Bayesian inference and data modelling

http://www.aims.ac.za/~mackay/inference/

On Thursday 8th December 2006, we will have two special lectures on Bayesian Inference: one by David, and one by Geoffrey. The first will take place in the Epidemiology slot at 4pm-5pm, and the second at 6.45pm-8pm. We encourage you all to come.

In the first class on Bayesian inference and probabilistic data modelling, we will discuss the following two problems.

Please think about these problems beforehand.

1. The three doors problem
   On a TV game show, a contestant is told the rules as follows:

   There are three doors, labelled 1, 2, 3. A single prize has been hidden behind one of them. You get to select one door. Initially your chosen door will not be opened. Instead, the gameshow host will open one of the other two doors, and he will do so in such a way as not to reveal the prize. For example, if you first choose door 1, he will then open one of doors 2 and 3, and it is guaranteed that he will choose which one to open so that the prize will not be revealed.
   At this point, you will be given a fresh choice of door: you can either stick with your first choice, or you can switch to the other closed door. All the doors will then be opened and you will receive whatever is behind your final choice of door.

   Imagine that the contestant chooses door 1 first; then the gameshow host opens door 3, revealing nothing behind the door, as promised. Should the contestant (a) stick with door 1, or (b) switch to door 2, or (c) does it make no difference?

2. An epidemiology inference problem: inference from observed waiting times
   A very simple disease model says that the number of people who are infective, I, decreases exponentially in accordance with the equation
   dI/dt = - gamma I
   Imagine that you want to infer the parameter gamma. The experimental data describe the durations of infection (waiting times) of some infective individuals.
   Assume N individuals are made infective, then their durations of infection {t_n} are measured. (That is, t_n is the time taken for individual n to leave the infective compartment.)
   Given these measurements {t_n}, what do you think gamma is?

   As a concrete example, the data might be:

   {t₁ = 0.9, t₂ = 0.2, t₃ = 1.5, t₄ = 1.1}.

3. If there is time, we will also look at A more complex epidemiology inference problem:
   Imagine that the exact durations of infection are not known for all individuals. Perhaps there is an observation interval (t_min, t_max), and only durations t_n that fall in that interval are measured; for all other durations, all that is recorded is whether t_n was less than t_min, or greater than t_max.
   

   As a concrete example, the data (given an observation interval t_min=0.1, t_max=1.2) might be:

   {t₁ = 0.9, t₂ = 0.2, t₃ > t_max, t₄ = 1.1}.