Algebraic Identification of Binary-Valued Hidden Markov Processes - PowerPoint PPT Presentation

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Identification of Binary-Valued Hidden Markov Processes Alexander Schönhuth CWI, Amsterdam SIAM AG 11 8 Oct 2011

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Guideline Introduction Hidden Markov Chains Complete Identification Algebraic Statistics Introduction The Hidden Markov Model The Main Theorem The Hankel Matrix Invariants Computations Outlook

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Hidden Markov Processes Alternative Formulation 0.5 0.45 0.25 0.3 0.25 0.25 • Initial probabilities π = ( 0 . 8 , 0 . 2 ) a b c a b c • Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 � 0 . 3 � 0.5 0 . 7 1 2 = 0.5 0.3 0 . 5 0 . 5 0.7 • Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START Let � e 1 x � 0 O x := , T x := O x · M , x = a , b , c e 2 x 0 then ( v := a 1 ... a n , T v := T a 1 ... T a n ) � 1 � P X ( X 1 = a 1 , ..., X n = a n ) = π ′ T v 1

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Identification of HMPs [Blackwell and Koopmans (1957), Gilbert (1959)] Complete Identification Problem : Let ( X t ) be a stochastic process which takes values in a finite set Σ . • [Decision] Decide whether ( X t ) is a hidden Markov process. • [Inference] If yes, infer its parameters.

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Identification of HMPs [Blackwell and Koopmans (1957), Gilbert (1959)] Complete Identification Problem : Let ( X t ) be a stochastic process which takes values in a finite set Σ . • [Decision] Decide whether ( X t ) is a hidden Markov process. • [Inference] If yes, infer its parameters.

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Identification of HMPs [Blackwell and Koopmans (1957), Gilbert (1959)] Complete Identification Problem : Let ( X t ) be a stochastic process which takes values in a finite set Σ . • [Decision] Decide whether ( X t ) is a hidden Markov process. • [Inference] If yes, infer its parameters. Related Work : • Generic solutions for stationary processes available [e.g. Finesso et al. (2010), Vidyasagar (2011)] • Cone-based reformulation for arbitrary processes [Heller (1965)] • Identifiability [e.g. Allman et al. (2009)] • Efficient equivalence test [Faigle, AS (2011)]

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Pitfall Lemma: • Every probability distribution P : Σ n → [ 0 , 1 ] is due to a hidden Markov process (on | Σ n | hidden states). • In case of a “stationary” P , the process can be chosen stationary. Therefore: • The answer to the question whether a finite amount of data is due to a hidden Markov process is always ’Yes’. • The restriction to stationary processes holds no promise for any better in that respect.

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Finite Identification Let P : Σ n → [ 0 , 1 ] be a probability distribution over Σ -strings of finite length n . Finite Identification : • [Decision] Decide whether P is induced by a hidden Markov process process on at most ⌊ n + 1 2 ⌋ hidden states. • [Inference] If yes, infer the possibly unique parametrization. Remark: Hidden Markov process on d hidden states is uniquely determined by its distribution on strings of length 2 d − 1. Algebraic statistics can give answers!

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Generic Identification Result Let H d , + be the set of (stochastic) d -hidden state HMP parametrizations. Theorem : Let | Σ | = 2, d ≤ n + 1 and P : Σ n → [ 0 , 1 ] be a probability distribution. 2 There is a an algebraic variety N d such that dim N d < dim H d , + and an algorithm A such that  ’HMP on d hidden states’ P ∈ f n , d (( H d , + \ N d )   A ( P ) = ’Cannot decide’ P ∈ f n , d ( N d ) .   ’No HMP on d hidden states’ else In the first case, A also outputs the parametrization, which is unique up to permutation of hidden states.

