The Greek Stochastics γ conference was held last week in Crete and focused this year on MCMC methods (last year was of statistics for biology). The schedule allowed little time for presentations from participants, but there were several “short courses”. In particular, Gareth Roberts gave a rather illuminating explanation on convergence of Gibbs samplers, in which each step of the sampler is viewed as a projection in a functional space.
In a Gibbs sampler with two steps (1. update given ; 2. update given ), step 1 is a projection (recall that is a projection iff , and it is clear than performing step 1 twice in a row is the same as performing it only once); the same is true of step 2. The spaces we are projecting on are ugly, but if you view them as lines, it is easy to see that the successive iterations converge to the intersection of the two lines, which corresponds to a fixed point, as shown in this very ugly figure. The convergence is faster if the two lines are close to orthogonal, which corresponds to correlation close to in the Gibbs sampler. Apparently, this line of thought was initiated by Yael Amit. I find it very helpful.
Another fascinating talk was given by David Spiegelhalter (the “general interest” talk), on communicating in Statistics. David studies the question “Why do people find probability unintuitive and difficult?”; he suggests that the answer in “because probability in unintuitive and difficult”. He gave a startling example: in a phone survey, if you ask which is the greater risk between 1/100, 1/10 and 1/1000, about 25-30% of people give the wrong answer. Media reports often use such notation, with the numerator fixed to 1. The message would be better understood if probabilities were given with a fixed denominator (1/100, 10/100, 0.1/100). David had many other suggestions on explaining risk and uncertainty to non-statisticians, all very useful.