Data-Discriminants of Likelihood Equations Jose Israel Rodriguez 1 and Xiaoxian Tang 2 1 University of Notre Dame, United States of America 2 National Institute For Mathematical Sciences (NIMS), Republic of Korea ISSAC 2015 Bath, UK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Assume p i is probability of observing side i ( i = 1 , 2 , 3 , 4) the die is unfair ( ⇔ ∃ j such that p j is not 25%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Assume p i is probability of observing side i ( i = 1 , 2 , 3 , 4) the die is unfair ( ⇔ ∃ j such that p j is not 25%) Given Constraints on p 1 , p 2 , p 3 and p 4 { ( p 1 , p 2 , p 3 , p 4 ) ∈ R 4 > 0 | Σ 4 i =1 p i = 1 } We artificially assume p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Assume p i is probability of observing side i ( i = 1 , 2 , 3 , 4) the die is unfair ( ⇔ ∃ j such that p j is not 25%) Given Constraints on p 1 , p 2 , p 3 and p 4 { ( p 1 , p 2 , p 3 , p 4 ) ∈ R 4 > 0 | Σ 4 i =1 p i = 1 } We artificially assume p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 Data Record We toss the die 100 times and record the times of getting each side e.g. [ u 1 = 11 , u 2 = 24 , u 3 = 15 , u 4 = 50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Assume p i is probability of observing side i ( i = 1 , 2 , 3 , 4) the die is unfair ( ⇔ ∃ j such that p j is not 25%) Given Constraints on p 1 , p 2 , p 3 and p 4 { ( p 1 , p 2 , p 3 , p 4 ) ∈ R 4 > 0 | Σ 4 i =1 p i = 1 } We artificially assume p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 Data Record We toss the die 100 times and record the times of getting each side e.g. [ u 1 = 11 , u 2 = 24 , u 3 = 15 , u 4 = 50] Question For given constraints and data, how to estimate p 1 , p 2 , p 3 and p 4 which BEST explains the data? Answer Maximize likelihood function p 11 1 p 24 2 p 15 3 p 50 subjected to given constraints 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p 11 1 p 24 2 p 15 3 p 50 subjected to given constraints? 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p 11 1 p 24 2 p 15 3 p 50 subjected to given constraints? 4 Answer. It is equivalent to maximize log( p 11 1 p 24 2 p 15 3 p 50 4 ). By the Lagrange Multiplier Method, we solve p 1 λ 1 + p 1 λ 2 − 11 = 0 p 2 λ 1 + 2 p 2 λ 2 − 24 = 0 p 3 λ 1 + 3 p 3 λ 2 − 15 = 0 p 4 λ 1 − 4 p 4 λ 2 − 50 = 0 p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 p 1 + p 2 + p 3 + p 4 − 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p 11 1 p 24 2 p 15 3 p 50 subjected to given constraints? 4 Answer. It is equivalent to maximize log( p 11 1 p 24 2 p 15 3 p 50 4 ). By the Lagrange Multiplier Method, we solve p 1 λ 1 + p 1 λ 2 − 11 = 0 p 2 λ 1 + 2 p 2 λ 2 − 24 = 0 p 3 λ 1 + 3 p 3 λ 2 − 15 = 0 p 4 λ 1 − 4 p 4 λ 2 − 50 = 0 p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 p 1 + p 2 + p 3 + p 4 − 1 = 0 and get 3 solutions [ p 1 = 1 . 2691 , p 2 = − 0 . 2903 , p 3 = − 0 . 0862 , p 4 = 0 . 1075 , λ 1 = 100 , λ 2 = − 91 . 3324], [ p 1 = 0 . 1857 , p 2 = 1 . 2980 , p 3 = − 0 . 6737 , p 4 = 0 . 1901 , λ 1 = 100 , λ 2 = − 40 . 7547], [ p 1 = 0 . 1232 , p 2 = 0 . 3057 , p 3 = 0 . 2214 , p 4 = 0 . 3497 , λ 1 = 100 , λ 2 = − 10 . 7463]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p u 1 1 p u 2 2 p u 3 3 p u 4 subjected to given constraints? 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p u 1 1 p u 2 2 p u 3 3 p u 4 subjected to given constraints? 4 Answer. We solve p 1 λ 1 + p 1 λ 2 − u 1 = 0 p 2 λ 1 + 2 p 2 λ 2 − u 2 = 0 p 3 λ 1 + 3 p 3 λ 2 − u 3 = 0 p 4 λ 1 − 4 p 4 λ 2 − u 4 = 0 p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 p 1 + p 2 + p 3 + p 4 − 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question How to maximize likelihood function p u 1 1 p u 2 2 p u 3 3 p u 4 subjected to given constraints? 4 Answer. We solve p 1 λ 1 + p 1 λ 2 − u 1 = 0 p 2 λ 1 + 2 p 2 λ 2 − u 2 = 0 p 3 λ 1 + 3 p 3 λ 2 − u 3 = 0 p 4 λ 1 − 4 p 4 λ 2 − u 4 = 0 p 1 + 2 p 2 + 3 p 3 − 4 p 4 = 0 p 1 + p 2 + p 3 + p 4 − 1 = 0 Remark For general [ u 1 , u 2 , u 3 , u 4 ], the system has 3 complex solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question For which u i , the system has 0 , 1 , 2 and 3 REAL/POSITIVE solutions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
Motivation First Example Question For which u i , the system has 0 , 1 , 2 and 3 REAL/POSITIVE solutions? Answer. Use real quantifier elimination/real root classification tools. For example, by RealRootClassification in Maple2015 [C. Chen, J. H. Davenport, J. P. May, M. M. Maza, B. Xia and R. Xiao, 2010], for any ( u 1 , u 2 , u 3 , u 4 ) ∈ R 4 > 0 , • D ( u 1 , u 2 , u 3 , u 4 ) > 0 ⇒ 3 distinct real solutions and 1 of them is positive; • D ( u 1 , u 2 , u 3 , u 4 ) < 0 ⇒ 1 real solution and it is positive. where D = u 1 u 2 u 3 u 4 ( u 1 + u 2 + u 3 + u 4 )(441 u 14 + 4998 u 13 u 2 + 20041 u 12 u 22 + 33320 u 1 u 23 + 19600 u 24 − 756 u 13 u 3 + 20034 u 12 u 2 u 3 + 83370 u 1 u 22 u 3 + 79800 u 23 u 3 − 5346 u 12 u 32 + 55890 u 1 u 2 u 32 + 119025 u 22 u 32 + 4860 u 1 u 33 + 76950 u 2 u 33 + 18225 u 34 − 1596 u 13 u 4 − 11116 u 12 u 2 u 4 − 17808 u 1 u 22 u 4 + 4480 u 23 u 4 + 7452 u 12 u 3 u 4 − 7752 u 1 u 2 u 3 u 4 + 49680 u 22 u 3 u 4 − 17172 u 1 u 32 u 4 + 71460 u 2 u 32 u 4 + 27540 u 33 u 4 + 2116 u 12 u 42 + 6624 u 1 u 2 u 42 − 4224 u 22 u 42 − 9528 u 1 u 3 u 42 + 15264 u 2 u 3 u 42 + 14724 u 32 u 42 − 1216 u 1 u 43 − 512 u 2 u 43 + 3264 u 3 u 43 + 256 u 44 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jose Israel Rodriguez and Xiaoxian Tang Data-Discriminants of Likelihood Equations
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