Luigi Spezia Biomathematics & Statistics Scotland Aberdeen - PowerPoint PPT Presentation

Luigi Spezia Biomathematics & Statistics Scotland Aberdeen BAYESIAN VARIABLE SELECTION BAYESIAN VARIABLE SELECTION IN MARKOV MIXTURE MODELS Luigi Spezia – Bayesian variable selection in Markov mixture models Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Based on joint works with... R b Roberta Paroli t P li (U i (Università Cattolica SC, Milano) ità C tt li SC Mil ) Mark Brewer Mark Brewer (Bi (Biomathematics & Statistics Scotland) th ti & St ti ti S tl d) Susan Cooksley Susan Cooksley (The James Hutton Institute) (The James Hutton Institute) Christian Birkel Christian Birkel (University of Aberdeen) (University of Aberdeen) Luigi Spezia – Bayesian variable selection in Markov mixture models 1 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Overview 1) Markov mixture models : Hidden Markov models Markov switching autoregressive models State-space models with regime-switching Markov mixture transition distribution models Mixed Hidden Markov models Spatial hidden Markov models . . . 2) Variable selection methods: Stochastic Search Variable Selection Stochastic Search Variable Selection Kuo and Mallick’s method Gibbs Variable Selection Metropolized Kuo-Mallick Luigi Spezia – Bayesian variable selection in Markov mixture models 2 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Overview 3) Simulation results: Non-homogeneous hidden Markov model Markov switching autoregressive models + covariates g g 4) Three applications: Bernoulli non-homogeneous hidden Markov model Non homogeneous Markov switching autoregressive Non-homogeneous Markov switching autoregressive models + covariates Luigi Spezia – Bayesian variable selection in Markov mixture models 3 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Markov mixture models Hidden Markov models Markov switching autoregressive models State-space models with regime-switching Markov mixture transition distribution models Mixed Hidden Markov models Mixed Hidden Markov models Spatial hidden Markov models m m y t ∼ ∑ ∑ ω j p(y t | θ j ) ω j = 1 j 1 j=1 j=1 j 1 m m y t ∼ ∑ y t ∼ ∑ ∑ ∑ ω i j p(y t | θ j ) ω i,j p(y t | θ j ) ω i j = 1 ω i,j = 1 j=1 j=1 Luigi Spezia – Bayesian variable selection in Markov mixture models 4 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Hidden Markov models hyperparametrs θ x_t Ω X t 1 X t -1 X t X t X t+1 X t+1 Y t -1 Y t Y t+1 Luigi Spezia – Bayesian variable selection in Markov mixture models 5 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Non-homogeneous hidden Markov models hyperparametrs θ x_t Ω t Z t X t 1 X t -1 X t X t X t+1 X t+1 Y t -1 Y t Y t+1 Luigi Spezia – Bayesian variable selection in Markov mixture models 5 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Markov switching autoregressive models hyperparametrs θ x_t Ω X t 1 X t -1 X t X t X t+1 X t+1 Y t -1 Y t Y t+1 Luigi Spezia – Bayesian variable selection in Markov mixture models 5 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Markov switching autoregressive models + covariates hyperparametrs θ x_t W t Ω X t 1 X t -1 X t X t X t+1 X t+1 Y t -1 Y t Y t+1 Luigi Spezia – Bayesian variable selection in Markov mixture models 5 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Markov switching autoregressive models + covariates hyperparametrs θ x_t W t Ω t Z t X t 1 X t -1 X t X t X t+1 X t+1 Y t -1 Y t Y t+1 Luigi Spezia – Bayesian variable selection in Markov mixture models 5 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Variable selection methods Stochastic Search Variable Selection (George and McCullogh, 1993) SSVS SSVS Kuo and Mallick’s method (Kuo and Mallick 1998) Kuo and Mallick s method (Kuo and Mallick, 1998) KM Gibbs Variable Selection (Dellaportas, Forster, Ntzoufras, 2000) GVS Metropolized-Kuo-MallicK (Paroli and Spezia, 2008) MKMK Luigi Spezia – Bayesian variable selection in Markov mixture models 6 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Variable selection methods w h = (w 1,h ,..., w t,h ,..., w T,h ) h=1,...,q γ h = 1 ⇒ w h included γ h = 0 ⇒ w h excluded 0 e cl ded There are 2 q possible models to select Th 2 ibl d l l The best model is identified by its highest posterior probability, that is the subset of covariates corresponding to the vector ( γ 1 ,..., γ h ,..., γ q ) with the highest frequence of the vector ( ) with the highest frequence of appearence in the MCMC sample Luigi Spezia – Bayesian variable selection in Markov mixture models 7 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

