(This is a somewhat more technical post than usual. If you just want the gist, skip to the visualization.)
N customers enter an Indian buffet restaurant, one after another. It has a seemingly endless array of dishes. The first customer fills her plate with a Poisson(α) number of dishes. Each successive customer i tastes the previously sampled dishes in proportion to their popularity (the number of previous customers who have sampled the kth dish, m_k, divided by i). The ith customer then samples a Poisson(α) number of new dishes.
That's the basic idea behind the Indian Buffet Process (IBP). On Monday Eli Bingham and I gave a presentation on the IBP in our machine learning seminar at Duke, taught by Katherine Heller. The IBP is used in Bayesian non-parametrics to put a prior on (exchangeability classes of) binary matrices. The matrices usually represent the presence of features ("dishes" above, or the columns of the matrix) in objects ("customers," or the rows of the matrix). The culinary metaphor is used by analogy to the Chinese Restaurant Process.
Although the visualizations in the main paper summarizing the IBP are good, I thought it would be helpful to have an interactive visualization where you could change α and N to see how what a random matrix with those parameters looks like. For this I used Shiny, although it would also be fun to do in d3.
One realization of the IBP, with α=10.
In the example above, the first customer (top row) sampled seven dishes. The second customer sampled four of those seven dishes, and then four more dishes that the first customer did not try. The process continues for all 10 customers. (Note that this matrix is not sorted into its left-ordered-form. It also sometimes gives an error if α << N, but I wanted users to be able to choose arbitrary values of N so I have not changed this yet.) You can play with the visualization yourself here.
Interactive online visualizations like this can be a helpful teaching tool, and the process of making them can also improve your own understanding of the process. If you would like to make another visualization of the IBP (or another machine learning tool that lends itself to graphical representation) I would be happy to share it here. I plan to add the Chinese restaurant process and a Dirichlet process mixture of Gaussians soon. You can find more about creating Shiny apps here.