## Lancelot James’ general theory for IBPs

February 9, 2015

When I visited Hong Kong last December, Lancelot James gave me a tutorial on his generalised theory of Indian Buffet Processes, which includes all sorts of interesting extensions that are starting to appear in the machine learning community.

The theory is written up in his recent Arxiv report, “Poisson Latent Feature Calculus for Generalized Indian Buffet Processes.”  I’m a big fan of Poisson processes, so its good to see the general theory written up, including a thorough posterior analysis.  In my recent tutorial (given at Aalto) I’ve included a simplified version of this with the key marginal posterior for a few well known cases.  Piyush Rai from Duke also has great summary slides as part of Lawrence Carin’s reading group at Duke.

Within this framework, you can now handle variations like the Beta-negative-Binomial Process of Mingyuan Zhou and colleagues presented at AI Stats 2012 (Arxiv version here), for which many different variations can be developed.

I’ve presented the general case as an application of improper priors because the Poisson process theory in this case is mostly just formalising their use for developing “infinite parameter vectors”.  Lancelot uses his Poisson Process Partition Calculus to develop a clean, general theory.  For me it was interesting to see that his results also apply to multivariate and auxiliary parameter vectors:

• an infinite vector of Gamma scales $\vec\beta$ created using a Gamma process can have an auxiliary vector of Gamma shapes $\vec\alpha$ sampled pointwise, so $\alpha_k$ is associated with $\beta_k$
• most of the scales are infinitesimal
• most of the shapes are not infinitesimal
• an infinite set of $K+1$-dimensional multinomial’s can be created where the first $K$ dimensions are generated using a Dirichlet style Poisson rate:
• most of the multinomials will have infinitesimal values in the first $K$ dimensions
• the Beta process corresponds to the case where $K=1$

### One comment

1. […] they attribute to Teh and Görür (NIPS 2009) but its easily derived using Dirichlet marginals and Lancelot James’ general formulas for IBPs.  This is the so-called IBP compound Dirichlet. From there, its easy to derive a collapsed Gibbs […]