Hierarchical Beta Process and the Indian Buffet Process

by R. Thibaux and M. I. Jordan

Discussion led by

Qi An

Outline

• Introduction• Indian buffet process (IBP)• Beta process (BP)• Connections between IBP and BP• Hierarchical beta process (hBP)• Application to document classification• Conclusions

Introduction

• Mixture models – Each data is drawn from

one mixture component– Number of mixture

components is not set a prior

– Distribution over partitions

• Factorial models– Each data is associated

with a set of latent Bernoulli variables

– Cardinality of the set of features can vary

– A “featural” description of objects

– A natural way to define interesting topologies on cluster

– May be appropriate for large number of clusters

VS.

Beta process

Beta process is a special case of independent increment process, or Levy process,

If we draw a set of points from a Poisson process with base measure v, then

]1,0[),( ii pw

When the base measure B0 is discrete: , then B has atoms at the same locations with

i

wi iqB 0

i

wi ipB ))1(,(~ iii qccqBetap

As the representation shows, B is discrete with probability one.

Levy process can be characterized by Levy measure. For beta process, it is

Bernoulli process

Here, Ω can be viewed as a set of potential features and the random measure B defines the probability that X can possess particular feature.

In Indian buffet process, X is the customer and its features are the dishes the customer taste.

Connections between IBP and BPIt is proven that the observations from a beta process satisfy

Procedure:

The first customer will try Poi(γ) number of dishes (feature). After that , the new observation can taste previous dish j with probability and then try a number of new features

As a result, beta process is a two-parameter (c, γ) generalization of the Indian buffet process.

IBP=BP(c=1, γ=α)

where is the total mass

The total number of unique dishes can be roughly represented as

This quantity becomes Poi(γ) if c0 (all customers share the same dishes) or Poi(n γ) if c∞ (no sharing).

An algorithm to generate beta process

Authors propose to generate an approximation, , of B

Let For each step n≥1

Hierarchical beta processConsider a document classification problem. We have a training data set X, which is a list of documents. Each document is classified by one of n topics. We model a document by the set of words it contains. We assume document Xi,j is generated by including each word w independently with a probability pj

w specific to topic j. These probabilities form a discrete measure Aj over all word space Ω. We can put a beta process BP(cj,B) prior on Aj.

Since we want the sharing across different topics, B has to be discrete. We thus put a beta process prior BP(c0,B0) on B, which allows sharing the same atoms among topics.

The HBP model can be summarized as:

This model can be solved with Monte Carlo inference algorithm.

Applications

• Authors applied the hierarchical beta process to a document classification problem

• Compare it to the Naïve Bayes (with Laplace smoothing) results

• The hBP model can obtain 58% result while the best Naïve Bayes result is 50%

Conclusions

• The beta process is shown to be suitable for nonparametric Bayesian factorial modeling

• The beta process can be extended to a recursively-defined hierarchy of beta process

• Compared to the Dirichlet process, the beta process has the potential advantage of being an independent increments process

• More work on inference algorithm is necessary to fully exploit beta process models.