Stick-Breaking Constructions Patrick Dallaire June 10th, 2011

Outline Introduction of the Stick-Breaking process

Outline Introduction of the Stick-Breaking process Presentation of fundamental representation

Outline Introduction of the Stick-Breaking process Presentation of fundamental representation The Dirichlet process The Pitman-Yor process The Indian buffet process

Outline Introduction of the Stick-Breaking process Presentation of fundamental representation The Dirichlet process The Pitman-Yor process The Indian buffet process Definition of the Beta process

Outline Introduction of the Stick-Breaking process Presentation of fundamental representation The Dirichlet process The Pitman-Yor process The Indian buffet process Definition of the Beta process A Stick-Breaking construction of Beta process

Outline Introduction of the Stick-Breaking process Presentation of fundamental representation The Dirichlet process The Pitman-Yor process The Indian buffet process Definition of the Beta process A Stick-Breaking construction of Beta process Conclusion and current work

The Stick-Breaking process

The Stick-Breaking process Assume a stick of unit length

The Stick-Breaking process Assume a stick of unit length

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut

The Stick-Breaking process Assume a stick of unit length At each iteration, a part of the remaining stick is broken by sampling the proportion to cut How should we sample these proportions?

Beta random proportions Let be the proportion to cut at iteration

Beta random proportions Let be the proportion to cut at iteration The remaining length can be expressed as

Beta random proportions Let be the proportion to cut at iteration The remaining length can be expressed as Thus, the broken part is defined by

Beta random proportions Let be the proportion to cut at iteration The remaining length can be expressed as Thus, the broken part is defined by We first consider the case where

Beta distribution The Beta distribution is a density function on Parameters and control its shape

The Dirichlet process

The Dirichlet process Dirichlet processes are often used to produce infinite mixture models

The Dirichlet process Dirichlet processes are often used to produce infinite mixture models Each observation belongs to one of the infinitely many components

The Dirichlet process Dirichlet processes are often used to produce infinite mixture models Each observation belongs to one of the infinitely many components The model ensures that only a finite number of components have appreciable weight

The Dirichlet process A Dirichlet process, , can be constructed according to a Stick-Breaking process Where is the base distribution and is a unit mass at

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

Construction demo

The Pitman-Yor process

The Pitman-Yor process A Pitman-Yor process, , can be constructed according to a Stick-Breaking process Where and

Evolution of the Beta cuts The parameter controls the speed at which the Beta distribution changes

Evolution of the Beta cuts The parameter controls the speed at which the Beta distribution changes The parameter determines initial shapes of the Beta distribution

Evolution of the Beta cuts The parameter controls the speed at which the Beta distribution changes The parameter determines initial shapes of the Beta distribution When , there is no changes over time and its called a Dirichlet process

Evolution of the Beta cuts The parameter controls the speed at which the Beta distribution changes The parameter determines initial shapes of the Beta distribution When , there is no changes over time and its called a Dirichlet process MATLAB DEMO

The Indian Buffet process

The Indian Buffet process The Indian Buffet process was initially used to represent latent features

The Indian Buffet process The Indian Buffet process was initially used to represent latent features Observations are generated according to a set of unknown hidden features

The Indian Buffet process The Indian Buffet process was initially used to represent latent features Observations are generated according to a set of unknown hidden features The model ensure that only a finite number of features have appreciable probability

The Indian Buffet process Recall the basic Stick-Breaking process

The Indian Buffet process Recall the basic Stick-Breaking process

The Indian Buffet process Recall the basic Stick-Breaking process Here, we only consider the remaining parts

The Indian Buffet process Recall the basic Stick-Breaking process Here, we only consider the remaining parts

The Indian Buffet process Recall the basic Stick-Breaking process Here, we only consider the remaining parts Each value corresponds to a feature probability of appearance

Summary

Summary The Dirichlet process induces a probability over infinitely many classes

Summary The Dirichlet process induces a probability over infinitely many classes This is the underlying de Finetti mixing distribution of the Chinese restaurant process

De Finetti theorem It states that the distribution of any infinitely exchangeable sequence can be written where is the de Finetti mixing distribution

Summary The Dirichlet process induces a probability over infinitely many classes This is the underlying de Finetti mixing distribution of the Chinese restaurant process The Indian Buffet process induces a probability over infinitely many features

Summary The Dirichlet process induces a probability over infinitely many classes This is the underlying de Finetti mixing distribution of the Chinese restaurant process The Indian Buffet process induces a probability over infinitely many features Its underlying de Finetti mixing distribution is the Beta process

The Beta process

The Beta process This process

Beta with Stick-Breaking The Beta distribution has a Stick-Breaking representation which allows to sample from

Beta with Stick-Breaking The Beta distribution has a Stick-Breaking representation which allows to sample from The construction is

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking

Beta with Stick-Breaking The Beta distribution has a Stick-Breaking representation which allows to sample from The construction is

The Beta process A Beta process is defined as as , and is a Beta process

Stick-Breaking the Beta process The Stick-Breaking construction of the Beta process is such that

Stick-Breaking the Beta process Expending the first terms

Conclusion We briefly described various Stick-Breaking constructions for Bayesian nonparametric priors These constructions help to understand the properties of each process It also unveils connections among existing priors The Stick-Breaking process might help to construct new priors

Current work Applying a Stick-Breaking process to select the number of support points in a Gaussian process Defining a stochastic process for unbounded random directed acyclic graph Finding its underlying Stick-Breaking representation