Asymptotics for hidden Markov models with covariates Jens Ledet - PowerPoint PPT Presentation

Introduction and Examples Results Outline of proof Sunspot Asymptotics for hidden Markov models with covariates Jens Ledet Jensen Department of Mathematical Sciences Aarhus University New Frontiers in Applied Probability, August 2011

Introduction and Examples Results Outline of proof Sunspot Outline of talk Two examples of correlated count data with covariates Results: old and new Outline of proof: Mixing properties Central limit theorem for “score” Uniform convergence of “information” Result: asymptotic normality of parameter estimate

Introduction and Examples Results Outline of proof Sunspot Hay and Pettitt Bayesian analysis of a time series of counts with covariates: an application to the control of an infectious disease Biostatistics 2001 Counts: Monthly ESBL bacteria producing Klebsiella pneumonia in an Australian hospital (resistant to many antibiotics, first outbreak in Denmark: 2007) Covariate: the amount of antibiotic cephalosporins used, lagged 3 months

Introduction and Examples Results Outline of proof Sunspot Plot of data 10 8 Klebsiella 6 4 2 0 0 10 20 30 40 50 60 70 month Cephalosporin, lag3 0.20 0.10 0.00 0 10 20 30 40 50 60 70 month

Introduction and Examples Results Outline of proof Sunspot Model y i : count, z i :covariate y i | µ i ∼ Poisson ( µ i ) x i = φ x i − 1 + N ( 0 , σ 2 ) µ i = exp ( β z i + x i ) , Homogeneous hidden Markov and non-homogeneous emission probabilitites Fully bayesian analysis using MCMC: posterior quantities: parameter mean 2.5% 97.5% β 6.9 1.7 11.1

Introduction and Examples Results Outline of proof Sunspot My run 1 Discretize hidden state space ( x i ): 41 points (truncated AR(1)) ˆ β = 5 . 0, likelihood ratio test, β = 0: 5.5% red: β = 0, MAP of 10 exp ( x i ) 8 blue: β = 5, MAP of Klebsiella 6 exp ( β z i + x i ) 4 green: β = 5, MAP of 2 exp ( x i ) 0 0 10 20 30 40 50 60 70 month

Introduction and Examples Results Outline of proof Sunspot My run 2 x i = (˜ x i , u i ) is a Markov chain on { 0 . 5 , 1 , 2 , . . . , 9 } × {− 1 , 0 , 1 } first coordinate: hidden mean second coordinate: increase or decrease at last step − 1 0 1 − 1 1 − α α 0 u i : 0 ρ ( 1 − γ ) γ ( 1 − ρ )( 1 − γ ) 1 0 α 1 − α ) ˜ x i : decrease: i → i − 1 or i − 2 or i − 3 increase: i → i + 1 or i + 2 or i + 3

Introduction and Examples Results Outline of proof Sunspot My run 2 ˆ β = 0 . 9, likelihood ratio test, β = 0: 40% 10 red: β = 0, MAP of ˜ x i 8 blue: β = 0 . 9, MAP of Klebsiella 6 ˜ x i exp ( β z i ) 4 green: β = 0 . 9, MAP of 2 ˜ x i 0 0 10 20 30 40 50 60 70 month Model problem: roles of covariate and hidden variable are not well separated

Introduction and Examples Results Outline of proof Sunspot Jørgensen, Lundbye-Christensen, Song, Sun A longitudinal study of emergency room visits and air pollution for Prince Gorge, British Columbia Statistics in Medicine 1996 Counts: daily counts of emergency room visits for four repiratory diseases Covariates: 4 meteorological ( ˜ z ) and 2 air pollution ( z )

Introduction and Examples Results Outline of proof Sunspot Plot of Data

Introduction and Examples Results Outline of proof Sunspot Plot of Data

Introduction and Examples Results Outline of proof Sunspot Model y it | x t ∼ Poisson ( a it x t ) a it = exp ( α i ˜ z i ) x t | x t − 1 ∼ Gamma ( E = b t x t − 1 , Var = b 2 t x t − 1 σ 2 ) b t = exp ( β ( z t − z t − 1 )) Non-homogeneous hidden Markov and non-homogeneous emission probabilitites Analysis via approximate Kalman filter

Introduction and Examples Results Outline of proof Sunspot General model in this talk x i : non-homogeneous Markov chain, not observed transition density: p i ( x i | x i − 1 ; θ ) y i : conditionally independent given ( x 1 , . . . , x n ) , observed conditional distribution depends on x i only emission density: g i ( y i | x i ; θ ) Covariates: enters through the index i on p and g

Introduction and Examples Results Outline of proof Sunspot Papers: setup state spaces: hidden observed Baum and Petrie 1966 finite finite Bickel, Ritov and Rydén 1998 finite general Jensen and Petersen 1999 ∼ general general Douc, Moulines and Rydén 2004 ∼ general general, AR(1) Jensen 2005 finite finite Fuh 2006 (general) (general) All except J 2005: homogeneous Markov, homogeneous emission Result: there exists solution ˆ θ to likelihood equations with √ n (ˆ θ − θ ) asymptotically normal

Introduction and Examples Results Outline of proof Sunspot Fuh, Ann.Statist. 2006 Appears very general Example from paper: x i is AR(1), y i = x i + N ( 0 , 1 ) But: there are serious errors in the paper results cannot be trusted

