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Bayesian Sets. Zoubin Ghahramani and Kathertine A. Heller NIPS 2005. Presented by Qi An Mar. 17 th , 2006. Outline. Introduction Bayesian Sets Implementation Binary data Exponential families Experimental results Conclusions. Introduction. Inspired by “Google TM Sets”

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bayesian sets

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
bayesian sets1
Bayesian Sets
  • 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:
bayesian sets2
Bayesian Sets
  • 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’.
sparse binary data
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


sparse binary data1
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:



exponential families
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
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:
  • 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.