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# 2016 7-13

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Structure Learning and Universal Coding when Missing Values Exist
ISIT 2016, Barcelona, Spain, July 2016.

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### 2016 7-13

1. 1. Structure Learning and Universal Coding when Missing Values Exist Joe Suzuki Osaka University
2. 2. m missing values out of nNn samples N variables Bayesian Model Selection with Missing Values: On what variables each variable depends ? P(Data) is constant Max P(M|Data) ⇔ Max P(M,Data)Choose M with Max P(M|Data) Compute P(M,Data) values (exponential with m) Exponential Computation with Nm i.i.d
3. 3. From Data to Graphical Model N=3 can be missing • Bayesian Network (DAG) • Markov Network (undirected) • Forest • Spanning Tree Model selection Exponential with N still exponential with m
4. 4. Contributions If the true model is expressed by a forest, and even if some values in the data are missing, 1. a model M with the max P(M|Data) is obtained in O(N^2) 2. If missing occurs stationary ergodic, maximizing P(M|Data) does not imply consistent estimation of M 3. An Exact Universal coding reducndancy in a precise form In this talk, we prove those statements one by one.
5. 5. Chow-Liu (1968)
6. 6. Universal Measures
7. 7. From Data: Structure Learning (Suzuki 93, 12)
8. 8. Generalization: missing values are present
9. 9. Proof Sketch: expand the forest from root
10. 10. Definitions
11. 11. Universal coding of forest data with missing values We generalize into the case that missing values are stationary ergodic by introducing
12. 12. Conclusions If the true model is expressed by a forest, and even if some values in the data are missing, 1. a model M with the max P(M|Data) is obtained in O(N^2) 2. If missing occurs stationary ergodic, maximizing P(M|Data) does not imply consistent estimation of M 3. An Exact Universal coding reducndancy in a precise form Future work: more general class of graphical models