Hierarchical beta process and the indian buffet process
This presentation is the property of its rightful owner.
Sponsored Links
1 / 14

Hierarchical Beta Process and the Indian Buffet Process PowerPoint PPT Presentation


  • 193 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Hierarchical Beta Process and the Indian Buffet Process

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Hierarchical beta process and the indian buffet process

Hierarchical Beta Process and the Indian Buffet Process

by R. Thibaux and M. I. Jordan

Discussion led by

Qi An


Outline

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

Introduction

VS.


Beta process

Beta process

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

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

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

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

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


Bernoulli process

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 bp

Connections between IBP and BP

It 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

where is the total mass

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

IBP=BP(c=1, γ=α)


Hierarchical beta process and the indian buffet process

The total number of unique dishes can be roughly represented as

This quantity becomes Poi(γ) if c0 (all customers share the same dishes) or Poi(n γ) if c∞ (no sharing).


An algorithm to generate beta process

An algorithm to generate beta process

Authors propose to generate an approximation, , of B

Let For each step n≥1


Hierarchical beta process

Hierarchical beta process

Consider 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 pjw 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

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

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.


  • Login