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Bayesian Sets

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Bayesian Sets

Zoubin Ghahramani and Kathertine A. Heller

NIPS 2005

Presented by Qi An

Mar. 17th, 2006

- Introduction
- Bayesian Sets
- Implementation
- Binary data
- Exponential families

- Experimental results
- Conclusions

- Inspired by “GoogleTM Sets”
- What do Jesus and Darwin have 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

- 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.

- 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

- Clustering on demand

- Very simple algorithm
- Given and , we aim to rank the elements 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:

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

- 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

- 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

- 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:

- 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:

- 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.