Gene Network inference for high-dimensional problems A. Mohammadi - PowerPoint PPT Presentation

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES Gene Network inference for high-dimensional problems A. Mohammadi and E. Wit March 28, 2013

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES M OTIVATION Flow cytometry data with 11 proteins from Sachs et al. (2005)

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES R ESULT FOR C ELL SIGNALING DATA

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES P ROBLEM IN B AYESIAN GRAPH ESTIMATION p ( G ) p ( data | G ) p ( G | data ) = G ∈G p ( G ) p ( data | G ) � Trans-dimensional MCMC in general ◮ Reversible-jump MCMC ◮ Birth-death MCMC Our solution ◮ We proposed birth-death MCMC method for undirected graph estimation ◮ Implement to R : BDgraph package

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES Gaussian graphical model Respect to graph G = ( V , E ) as N p ( 0 , Σ) | K = Σ − 1 is positive definite based on G � � M G = Pairwise Markov property X i ⊥ X j | X V \{ i , j } ⇔ k ij = 0 ,

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES B IRTH - DEATH PROCESS ◮ Spatial birth-death process: Preston (1976) ◮ Birth-death MCMC: Stephens (2000) in mixture models Birth-death process in GGM ◮ Adding new edge in birth and deleting edge in death time

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S IMPLE CASE

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S IMPLE CASE

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S IMPLE CASE

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES C ANVERGENCY Preston (1976): Backward Kolmogorov Under Balance condition, process converges to unique stationary distribution. Mohammadi and Wit (2013): BDMCMC in GGM Stationary distribution = Posterior distribution of (G,K) So, relative sojourn time in graph G = p ( G | data )

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES P ROPOSED BDMCMC ALGORITHM Step 1 : (a). Calculate birth and death rates β ξ ( K ) = λ b , new link ξ = ( i , j ) b ξ ( k ξ ) p ( G − ξ , K − ξ | x ) δ ξ ( K ) = λ b , existing link ξ = ( i , j ) p ( G , K | x ) (b). Calculate waiting time, (c). Simulate type of jump, birth or death Step 2: Sampling new precision matrix: K + ξ or K − ξ

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES P ROPOSED PRIOR DISTRIBUTIONS Prior for graph ◮ Discrete Uniform ◮ Truncated Poisson according to number of links Prior for precision matrix ◮ G-Wishart: W G ( b , D ) − 1 � � p ( K | G ) ∝ | K | ( b − 2 ) / 2 exp 2 tr ( DK ) − 1 � � � | K | ( b − 2 ) / 2 exp I G ( b , D ) = 2 tr ( DK ) dK P G

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES C OMPUTING DEATH RATES � | K − ξ | I G ( b , D ) � ( b ∗ − 2 ) / 2 − tr ( D ∗ ( K − ξ − K )) / 2 � � δ ξ ( K ) = γ b exp | K | I G − ξ ( b , D ) E G [ f T ( ψ ν )] I G − ξ ( b , D ) = 2 √ π t ii t jj I G ( b , D ) Γ(( b + ν i ) / 2 ) Γ(( b + ν i − 1 ) / 2 ) E G − ξ [ f T ( ψ ν )] Plot for ratio of normalizing constants 1.20 1.15 ratio of expectation 1.10 1.05 1.00 0 50 100 150 200 250 300 number of nodes

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES R PACKAGE BDgraph package ◮ Graph estimation for high-dimensional cases ◮ Graph estimation for low-dimensional cases

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S IMULATION : 8 NODES N 8 ( 0 , Σ) | K = Σ − 1 ∈ P G � � M G =   1 . 5 0 0 0 0 0 . 4 1 . 5 0 0 0 0 0     1 . 5 0 0 0 0      . 5  1 0 0 0   K =   1 . 5 0 0     . 5 1 0     1 . 5     1

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S OME RESULT Effect of Sample size Number of data 20 30 40 60 80 100 p(true graph | data) 0.018 0.067 0.121 0.2 0.22 0.35 false positive 0 0 0 0 0 0 false negative 1 0 0 0 0 0

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S IMULATION : C IRCLE GRAPH WIHT 120 NODES N 120 ( 0 , Σ) | K = Σ − 1 ∈ P G � � M G = , ◮ n = 2000 ≪ 7260 ◮ Priors: K ∼ W G ( 3 , I 120 ) and G ∼ TU ( all possible graphs ) ◮ 10000 iterations and 5000 iterations as burn-in Result ◮ Time: 4 hours ◮ p(true graph | data) = 0 . 09 which is most probable graph

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES S UMMARY M OHAMMADI , A. AND E. C. W IT (2013) Gaussian graphical model determination based on birth-death MCMC inference, arXiv preprint arXiv:1210.5371v4