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Updating Probabilities with Data and Moments

This article by Adom Giffin and Ariel Caticha explores the application of maximum entropy method for updating probabilities in the presence of data and moments. It includes a review of maxent with data, a multinomial example, numerical results, and discussions on constraints and their commutability.

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Updating Probabilities with Data and Moments

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  1. Adom Giffin - MaxEnt 2007

  2. Updating Probabilities with Data and Moments Adom Giffin and Ariel Caticha Department of Physics University at Albany-SUNY Adom Giffin - MaxEnt 2007

  3. Overview • Review of ME with data • ME with data and moments • A multinomial example • Numerical results • Constraints are not commutable In general Adom Giffin - MaxEnt 2007

  4. Review Bayes Rule Adom Giffin - MaxEnt 2007

  5. ME - Updating Entropy Adom Giffin - MaxEnt 2007

  6. ME with Data Constraint Constraints: Solution: Adom Giffin - MaxEnt 2007

  7. ME with Data Constraint A. Caticha and A. Giffin, “Updating Probabilities", Bayesian Inference and Maximum Entropy Methods in Science and Engineering, ed. by Ali Mohammad-Djafari, AIP Conf. Proc. 872, 31 (2006). Adom Giffin - MaxEnt 2007

  8. Data and Moments Constraints: Adom Giffin - MaxEnt 2007

  9. Data and Constraints Solution: Adom Giffin - MaxEnt 2007

  10. Multinomial Example (An unfair die) Adom Giffin - MaxEnt 2007

  11. Multinomial Example (k sided die) Constraints: where and Adom Giffin - MaxEnt 2007

  12. Multinomial Example (k sided die) Solution: Adom Giffin - MaxEnt 2007

  13. Multinomial Example (k sided die) The Multinomial Likelihood where and Adom Giffin - MaxEnt 2007

  14. Multinomial Example (k sided die) The Moment Constraint (Bias) Adom Giffin - MaxEnt 2007

  15. Multinomial Example (k sided die) k Sided Die Solution – (k-1) simplex Adom Giffin - MaxEnt 2007

  16. Multinomial Example (3 sided die) Adom Giffin - MaxEnt 2007

  17. Multinomial Example (3 sided die) The Multinomial Likelihood Adom Giffin - MaxEnt 2007

  18. Multinomial Example (3 side die) The Moment Constraint (Bias) Adom Giffin - MaxEnt 2007

  19. Multinomial Example (3 sided die) 3 Sided Die Solution – 2 simplex Adom Giffin - MaxEnt 2007

  20. 3 Sided Die - <m1> = 2<m3> Adom Giffin - MaxEnt 2007

  21. Applications • Econometrics • Ecology • ??? Adom Giffin - MaxEnt 2007

  22. Data and Constraints Solution: Adom Giffin - MaxEnt 2007

  23. Adom Giffin - MaxEnt 2007

  24. Normalization Adom Giffin - MaxEnt 2007

  25. Commutivity of Constraints In general, constraints do not commute Constraint 2 Constraint 1 Adom Giffin - MaxEnt 2007

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