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 BIRTHDEATH PROCESS Spacial birthdeath process: Preston (1976) Birthdeath MCMC: Stephen (2000) in mixture models
6 BIRTHDEATH MCMC DESIGN General birthdeath process Continuous Markov process Birth and death events are independent Poisson processes Time of birth or death event is exponentially distributed Birthdeath 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 GWishart: 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 GWishart distribution Block Gibbs sampler Edgewise block Gibbs sampler According to maximum cliques MetropolisHastings 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 highdimensional 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 highdimensional graphs bdmcmc.low : for lowdimensional 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 burnin 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 birthdeath MCMC inference, arxiv preprint arxiv: v4 WANG, H. AND S. LI (2012) Efficient Gaussian graphical model determination under GWishart prior distributions. Electronic Journal of Statistics, 6: ATAYKAYIS, 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 birthanddeath processes. Bull. Inst. Internat. Statist., 34:
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