Network determination based on birth-death MCMC inference

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES<strong>Network</strong> <strong>determination</strong> <strong>based</strong> on birth-deathMCMC inferenceA. Mohammadi and E. WitFebruary 4, 2013

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESWHAT IS TALK ABOUT ?Problem◮ High-dimensional cases: p(p − 1)/2 ≫ n◮ Bayesian approches : Not fast◮ glasso : Sensitivity to tuning parametersSolution◮ We proposed Bayesian method which is fast and accurate◮ Implement to R package: BDgraphTrans-dimensional MCMC◮ Reversible-jump MCMC◮ Birth-Death MCMC

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES<strong>Network</strong>Gaussian graphical model with respect to graph G = (V, E) asM G ={}N p (0, Σ) | K = Σ −1 is positive definite <strong>based</strong> on GPairwise Markov propertyX i ⊥X j | X V\{i,j} ⇔ k ij = 0,

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESBIRTH-DEATH PROCESS◮ Spacial birth-death process: Preston (1976)◮ Brith-death MCMC: Stephen (2000) in mixture models

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESBIRTH-DEATH MCMC DESIGNGeneral birth-death process◮ Continuous Markov process◮ Birth and death events are independent Poisson processes◮ Time of birth or death event is exponentially distributedBirth-death process in GGM◮ Adding new edge in birth time and deleting edge in deathtime

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMPLE CASE

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESBALANCE CONDITIONPreston (1976): Backward KolmogorovIf balance conditions are hold, process converges to uniquestationary distribution.

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESPROPOSED BIRTH-DEATH MCMC ALGORITHMProposal birth and death ratesβ ξ (K) = λ b , new link ξ = (i, j)δ ξ (K) = b ξ(k ξ )p(G − ξ , K− ξ |x)λ b , existing link ξ = (i, j)p(G, K|x)Proposal birth-death MCMC algorithmStarting with initial graph:Step 1: (a). Calculate birth and death rates(b). Calculate waiting time, λ(K) = 1/(β(K) + δ(K))(c). Simulate type of jump, birth or deathStep 2: Sampling from new precision matrix: K + ξor K− ξ

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSAMPLING BIRTH-DEATH MCMC ALGORITHM

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES◮ Birth-death MCMC algorithm for general case ...

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSPECIFIC ELEMENT OF BDMCMC METHOD◮ Prior distributions◮ Computing death rates◮ Sampling from precision matrix

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESPROPOSED PRIOR DISTRIBUTIONSPrior for graph◮ Discrete Uniform◮ Truncated Poisson according to number of linksPrior for precision matrix◮ G-Wishart: W G (b, D)p(K|G) ∝ |K| (b−2)/2 exp{− 1 }2 tr(DK)∫I G (b, D) = |K| (b−2)/2 exp{− 1 }P G2 tr(DK) dK

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESG-WISHART DISTRIBUTIONSampling from G-Wishart distribution◮ Accept-reject algorithm◮ Metropolis-Hastings algorithm◮ Block Gibbs sampler◮ According to maximum cliques◮ Edgewise block Gibbs sampler

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESCOMPUTING DEATH RATESδ ξ (K) = p(G− ξ , K− ξ |x)p(G, K|x) γ bb ξ (k ξ )=I G (b, D)(|K −ξ |(b, D) |K|I G−ξ) (b ∗ −2)/2 {exp − 1 }2 tr(D∗ (K −ξ − K)) γ b b ξLimitationAlgorithm is very slow !!

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESRatio of normalizing constantI G (b, D)I G −ξ(b, D) = 2√ Γπt ii t jjΓ( ) b+νi2( b+νi −12) E G [f T (ψ ν )]E G −ξ [f T (ψ ν )]Plot for ratio of normalizing constantsratio of expectation1.00 1.05 1.10 1.15 1.200 50 100 150 200 250 300number of nodes

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESDeath rates for high-dimensional casesδ ξ (K) = 2 √ πt ii t jjΓ( ) b+νi2( ) b+νi −12Γ{× exp − 1 2 tr(D∗ (K −ξ − K))) (b ∗ −2)/2(|K −ξ ||K|}γ b b ξ (k ξ )R packageWe compile our method into BDgraph package which isavailable from CRAN web site

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMULATION: 8 NODES}M G ={N 8 (0, Σ)|K = Σ −1 ∈ P G⎡⎤1 .5 0 0 0 0 0 .41 .5 0 0 0 0 01 .5 0 0 0 0K =1 .5 0 0 01 .5 0 01 .5 0⎢⎥⎣1 .5⎦1

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSOME RESULTEffect of Sample sizeNumber of data 20 30 40 60 80 100 150p(true graph | data) 0.018 0.067 0.121 0.2 0.22 0.35 0.43false discovery 1 0 0 0 0 0 0false negative 0 0 0 0 0 0 0

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSOME RESULT⎡⎤1 1 0.03 0.06 0.02 0.02 0.03 11 1 0.04 0.03 0.02 0.03 0.031 1 0.06 0.04 0.06 0.03ˆp ξ =1 1 0.05 0.04 0.031 1 0.05 0.13.1 1 0.13⎢⎥⎣1 1 ⎦1⎡⎤1.3 0.6 0 0 0 0 0 0.51.4 0.5 0 0 0 0 01 0.5 0 0 0 0ˆK =1.2 0.6 0 0 01.3 0.4 0 0.0.9 0.5 0⎢⎥⎣0.9 0.4⎦

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESSIMULATION: 120 NODESM G ={N 120 (0, Σ)|K = Σ −1 ∈ P G},◮ n = 1000 ≪ 7260◮ Priors: K ∼ W G (3, I 120 ) and G ∼ TU(all possible graphs)◮ 10000 iterations and 5000 iterations as burn-inResult◮ Time 190 minutes◮ p(true graph | data) = 0.41 which is most probable graph

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESCELL SIGNALING DATAFlow cytometry data with 11 proteins from Sachs et al. (2005)(Left) Result from our algorithm(Right) Result from Sachs et al (2005)Friedman et al (2008): full graph according to g-lasso

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESCONCLUSION

NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLESThanks for your attentionReferencesMOHAMMADI, A. AND E. C. WIT (2012) Efficient birth-death MCMC inference forGaussian graphical models. arXiv preprint arXiv:1210.5371WANG, H. AND S. LI (2012) Efficient Gaussian graphical model <strong>determination</strong> underG-Wishart prior distributions. Electronic Journal of Statistics, 6:168-198ATAY-KAYIS, A. AND H. MASSAM (2005) A Monte Carlo method for computing themarginal likelihood in nondecomposable Gaussian graphical models. Biometrika Trust,92(2):317-335PRESTON, C. J. (1976) Special birth-and-death processes. Bull. Inst. Internat. Statist.,34:1436-1462