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Course: Neural Networks, Instructor: Professor L.Behera.

Course: Neural Networks, Instructor: Professor L.Behera. Dirichlet Process. -Joy Bhattacharjee, Department of ChE, IIT Kanpur. Johann Peter Gustav Lejeune Dirichlet. What is Dirichlet Process ?. The Dirichlet process is a stochastic process used in Bayesian nonparametric

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Course: Neural Networks, Instructor: Professor L.Behera.

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  1. Course: Neural Networks, Instructor: Professor L.Behera. Dirichlet Process -Joy Bhattacharjee,Department of ChE, IIT Kanpur. Johann Peter Gustav Lejeune Dirichlet

  2. What is Dirichlet Process ? The Dirichlet process is a stochastic process used in Bayesian nonparametric models of data, particularly in Dirichlet process mixture models (also known as infinite mixture models). It is a distribution over distributions, i.e. each draw from a Dirichlet process is itself a distribution. It is called a Dirichlet process because it has Dirichlet distributed finite dimensional marginal distributions.

  3. What is Dirichlet Process ? The Dirichlet process is a stochastic process used in Bayesian nonparametric models of data, particularly in Dirichlet process mixture models (also known as infinite mixture models). It is a distribution over distributions, i.e. each draw from a Dirichlet process is itself a distribution. It is called a Dirichlet process because it has Dirichlet distributed finite dimensional marginal distributions.

  4. What is Dirichlet Process ? The Dirichlet process is a stochastic process used in Bayesian nonparametric models of data, particularly in Dirichlet process mixture models (also known as infinite mixture models). It is a distribution over distributions, i.e. each draw from a Dirichlet process is itself a distribution. It is called a Dirichlet process because it has Dirichlet distributed finite dimensional marginal distributions.

  5.  xi N Dirichlet Priors • A distribution over possible parameter vectors of the multinomial distribution • Thus values must lie in the k-dimensional simplex • Beta distribution is the 2-parameter special case • Expectation • A conjugate prior to the multinomial

  6. What is Dirichlet Distribution ? Methods to generate Dirichlet distribution : Polya’s Urn Stick Breaking Chinese Restaurant Problem

  7. Samples from a DP

  8. Dirichlet Distribution

  9. Polya’s Urn scheme: Suppose we want to generate a realization of Q Dir(α). To start, put i balls of color i for i = 1; 2; : : : ; k; in an urn. Note that i > 0 is not necessarily an integer, so we may have a fractional or even an irrational number of balls of color i in our urn! At each iteration, draw one ball uniformly at random from the urn, and then place it back into the urn along with an additional ball of the same color. As we iterate this procedure more and more times, the proportions of balls of each color will converge to a pmf that is a sample from the distribution Dir(α).

  10. Mathematical form:

  11. Stick Breaking Process The stick-breaking approach to generating a random vector with a Dir(α) distribution involves iteratively breaking a stick of length 1 into k pieces in such a way that the lengths of the k pieces follow a Dir(α) distribution. Following figure illustrates this process with simulation results.

  12. 0 0.4 0.4 0.6 0.5 0.3 0.3 0.8 0.24 G0 Stick Breaking Process

  13. Chinese Restaurant Process

  14. Chinese Restaurant Process

  15. Nested CRP • To generate a document given a tree with L levels • Choose a path from the root of the tree to a leaf • Draw a vector  of topic mixing proportions from an L-dimensional Dirichlet • Generate the words in the document from a mixture of the topics along the path, with mixing proportions 

  16. Nested CRP Day 1 Day 2 Day 3

  17. Properties of the DP • Let (,) be a measurable space, G0 be a probability measure on the space, and  be a positive real number • A Dirichlet process is any distribution of a random probability measure G over (,) such that, for all finite partitions (A1,…,Ar) of , • Draws G from DP are generally not distinct • The number of distinct values grows with O(log n)

  18. In general, an infinite set of random variables is said to be infinitely exchangeable if for every finite subset {xi,…,xn} and for any permutation  we have • Note that infinite exchangeability is not the same as being independent and identically distributed (i.i.d.)! • Using DeFinetti’s theorem, it is possible to show that our draws  are infinitely exchangeable • Thus the mixture components may be sampled in any order.

  19. Mixture Model Inference • We want to find a clustering of the data: an assignment of values to the hidden class variable • Sometimes we also want the component parameters • In most finite mixture models, this can be found with EM • The Dirichlet process is a non-parametric prior, and doesn’t permit EM • We use Gibbs sampling instead

  20. Finite mixture model

  21. Infinite mixture model

  22. DP Mixture model

  23. 19 5 18 5 17 5 16 8 15 8 14 8 13 8 12 8 11 9 10 9 9 9 8 10 7 10 6 10 5 10 4 12 3 12 2 15 1 16 20 0 Agglomerative Clustering • Pros: Doesn’t need generative model (number of clusters, parametric distribution) • Cons: Ad-hoc, no probabilistic foundation, intractable for large data sets Num Clusters Max Distance

  24. Mixture Model Clustering • Examples: K-means, mixture of Gaussians, Naïve Bayes • Pros: Sound probabilistic foundation, efficient even for large data sets • Cons: Requires generative model, including number of clusters (mixture components)

  25. Applications • Clustering in Natural Language Processing • Document clustering for topic, genre, sentiment… • Word clustering for Part of Speech(POS), Word sense disambiguation(WSD), synonymy… • Topic clustering across documents • Noun coreference: don’t know how many entities are there • Other identity uncertainty problems: deduping, etc. • Grammar induction • Sequence modeling: the “infinite HMM” • Topic segmentation) • Sequence models for POS tagging • Society modeling in public places • Unsupervised machine learning

  26. References: • Bela A. Frigyik, AmolKapila, and Maya R. Gupta , University of Washington, • Seattle, UWEE Technical report : Introduction to Dirichlet distribution and related processes, report number UWEETR2010-0006. • Yee WhyeTeh, University College London : Dirichlet Process • Khalid-El-Arini, Select Lab meeting, October 2006. • TegGranager, Natural Language Processing, Stanford University : Introduction to Chinese Restaurant problem and Stick breaking scheme. • Wikipedia

  27. Questions ? • Suggest some distributions that can use Dirichlet process to find classes. • What are the applications in finite mixture model? • Comment on: The DP of a cluster is also a Dirichlet distribution.

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