Bayesian model selection in graphs by using BDgraph package

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1 Bayesian model selection in graphs by using BDgraph package A. Mohammadi and E. Wit March 26, 2013

2 MOTIVATION Flow cytometry data with 11 proteins from Sachs et al. (2005)

3 RESULT FOR CELL SIGNALING DATA

4 Gaussian graphical model Respect to graph G = (V, E) as M G = { } N p (0, Σ) K = Σ 1 is positive definite based on G Pairwise Markov property X i X j X V\{i,j} k ij = 0,

5 BIRTH-DEATH PROCESS Spacial birth-death process: Preston (1976) Birth-death MCMC: Stephen (2000) in mixture models

6 BIRTH-DEATH MCMC DESIGN General birth-death process Continuous Markov process Birth and death events are independent Poisson processes Time of birth or death event is exponentially distributed Birth-death process in GGM Adding new edge in birth and deleting edge in death time

7 SIMPLE CASE

8 SIMPLE CASE

9 SIMPLE CASE

10 BALANCE CONDITION 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 = posterior probability of G

11 PROPOSED BDMCMC ALGORITHM Step 1: (a). Calculate birth and death rates β 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) (b). Calculate waiting time, (c). Simulate type of jump, birth or death Step 2: Sampling new precision matrix: K + ξ or K ξ

12 PROPOSED PRIOR DISTRIBUTIONS Prior for graph Discrete Uniform Truncated Poisson according to number of links Prior 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 G 2 tr(dk) dk

13 SPECIFIC ELEMENT OF BDMCMC Sampling from G-Wishart distribution Block Gibbs sampler Edgewise block Gibbs sampler According to maximum cliques Metropolis-Hastings algorithm Computing death rates δ 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

14 Ratio of normalizing constant I G (b, D) I G ξ(b, D) = 2 Γ πt ii t jj Γ ( ) b+νi 2 ( b+νi 1 2 ) E G [f T (ψ ν )] E G ξ [f T (ψ ν )] Plot for ratio of normalizing constants ratio of expectation number of nodes

15 Death rates for high-dimensional cases δ ξ (K) = 2 πt ii t jj Γ ( ) b+νi 2 ( ) b+νi 1 2 Γ { exp 1 2 tr(d (K ξ K)) ) (b 2)/2 ( K ξ K } γ b b ξ (k ξ ) BDgraph package bdmcmc.high : for high-dimensional graphs bdmcmc.low : for low-dimensional graphs

16 SIMULATION: 8 NODES } M G = {N 8 (0, Σ) K = Σ 1 P G K =

17 SOME RESULT Effect of Sample size Number of data p(true graph data) false positive false negative

18 SIMULATION: 120 NODES M G = {N 120 (0, Σ) K = Σ 1 P G }, n = Priors: K W G (3, I 120 ) and G TU(all possible graphs) iterations and 5000 iterations as burn-in Result Time 4 hours p(true graph data) = 0.09 which is most probable graph

19 SUMMARY

20 Thanks for your attention References MOHAMMADI, A. AND E. C. WIT (2013) Gaussian graphical model determination based on birth-death MCMC inference, arxiv preprint arxiv: v4 WANG, H. AND S. LI (2012) Efficient Gaussian graphical model determination under G-Wishart prior distributions. Electronic Journal of Statistics, 6: ATAY-KAYIS, A. AND H. MASSAM (2005) A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models. Biometrika Trust, 92(2): PRESTON, C. J. (1976) Special birth-and-death processes. Bull. Inst. Internat. Statist., 34:

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