The Phylogenetic Indian Buffet Process: A Non-Exchangeable Nonparametric Prior for Latent Features

By: Kurt T. Miller, Thomas L. Griffiths and Michael I. JordanICML 2008

Presented by: John Paisley

Duke University, ECE

Motivation

• Nonparametric models are often used with the assumption of exchangeability. – The Indian Buffet Process is an example

• Sometimes, non-exchangeable models might be more appropriate.– The Phylogenetic Indian Buffet Process– Similar to the IBF, but uses additional information of

how related diners are with each other.– These relationships are captured in a tree structure.

Indian Buffet Process

Phylogenetic Indian Buffet Process

• Uses a tree to model columns zk

• This is done as follows:– Assign the root node to be zero

– Along an edge of distance t, let this change to a 1 with probability , where

. The distance from every leaf to the root is 1.

– If a 0 is changed to a 1 along a path to a node, all subsequent nodes are 1 and therefore so are the leaves.

Sampling Issues

• For (1), use the sum-product algorithm (Pearl, 1988).

• For (2), use the chain rule of probability.

• An MCMC inference algorithm is given in detail.

Experimental Results• Elimination by Aspects (EBA) model

– A Choice Model

• Let there be i objects and zik indicate the ith object has the kth feature. Let each feature have a weight, wk. The EBA model defines the probability of choosing object I over j as

• The likelihood of an observation matrix, X, is

• This has been modeled using the IBP.

Experimental Results• Consider now an underlying tree

structure to this model.

• Preference trees: Out of 9 personalities, 3 movie stars, 3 athletes and 3 politicians, people made the 36 pairwise choices of whom they would rather spend time with. Here, L is the length of the edge of each general category to a leaf.

• A soft version of this tree is modeled with the pIBP using data generated from this model with L = 0.1

Experimental Results

• Example results: As the number of samples decreases, the pIBP is able to infer the structure better than the IBP because of the prior.

Experimental Results

• As can be seen, the additional structure in the model produces better results.