Robustifying the Viterbi Algorithm Cedric De Boom, Jasper De Bock, - PowerPoint PPT Presentation

Robustifying the Viterbi Algorithm Cedric De Boom, Jasper De Bock, Arthur Van Camp, Gert de Cooman PGM 2014, Utrecht

A Toy Example Weather estimation ? 2

A Toy Example Weather estimation ? 3

A Toy Example Weather estimation H ? L 4

A Toy Example Weather estimation H ? 5

A Toy Example Weather estimation X 1 X 2 X 3 ? ? ? H L L O 1 O 2 O 3 6

Hidden Markov Model Local models X 1 X 2 X 3 ? ? ? p 1 ( X 1 ) p 2 ( X 2 | X 1 ) p 3 ( X 3 | X 2 ) H L L O 1 O 2 O 3 q 1 ( O 1 | X 1 ) q 2 ( O 2 | X 2 ) q 3 ( O 3 | X 3 ) 7

Hidden Markov Model Local models Local models p 1 ( X 1 ) p 2 ( X 2 | X 1 ) p 3 ( X 3 | X 2 ) q 1 ( O 1 | X 1 ) q 2 ( O 2 | X 2 ) q 3 ( O 3 | X 3 ) 8

Hidden Markov Model Global model Local models p 1 ( X 1 ) p 2 ( X 2 | X 1 ) p 3 ( X 3 | X 2 ) q 1 ( O 1 | X 1 ) q 2 ( O 2 | X 2 ) q 3 ( O 3 | X 3 ) Global model p ( X 1:3 , O 1:3 ) = p 1 ( X 1 ) q 1 ( O 1 | X 1 ) 3 Y p i ( X i | X i − 1 ) q i ( O i | X i ) i =2 9

Hidden Markov Model Estimating the hidden sequence Global model p ( X 1:3 , O 1:3 ) = p 1 ( X 1 ) q 1 ( O 1 | X 1 ) 3 Y p i ( X i | X i − 1 ) q i ( O i | X i ) i =2 arg max x 1:3 p ( x 1:3 | o 1:3 ) 10

Hidden Markov Model Estimating the hidden sequence Global model p ( X 1:3 , O 1:3 ) = p 1 ( X 1 ) q 1 ( O 1 | X 1 ) 3 Y p i ( X i | X i − 1 ) q i ( O i | X i ) i =2 p ( x 1:3 , o 1:3 ) arg max x 1:3 p ( x 1:3 | o 1:3 ) = arg max p ( o 1:3 ) x 1:3 10

Hidden Markov Model Estimating the hidden sequence Global model p ( X 1:3 , O 1:3 ) = p 1 ( X 1 ) q 1 ( O 1 | X 1 ) 3 Y p i ( X i | X i − 1 ) q i ( O i | X i ) i =2 p ( x 1:3 , o 1:3 ) arg max x 1:3 p ( x 1:3 | o 1:3 ) = arg max p ( o 1:3 ) x 1:3 = arg max x 1:3 p ( x 1:3 , o 1:3 ) 10

Hidden Markov Model Estimating the hidden sequence Global model p ( X 1:3 , O 1:3 ) = p 1 ( X 1 ) q 1 ( O 1 | X 1 ) 3 Y p i ( X i | X i − 1 ) q i ( O i | X i ) i =2 p ( x 1:3 , o 1:3 ) arg max x 1:3 p ( x 1:3 | o 1:3 ) = arg max p ( o 1:3 ) x 1:3 = arg max x 1:3 p ( x 1:3 , o 1:3 ) Viterbi algorithm (1967) 10

Hidden Markov Model Viterbi algorithm Viterbi algorithm (1967) Recursive 11

Hidden Markov Model Viterbi algorithm Viterbi algorithm (1967) Recursive Complexity O( nm 2 ) n : length of the sequence m : size of state space 11

Hidden Markov Model Viterbi algorithm Viterbi algorithm (1967) Recursive Complexity O( nm 2 ) n : length of the sequence m : size of state space Extendible to k -best Viterbi 11

Imprecise Hidden Markov Model Local models ? ? ? M X 2 | X 1 M X 3 | X 2 M X 1 H L L M O 1 | X 1 M O 2 | X 2 M O 3 | X 3 12

Imprecise Hidden Markov Model Local models Local models M X 1 M X 2 | X 1 M X 3 | X 2 M O 1 | X 1 M O 2 | X 2 M O 3 | X 3 13

