processes, and if so, the extent to which they differ. For example, consider two groups of people in a randomised controlled trial. People in group A are given a drug and people in group B are given a placebo. Sometime later a relevant physiological variable is measured for each person. We would then be interested in how this variable differed between the two groups.

The file SpotTheDifference.txt contains two data sets which you should use for this exercise. You can use the Gaussian model from Lab Sheet 1 as a starting point.

Use OpenBUGS to model each of the two data sets as Gaussian and run MCMC to infer all the parameters of these two Gaussians. In addition, set up variables for (i) the difference between the two means and (ii) the difference between the two standard deviations. (Hint: if mu1 and mu2 are the means, the statement diffmu <- mu1 - mu2 makes the variable diffmu their difference).

You should submit:
- A model specification file
- A PDF document containing:
o Plots of all the posterior densities obtained, including those for (i) the difference between the means and (ii) the difference between the standard deviations.
o A paragraph of text in which you draw conclusions. Specifically, you should comment on the strength of the evidence that the distributions differ, and on the ways in which they differ (if any). Back up your conclusions with your choice of statistics summarising the relevant posteriors.

To obtain full marks requires an appropriate model and inference set-up, and text that demonstrates an understanding of how to interpret the results obtained.

Part 2: Multiple observers

Before attempting this exercise, you should familiarise yourself with the Seven Scientists example from Lab Sheet 2.

Imagine you have a set of N objects of some kind, and that you ask M different observers to take a measurement from each of these objects. For example, you might ask M radiologists to measure the length of an organ in each of N medical images obtained from N different patients. The people doing the measuring have varying degrees of skill: some are more precise at measuring than others.

Using the Seven Scientists model from Lab Sheet 2 as a starting point, specify a suitable model for this situation. The model should enable inference of the N quantities of interest as well as variables indicative of the skill level of each of the M observers.

The file MultipleObserversData.txt contains an example dataset for the case when N=20 and M=3. Use BUGS to run MCMC inference using the model you specified.

You should submit:
- A model specification file and an MCMC initialisation file
- A PDF document containing:
o A note explaining what you did to try to ensure adequate convergence of MCMC
o Plots of all the marginal posterior densities obtained along with their summary statistics.
o A paragraph of text in which you summarise and draw conclusions about the N quantities and the skill levels of the M observers. (This should demonstrate your understanding of how to interpret the results obtained).

Attachment:- Assignment.rar