Drunken rooks, etc.

I was telling my class the other day about the Gibbs sampler and Metropolis algorithm and was describing them as random walks–for example, with the Gibbs sampler you consider a drunk who is walking at random on the Manhattan street grid (in the hypothetical world in which the Manhattan street grid is rectilinear). But then I realized that’s not right, because a Gibbs jump can go as far as necessary in any direction–it is not limited to one step. It’s actually the path of a drunken rook! From there, it’s natural to think of the Metropolis algorithm as a drunken knight. Or maybe a drunken king, if the jumps are small.

To take this analogy further: parameter expansion (i.e., redundant parameterization) allows rotations of parameter space, thus allowing the rooks to move diagonally as well as rectilinearly–thus, a drunken queen! I don’t know if these analogies helped the students, but I like them.