Imprecise Hidden Markov Model Global model Local models M X 1 M X 2 | X 1 M X 3 | X 2 M O 1 | X 1 M O 2 | X 2 M O 3 | X 3 Global model ( 3 Y M = p i ( X i | X i − 1 ) q i ( O i | X i ) : i =1 ( ∀ k ∈ { 1 , 2 , 3 } ) p k ( ·| X k − 1 ) ∈ M X k | X k − 1 , q k ( ·| X k ) ∈ M O k | X k 14

Imprecise Hidden Markov Model Global model Local models M X 1 M X 2 | X 1 M X 3 | X 2 M O 1 | X 1 M O 2 | X 2 M O 3 | X 3 May contain infinitely many precise models! Global model ( 3 Y M = p i ( X i | X i − 1 ) q i ( O i | X i ) : i =1 ( ∀ k ∈ { 1 , 2 , 3 } ) p k ( ·| X k − 1 ) ∈ M X k | X k − 1 , q k ( ·| X k ) ∈ M O k | X k 14

Imprecise Hidden Markov Model Estimating the hidden sequence Partial order x 1:3 � ˆ x 1:3 , ( 8 p 2 M ) p ( x 1:3 | o 1:3 ) > p (ˆ x 1:3 | o 1:3 ) 15

Imprecise Hidden Markov Model Estimating the hidden sequence Partial order x 1:3 � ˆ x 1:3 , ( 8 p 2 M ) p ( x 1:3 | o 1:3 ) > p (ˆ x 1:3 | o 1:3 ) Set of maximal solutions opt max ( X 1:3 ) , { ˆ x 1:3 ∈ X 1:3 : ( ∀ x 1:3 ∈ X 1:3 ) x 1:3 ⌥ ˆ x 1:3 } 15

Imprecise Hidden Markov Model Estimating the hidden sequence Partial order x 1:3 � ˆ x 1:3 , ( 8 p 2 M ) p ( x 1:3 | o 1:3 ) > p (ˆ x 1:3 | o 1:3 ) Set of maximal solutions opt max ( X 1:3 ) , { ˆ x 1:3 ∈ X 1:3 : ( ∀ x 1:3 ∈ X 1:3 ) x 1:3 ⌥ ˆ x 1:3 } Indecision There may be multiple maximal solutions. 15

Imprecise Hidden Markov Model Rewriting the solution set Partial order x 1: n � ˆ n , ( 8 p 2 M ) p ( x 1: n | o 1: n ) > p (ˆ x 1: n | o 1: n ) x 1: n , 16

Imprecise Hidden Markov Model Rewriting the solution set Partial order x 1: n � ˆ n , ( 8 p 2 M ) p ( x 1: n | o 1: n ) > p (ˆ x 1: n | o 1: n ) x 1: n , ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) > p (ˆ x 1: n , o 1: n ) 16

Imprecise Hidden Markov Model Rewriting the solution set Partial order x 1: n � ˆ n , ( 8 p 2 M ) p ( x 1: n | o 1: n ) > p (ˆ x 1: n | o 1: n ) x 1: n , ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) > p (ˆ x 1: n , o 1: n ) ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) x 1: n , o 1: n ) > 1 p (ˆ 16

Imprecise Hidden Markov Model Rewriting the solution set Partial order x 1: n � ˆ n , ( 8 p 2 M ) p ( x 1: n | o 1: n ) > p (ˆ x 1: n | o 1: n ) x 1: n , ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) > p (ˆ x 1: n , o 1: n ) ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) x 1: n , o 1: n ) > 1 p (ˆ p ( x 1: n , o 1: n ) ⇔ min x 1: n , o 1: n ) > 1 p (ˆ p ∈ M 16

Imprecise Hidden Markov Model Rewriting the solution set Partial order x 1: n � ˆ n , ( 8 p 2 M ) p ( x 1: n | o 1: n ) > p (ˆ x 1: n | o 1: n ) x 1: n , ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) > p (ˆ x 1: n , o 1: n ) ⇔ ( ∀ p ∈ M ) p ( x 1: n , o 1: n ) x 1: n , o 1: n ) > 1 p (ˆ p ( x 1: n , o 1: n ) ⇔ min x 1: n , o 1: n ) > 1 p (ˆ p ∈ M What if p (ˆ x 1: n , o 1: n ) becomes zero? 16

Imprecise Hidden Markov Model Rewriting the solution set Partial order p ( x 1: n , o 1: n ) x 1: n � ˆ ⇔ min x 1: n , o 1: n ) > 1 x 1: n , p (ˆ p ∈ M 17