(Gaussian) Hidden Markov models -1 ) e t ∼ N (0; λ i (y t | x t = i) = µ i + e t (i = 1,…,m) P(x t = i | x t-1 = k) = ω k,i (y t | x t = i) ∼ N ( µ i + e t ; λ i ( -1 ) ) m ω j,i N ( µ i ; λ i -1 ) y t ∼ ∑ i=1 Luigi Spezia – Bayesian variable selection in Markov mixture models 8 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Non-Homogeneous Hidden Markov models Ω t = [ ω t [ j,i ] j, ] logit( ω t j,i ) = ln( ω t j,i / ω t i,i ) = z t ’ α j,i z t = (1,z t,1 ,…, z t,q )’ j, j, , j, , ,q α j,i = ( α 0(j,i) , α 1(j,i) ,…, α q(j,i) )’ if i ≠ j α i,j = 0 (q) if i = j α = 0 if i = j exp(z t ' α j,i ) ω t j ω j,i = j i = m 1 + ∑ exp(z t ' α j,i ) i=1 i=1 Luigi Spezia – Bayesian variable selection in Markov mixture models 9 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Stochastic Search Variable Selection exp(z t ' α j,i ) ω t (y t | x t =i)= µ i +e t j,i = m 1 + ∑ 1 + ∑ exp(z ' α ) exp(z t ' α j,i ) i=1 γ k(j) = 1 ⇒ z t-1 k included, if x t-1 = j γ k(j) γ = ( γ 1 ’, …, γ j ’,…, γ m ’)’; γ , j ( γ 1 , , γ j , , γ m ) ; t-1,k t-1 γ (j) = (1, γ 1(j) , …, γ q(j) )’ γ k(j) = 0 ⇒ z t-1,k excluded, if x t-1 = j µ i ∼ N (•; •) λ i ∼ G (•; •) α | γ ∼ N α j,i | γ j ∼ N q+1 (0; D γ (j) ) (0; D ) D γ (j) = diag[1, ( δ 1(j) τ 1(j) ) 2 ,…, δ k(j) τ k(j) ) 2 ,…,( δ q(j) τ q(j) ) 2 ] with δ k(j) = c k(j) if γ k(j) = 1 and δ k(j) = 0 and γ k(j) = 1; c k(j) and τ k(j) fixed ith δ if 1 d δ 0 d 1 d fi d ù γ k(j) ∼ Be (0.5) Luigi Spezia – Bayesian variable selection in Markov mixture models 10 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Kuo and Mallick’s method -1 ) e t ∼ N (0; λ i (y t | x t =i) = µ i + e t (i = 1,…,m) exp(z t ' diag[ γ j ] α j,i ) ω t j,i = m m exp(z t ' diag[ γ j ] α j,i ) 1 + ∑ i=1 µ i ∼ N (•; •) λ i ∼ G (•; •) α j,i ∼ N q+1 (•; •) j,i q 1 γ k(j) ∼ Be (0.5) Luigi Spezia – Bayesian variable selection in Markov mixture models 11 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Gibbs Variable Selection SSVS + KM -1 ) 1 e t ∼ N (0; λ i (y t | x t =i) + µ i + e t (i = 1,…,m) exp(z t ' diag[ γ j ] α j,i ) g[ γ j ] p( t j,i ) ω t = ω j,i = m exp(z t ' diag[ γ j ] α j,i ) 1 + ∑ i=1 i=1 µ i ∼ N (•; •) λ i ∼ G (•; •) α | γ ∼ N α j,i | γ j ∼ N q+1 (0; D γ (j) ) (0; D ) D γ (j) = diag[1, ( δ 1(j) τ 1(j) ) 2 ,…, δ k(j) τ k(j) ) 2 ,…,( δ q(j) τ q(j) ) 2 ] with δ k(j) = c k(j) if γ k(j) = 1 and δ k(j) = 0 and γ k(j) = 1; c k(j) and τ k(j) fixed ith δ if 1 d δ 0 d 1 d fi d ù γ k(j) ∼ Be (0.5) Luigi Spezia – Bayesian variable selection in Markov mixture models 12 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Metropolized-Kuo-MallicK -1 ) e t ∼ N (0; λ i (y t | x t =i) = µ i + e t (i = 1,…,m) exp(z t ' diag[ γ j ] α j,i ) ω t j,i = m m exp(z t ' diag[ γ j ] α j,i ) 1 + ∑ i=1 µ i ∼ N (•; •) λ i ∼ G (•; •) α j,i ∼ N q+1 (•; •) j,i q 1 γ k(j) ∼ Be (0.5) Luigi Spezia – Bayesian variable selection in Markov mixture models 13 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Simulations n = 500; q = 5; m = 2; 3 corr(Z 2 ; Z 4 ) = corr(Z 1 ; Z 5 ) = 0; 0.3; 0.7; 0.9 other corr = random; |random| ≤ corr(Z 2 ; Z 4 ) | ≤ th d | d (Z Z ) ex: n = 500; q = 5; m = 3; corr(Z 2 ; Z 4 ) = corr(Z 1 ; Z 5 ) = 0.7 SSVS KM SSVS KM GVS GVS MKMK MKMK state 1 (Z 4 ; Z 5 ) .04* .07* .13 .64 state 2 (Z 2 ; Z 3 ; Z 5 ) ( 2 ; 3 ; 5 ) .04* .08* .12 .44 state 3 (Z 2 ; Z 5 ) .03* .05* .10 .70 Luigi Spezia – Bayesian variable selection in Markov mixture models 14 Bayes 250 Workshop – Edinburgh, 5-7 September 2011

Markov switching autoregressive models p q (y t | x t =i) = µ i + ∑ φ τ (i) y t- τ + ∑ -1 ) θ h(i) w t,h + e t e t ∼ N (0; λ i τ =1 τ =1 h=1 h=1 (i = 1,…,m) P(x t = i | x t-1 = k) = ω k,i = k) = ω P(x = i | x p q (y t | x t =i) ∼ N ( µ i + ∑ N ( µ i ∑ θ h(i) w t,h ; λ i ) -1 ) φ τ (i) y t- τ ; + ∑ φ τ (i) y t- τ ; ∑ (y t | x t i) θ h(i) w t h ; λ i τ =1 h=1 p p m m q q ω j,i N ( µ i + ∑ ( -1 ) ) y t ∼ ∑ φ τ (i) y t- τ ; + ∑ θ h(i) z t,h ; λ i i=1 τ =1 h=1 Luigi Spezia – Bayesian variable selection in Markov mixture models 15 Bayes 250 Workshop – Edinburgh, 5-7 September 2011