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Undirected Graphical Models

Undirected Graphical Models. Yuan Yao Peking University. What’s a graphical model?. Markov property: Conditional Independence. Hammersley -Clifford Theorem. A clique is a complete subgraph A maximal clique is a clique where no other clique contains it

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Undirected Graphical Models

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  1. Undirected Graphical Models Yuan Yao Peking University

  2. What’s a graphical model?

  3. Markov property: Conditional Independence

  4. Hammersley-Clifford Theorem • A clique is a complete subgraph • A maximal clique is a clique where no other clique contains it • A joint probability admits the following factorization with cliques • where Z is the partition function

  5. Clique Factorization is not unique

  6. Example I: 西游记 • west.Rdata • 408-by-303 data matrix • The first column contains chapter ID (1,…,100) • 302 characters appeared {1,0} in 408 scenes (samples) • 16 main characters who appeared no less than 40 samples

  7. An Ising model Green edges：positive interactions Red edges: negative interactions

  8. Example II: 红楼梦 • dream.Rdata • 475-by-375 data matrix • 374 characters appeared {1,0} in 475 scenes (samples) • The first column is an indicator if the scene is in the first 80 chapters (by Xueqin Cao) or later (by E Gao) • 18 main characters who appeared no less than 30 scenes in the first 80 chapters

  9. Left: 曹雪芹前80回； Right：高鹗后40回

  10. Main Content • Gaussian Graphical Models for real random variables • Semiparametric Gaussian Copula Graphical Models • Ising Models (Boltzman Machine) for discrete random variables

  11. Gaussian Graphical Model

  12. Precision Matrix

  13. Sparsity in High Dimensional Statistics

  14. Gaussian Graphical Models

  15. Proof: Linear regression Y ~ Z whose coefficient:

  16. Sparse precision matrix estimation

  17. Neighborhood Selection

  18. Recall:

  19. Parallel LASSO

  20. Estimator and Symmetrization

  21. L1-penalized Maximum Likelihood Estimator (MLE)

  22. Graphical LASSO, also known as

  23. CLIME: motivation

  24. CLIME: Dantzig Selector

  25. CLIME as Linear Programming

  26. Symmetrization

  27. Nonconvex Penalized MLE

  28. SCAD Penalty

  29. Locally Linear Approximation:Adaptive LASSO

  30. Reference

  31. Normality?

  32. Semiparametric Gaussian Copula Model

  33. Nonparanormal Gaussian Model

  34. Semiparametric Gaussian Copula Model

  35. Conditional Independence

  36. Nonparametric Part: Estimate of the marginal monotone transform

  37. Rank Correlation

  38. Semiparametric Graphical LASSO R package: huge

  39. Ising Model

  40. A Brief History

  41. Ising Model

  42. Sparsity

  43. Boltzman Distribution

  44. Penalized MLE

  45. Sparsity Enforced Estimates

  46. Partition function is intractable

  47. Conditional Likelihood

  48. Neighborhood Selection: L1-regularized Logistic Regression

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