Imprecise Hidden Markov Model Rewriting the solution set Partial order p ( x 1: n , o 1: n ) x 1: n � ˆ ⇔ min x 1: n , o 1: n ) > 1 x 1: n , p (ˆ p ∈ M n p k ( x k | x k − 1 ) q k ( o k | x k ) Y ⇔ min x k ) > 1 p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p ∈ M k =1 17

Imprecise Hidden Markov Model Rewriting the solution set Partial order p ( x 1: n , o 1: n ) x 1: n � ˆ ⇔ min x 1: n , o 1: n ) > 1 x 1: n , p (ˆ p ∈ M n p k ( x k | x k − 1 ) q k ( o k | x k ) Y ⇔ min x k ) > 1 p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p ∈ M k =1 n p k ( x k | x k − 1 ) q k ( o k | x k ) Y min min x k ) > 1 ⇔ p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p k ( ·| X k − 1 ) ∈ M Xk | Xk − 1 q k ( ·| X k ) ∈ M Ok | Xk k =1 17

Imprecise Hidden Markov Model Rewriting the solution set Partial order p ( x 1: n , o 1: n ) x 1: n � ˆ ⇔ min x 1: n , o 1: n ) > 1 x 1: n , p (ˆ p ∈ M n p k ( x k | x k − 1 ) q k ( o k | x k ) Y ⇔ min x k ) > 1 p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p ∈ M k =1 n p k ( x k | x k − 1 ) q k ( o k | x k ) Y min min x k ) > 1 ⇔ p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p k ( ·| X k − 1 ) ∈ M Xk | Xk − 1 q k ( ·| X k ) ∈ M Ok | Xk k =1 n Y χ k ( x k , x k − 1 , ˆ x k , ˆ x k − 1 ) ω k ( x k , ˆ x k , o k ) > 1 ⇔ k =1 17

Imprecise Hidden Markov Model Rewriting the solution set Partial order p ( x 1: n , o 1: n ) x 1: n � ˆ ⇔ min x 1: n , o 1: n ) > 1 x 1: n , p (ˆ p ∈ M n p k ( x k | x k − 1 ) q k ( o k | x k ) Y ⇔ min x k ) > 1 p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p ∈ M k =1 n p k ( x k | x k − 1 ) q k ( o k | x k ) Y min min x k ) > 1 ⇔ p k (ˆ x k | ˆ x k − 1 ) q k ( o k | ˆ p k ( ·| X k − 1 ) ∈ M Xk | Xk − 1 q k ( ·| X k ) ∈ M Ok | Xk k =1 n Y χ k ( x k , x k − 1 , ˆ x k , ˆ x k − 1 ) ω k ( x k , ˆ x k , o k ) > 1 ⇔ k =1 Can be calculated in advance 17

Imprecise Hidden Markov Model Rewriting the solution set Set of maximal solutions opt max ( X 1: n ) , { ˆ x 1: n 2 X 1: n : ( 8 x 1: n 2 X 1: n ) x 1: n 6� ˆ x 1: n } ⇔ n Y max χ k ( x k , x k − 1 , ˆ x k , ˆ x k − 1 ) ω k ( x k , ˆ x k , o k ) ≤ 1 ⇔ x 1: n ∈ X 1: n k =1 18

Imprecise Hidden Markov Model Rewriting the solution set Set of maximal solutions opt max ( X 1: n ) , { ˆ x 1: n 2 X 1: n : ( 8 x 1: n 2 X 1: n ) x 1: n 6� ˆ x 1: n } ⇔ n Y max χ k ( x k , x k − 1 , ˆ x k , ˆ x k − 1 ) ω k ( x k , ˆ x k , o k ) ≤ 1 ⇔ x 1: n ∈ X 1: n k =1 How do we calculate this set? 18

Imprecise Hidden Markov Model Rewriting the solution set Set of maximal solutions opt max ( X 1: n ) , { ˆ x 1: n 2 X 1: n : ( 8 x 1: n 2 X 1: n ) x 1: n 6� ˆ x 1: n } ⇔ n Y max χ k ( x k , x k − 1 , ˆ x k , ˆ x k − 1 ) ω k ( x k , ˆ x k , o k ) ≤ 1 ⇔ x 1: n ∈ X 1: n k =1 How do we calculate this set? MaxiHMM algorithm 18

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19

MaxiHMM algorithm General overview MaxiHMM algorithm 19