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Hidden Markov Processes Alternative Formulation 0.5 0.45 0.25 0.3 0.25 0.25 • Initial probabilities π = ( 0 . 8 , 0 . 2 ) a b c a b c • Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 � 0 . 3 � 0.5 0 . 7 1 2 = 0.5 0.3 0 . 5 0 . 5 0.7 • Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START Let � e 1 x � 0 O x := , T x := O x · M , x = a , b , c e 2 x 0 then ( v := a 1 ... a n , T v := T a 1 ... T a n ) � 1 � P X ( X 1 = a 1 , ..., X n = a n ) = π ′ T v 1

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Hidden Markov Model Definition : Hidden Markov Model for d hidden states and strings of length n over Σ : C | Σ | n H d − → f H d , n ( M , E , π ) = (( T a = O a M ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n . �→ M ∈ C d × d : M 1 d = 1 d E ∈ C d ×| Σ | : E 1 | Σ | = 1 d π ∈ C d

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Hidden Markov Model Definition : Hidden Markov Model for d hidden states and strings of length n over Σ : C | Σ | n H d − → f H d , n ( M , E , π ) = (( T a = O a M ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n . �→ M ∈ C d × d : M 1 d = 1 d E ∈ C d ×| Σ | : E 1 | Σ | = 1 d π ∈ C d = C d 2 + d ( | Σ |− 1 ) H d ∼

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Hidden Markov Model Definition : Hidden Markov Model for d hidden states and strings of length n over Σ : C | Σ | n H d − → f H d , n ( M , E , π ) = (( T a = O a M ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n . �→ M ∈ C d × d : M 1 d = 1 d O a := diag ( e 1 a , ..., e da ) E ∈ C d ×| Σ | : E 1 | Σ | = 1 d T a := O a · M π ∈ C d T v = v 1 ... v n := T v 1 ... T v n = C d 2 + d ( | Σ |− 1 ) H d ∼

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Hidden Markov Model Definition : Hidden Markov Model for d hidden states and strings of length n over Σ : C | Σ | n H d − → f H d , n ( M , E , π ) = (( T a = O a M ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n . �→ M ∈ C d × d : M 1 d = 1 d O a := diag ( e 1 a , ..., e da ) E ∈ C d ×| Σ | : E 1 | Σ | = 1 d T a := O a · M π ∈ C d T v = v 1 ... v n := T v 1 ... T v n = C d 2 + d ( | Σ |− 1 ) H d ∼ Wanted: Equations describing im ( f H d , n )

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Finitary Model Definition : Finitary Model of rank d for strings of length n C | Σ | n f M d , n : M d − → (( T a ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n �→ where M d = { (( T a ) a ∈ Σ , π ) ∈ C | Σ | d 2 + d | � = C | Σ | d 2 . T a 1 = 1 } ∼ a Remark : Finitary models reflect finitary processes.

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Algebraic Statistical Models The Finitary Model Definition : Finitary Model of rank d for strings of length n C | Σ | n f M d , n : M d − → (( T a ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n �→ where M d = { (( T a ) a ∈ Σ , π ) ∈ C | Σ | d 2 + d | � = C | Σ | d 2 . T a 1 = 1 } ∼ a Remark : Finitary models reflect finitary processes. Reminder : Hidden Markov Model on d hidden states for strings of length n C | Σ | n C d 2 +( | Σ |− 1 ) d f H d , n : − → (( T a = O a M ) a ∈ Σ ) , π ) ( π ′ T v 1 ) v ∈ Σ n �→ Hence im ( f H d , n ) ⊂ im ( f M d , n ) .

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Dimension Theorem : Let n ≥ 2 d − 1. ( | Σ | − 1 ) d 2 + d dim im f M d , n = dim im f H d , n ( | Σ | − 1 ) d + d 2 . =

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Dimension Theorem : Let n ≥ 2 d − 1. ( | Σ | − 1 ) d 2 + d dim im f M d , n = dim im f H d , n ( | Σ | − 1 ) d + d 2 . = Corollary : dim im f M d , n = d 2 + d = dim im f H d , n in case of | Σ | = 2 which implies im f M d , n = im f H d , n since both are irreducible. Finitary model and hidden Markov model are algebraic statistically equivalent for binary-valued alphabets!

Guideline Introduction Algebraic Statistics The Main Theorem Computations Outlook Main Theorem The Hankel Matrix • Let v ∈ Σ n . Write p ( v = v 1 ... v n ) := P ( X 1 = v 1 , ..., X n = v n ) . for stochastic process ( X t ) . • Write uv = u 1 ... u m v 1 ... v n ∈ Σ m + n for concatenation of u = u 1 ... u m ∈ Σ m and v = v 1 ... v n ∈ Σ n .