Transcript The Phylogenetic Indian Buffet Process: A Non

The Phylogenetic Indian Buffet Process: A NonExchangeable Nonparametric Prior for Latent Features
By: Kurt T. Miller, Thomas L. Griffiths and Michael I. Jordan
ICML 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.