Bayesian Sets

Zoubin Ghahramani and Kathertine A. Heller

NIPS 2005

Presented by Qi An

Mar. 17th, 2006

Outline

• Introduction

• Bayesian Sets

• Implementation – Binary data– Exponential families

• Experimental results

• Conclusions

Introduction

• Inspired by “GoogleTM Sets”• What do Jesus and Darwin hav

e in common?– Two different views on the origin

of man– There are colleges at Cambridge

University named after them

• The objective is to retrieve items from a concept of cluster, given a query consisting of a few items from that cluster

Introduction

• Consider a universe of items , which can be a set of web pages, movies, people or any other subjects depending on the application

• Make a query of small subset of items , which are assumed be examples of some cluster in the data.

• The algorithm provides a completion to the query set, . It presumably includes all the elements in and other elements in that are also in this cluster.

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Introduction

• View the problem from two perspectives:– Clustering on demand

• Unlike other completely unsupervised clustering algorithm, here the query provides supervised hints or constraints as to the membership of a particular cluster.

– Information retrieval • Retrieve the information that are relevant to the

query and rank the output by relevance to the query

Bayesian Sets

• Very simple algorithm• Given and , we aim to rank the ele

ments of by how well they would “fit into” a set which includes

• Define a score for each :

• From Bayes rule, the score can be re-written as:

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x

xx

p

Dpscore c

Dx

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),()(

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Bayesian Sets

• Intuitively, the score compares the probability that x and were generated by the same model with the same unknown parameters θ, to the probability that x and came from models with different parameters θ and θ’.

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Bayesian Sets

Sparse Binary Data

• Assume each item is a binary vector where each component is a binary variable from an independent Bernoulli distribution:

• The conjugate prior for a Bernoulli distribution is a Beta distribution:

• For a query

where

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cD

Sparse Binary Data

• The score can be computed as:

• If we take a log of the score and put the entire data set into one

large matrix X with J columns, we can compute a vector s of log scores for all points using a single matrix vector multiplication:

where

and

Exponential Families

• If the distribution for the model is not a Bernoulli distribution, but in the form of exponential families:

we can use the conjugate prior:

so that the score is:

Experimental results

• The experiments are performed on three different datasets: the Grolier Encyclopedia dataset, the EachMovie dataset and NIPS authors dataset.

• The running times of the algorithm is very fast on all three datasets:

Experimental results

Experimental results

Conclusions

• A simple algorithm which takes a query of a small set of items and returns additional items from belonging to this set.

• The score is computed w.r.t a statistical model and unknown model parameters are all marginalized out.

• With conjugate priors, the score can be computed exactly and efficiently.

• The methods does well when compared to Google Sets in terms of set completions.

• The algorithm is very flexible in that it can be combined with a wide variety of types of data and probabilistic model.