Entropic Inequalities and marginal problems (Fritz and Chavez)
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Aforementioned paper discusses 'marginal problem': Given distributions on certain subsets of N random variables, determine whether exists a joint distribution that marginalizes to given subset distributions.
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Entropic Inequalities and marginal problems (Fritz and Chavez)
- 1. Jayant Apte ASPITRG Entropic Inequalities and Marginal Problems* Jayant Apte ASPITRG Drexel University *FRITZ, T., & CHAVES, R. (2013). Entropic Inequalities and Marginal Problems. IEEE transactions on information theory, 59(2), 803-817.
- 2. Jayant Apte ASPITRG The three coins
- 3. 3 Jayant Apte ASPITRG The three coins
- 4. 4 Jayant Apte ASPITRG The three coins
- 5. 5 Jayant Apte ASPITRG The three coins
- 6. 6 Jayant Apte ASPITRG An attempt
- 7. 7 Jayant Apte ASPITRG An attempt
- 8. 8 Jayant Apte ASPITRG An attempt
- 9. 9 Jayant Apte ASPITRG An attempt But they are not!!!
- 10. 10 Jayant Apte ASPITRG Conclusion ● There is no joint distribution compatible with given marginals
- 11. 11 Jayant Apte ASPITRG Conclusion ● There is no joint distribution compatible with given marginals ● There is some systematic error in the coin flips
- 12. 12 Jayant Apte ASPITRG Conclusion ● There is no joint distribution compatible with given marginals ● There is some systematic error in the coin flips ● OR The coins are operated by some weird mechanism creating the required distribution as a function depending on which two are flipped together
- 13. 13 Jayant Apte ASPITRG Outline ● Marginal scenario and marginal model ● Contextuality ● Marginal problem ● Partial polymatroids ● Relationship between marginal models and partial polymatroids ● Marginal Problem for polymatroids ● Computations ● Inference of common ancestors in Bayesian Networks ● Contexuality of the n-Cycle marginal scenario ● Extention to continuous random variables
- 15. 15 Jayant Apte ASPITRG Marginal Scenario
- 16. 16 Jayant Apte ASPITRG Marginal Model S S'T
- 19. 19 Jayant Apte ASPITRG Marginal problem
- 20. 20 Jayant Apte ASPITRG A naive probability space approach
- 21. 21 Jayant Apte ASPITRG Example: 3 coins
- 22. 22 Jayant Apte ASPITRG Example: 3 coins This LP turns out to be infeasible
- 23. 23 Jayant Apte ASPITRG The entropy cone
- 24. 24 Jayant Apte ASPITRG Basic inequalities
- 26. 26 Jayant Apte ASPITRG Partial Polymatroids
- 27. 27 Jayant Apte ASPITRG Contexuality of partial polymatroids
- 28. 28 Jayant Apte ASPITRG Example: Partial polymatroids over Triangle inequality
- 29. 29 Jayant Apte ASPITRG Example: Modified 3-coins
- 30. 30 Jayant Apte ASPITRG References ● FRITZ, T., & CHAVES, R. (2013). Entropic Inequalities and Marginal Problems. IEEE transactions on information theory, 59(2), 803-817.