Introduction and Examples Results Outline of proof Sunspot Conditions on hidden variable All papers and here: 0 < σ − ≤ p i ( ·|· ; θ ) ≤ σ + < ∞ , θ ∈ B 0 upper bounds on log derivatives of p i ( ·|· ; θ ) moments of upper bound of log derivatives of g i ( y i |· ; θ ) Not covered: x i is an AR(1)

Introduction and Examples Results Outline of proof Sunspot Conditions on observed variable BRR 1998, JP 1999: g ( y | a ; θ ) condition on max a , b g ( y | b ; θ ) to control mixing properties of x | y

Introduction and Examples Results Outline of proof Sunspot Conditions on observed variable BRR 1998, JP 1999: g ( y | a ; θ ) condition on max a , b g ( y | b ; θ ) to control mixing properties of x | y DMR 2004: simple trick to avoid this (choosing a different dominating measure, dependent on i ) same trick used here: � for all i , y i , θ ∈ B 0 : 0 < g i ( y i | x i ; θ ) µ ( dx i )) < ∞ Covered: y i | x i ∼ poisson ( exp ( β z i + x i )) , x : finite state space, z : bounded

Introduction and Examples Results Outline of proof Sunspot Conditions: Estimating equation ˆ Previous papers: θ = MLE Find ˆ Here: θ by solving S n ( θ ) = � n � ψ i ( θ ; ¯ � i = 1 E θ x i , y i ) | y 1 , . . . , y n = 0 ¯ x i = ( x i − 1 , x i , x i + 1 ) , E θ ψ i ( θ ) = 0 x i , y i ) = D 1 log ( p i ( x i | x i − 1 ; θ ) g i ( y i | x i ; θ )) MLE: ψ i ( θ ; ¯ moments of upper bound of ψ i ( · ; · , y i ) and D ψ i ( · ; · , y i )

Introduction and Examples Results Outline of proof Sunspot Example: estimating equation x i : finite state y i | x i : Ising lattice field on { 1 , 2 , . . . , k } 2 � � g i ( y i | x i ) = c ( β ( x i )) exp β ( x i ) � u ∼ v y iu y iv , y iu ∈ {− 1 , 1 } c ( β ) is unknovn: use pseudolikelihood → ψ D 1 log g i ( y i | x i ) → D 1 log � u p i ( y iu | y i , ( − u ) , x i )

Introduction and Examples Results Outline of proof Sunspot Estimation: Quasilikelihood Zeger: 1988 Solve M ( θ )( y − µ ( θ )) � i h ( y i ) : mixing of y i ’s Asymptotics is ‘simple’: Here: MLE or Estimating equation: each term in sum depends on all y i ’s!

Introduction and Examples Results Outline of proof Sunspot Way of thinking (arbitrary silly covariate sequence) � � 1 � J n ( θ ) = − DS n ( θ ) , γ ( n , δ ) = sup θ ∈ B ( δ ) n ( J n ( θ ) − J n ( θ 0 )) � Assume: P (i) 1 n J n ( θ 0 ) − F n → 0, F n nonrandom, eigen ( F n ) > c 0 (ii) γ ( n , δ n ) P → 0 for any δ n → 0 √ n S n ( θ 0 ) G − 1 / 2 D 1 (iii) → N p ( 0 , I ) , c 1 < eigen ( G n ) < c 2 n Result: √ n (ˆ D n J n ) G − 1 / 2 → N p ( 0 , I ) for any consistent ˆ θ n − θ 0 )( 1 θ . n

Introduction and Examples Results Outline of proof Sunspot Mixing: basic Conditional process ( x 1 , . . . , x n ) | ( y 1 , . . . , y n ) General: density c � n k = 1 p k ( x k | x k − 1 ) g k ( x k ) transition density wrt µ : p k ( x k | x k − 1 ) g k ( x k ) a k ( x k ) / a k − 1 ( x k − 1 ) define µ k by d µ k � d µ ( x k ) = g k ( x k ) a k ( x k ) / g k ( z ) a k ( z ) µ ( dz ) transition density q k ( x k | x k − 1 ) wrt µ k : � p k ( x k | x k − 1 ) / p k ( z | x k − 1 ) µ k ( dz ) σ − σ + ≤ q k ( x k | x k − 1 ) ≤ σ + Bounds: from σ − σ − ≤ p k ( x k | x k − 1 ) ≤ σ + � σ − � 2 ≤ q k ( x k | x k − 1 , x k + 1 ) ≤ � σ + � 2 Two sided: σ + σ −

Introduction and Examples Results Outline of proof Sunspot Mixing Chain: c � n k = 1 p k ( x k | x k − 1 ) g k ( x k ) Let r < s and ρ = 1 − σ − /σ + , then sup u P ( x s ∈ A | x r = u ) − inf v P ( x s ∈ A | x r = v ) ≤ ρ s − r , ρ = 1 − ( σ − /σ + ) 2 , then Let r < s 1 ≤ s 2 < t and ˜ sup a , b P ( x s 2 s 1 ∈ B | x r = a , x t = b ) ρ s 1 − r + ˜ − inf u , v P ( x s 2 ρ t − s 2 s 1 ∈ B | x r = u , x t = v ) ≤ ˜ Iterative argument: Doob 1953! (Generalization: perhaps read and understand Meyn and Tweedie: Markov Chains and Stochastic Stability)

Introduction and Examples Results Outline of proof Sunspot Central limit theorem S n = � n i = 1 E ( ψ i | y 1 , . . . , y n ) Mixing properties of summands ? Not so obvious Instead: E ( ψ i | y n 1 ) − E ( ψ i | y i + l ρ l − 1 i − l ) | ≤ 4 ( sup ¯ x i ψ i )˜