Bayesian model selection in graphs by using BDgraph package A. - - PowerPoint PPT Presentation

bayesian model selection in graphs by using bdgraph
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Bayesian model selection in graphs by using BDgraph package A. - - PowerPoint PPT Presentation

G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES Bayesian model selection in graphs by using BDgraph package A. Mohammadi and E. Wit March 26, 2013 G RAPH S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES M OTIVATION Flow cytometry data with 11


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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

Bayesian model selection in graphs by using BDgraph package

  • A. Mohammadi and E. Wit

March 26, 2013

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

MOTIVATION

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

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

RESULT FOR CELL SIGNALING DATA

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

Gaussian graphical model

Respect to graph G = (V, E) as MG =

  • Np(0, Σ) | K = Σ−1 is positive definite based on G
  • Pairwise Markov property

Xi⊥Xj | XV\{i,j} ⇔ kij = 0,

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

BIRTH-DEATH PROCESS

◮ Spacial birth-death process: Preston (1976) ◮ Birth-death MCMC: Stephen (2000) in mixture models

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

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

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SIMPLE CASE

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SIMPLE CASE

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SIMPLE CASE

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

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

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

PROPOSED BDMCMC ALGORITHM

Step 1: (a). Calculate birth and death rates

βξ(K) = λb, new link ξ = (i, j) δξ(K) = bξ(kξ)p(G−

ξ , K− ξ |x)

p(G, K|x) λb, existing link ξ = (i, j) (b). Calculate waiting time, (c). Simulate type of jump, birth or death Step 2: Sampling new precision matrix: K+

ξ or K− ξ

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

PROPOSED PRIOR DISTRIBUTIONS

Prior for graph

◮ Discrete Uniform ◮ Truncated Poisson according to number of links

Prior for precision matrix

◮ G-Wishart: WG(b, D)

p(K|G) ∝ |K|(b−2)/2 exp

  • −1

2tr(DK)

  • IG(b, D) =
  • PG

|K|(b−2)/2 exp

  • −1

2tr(DK)

  • dK
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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

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

δξ(K) = p(G−

ξ , K− ξ |x)

p(G, K|x) γbbξ(kξ) = IG(b, D) IG−

ξ (b, D)

  • |K−ξ|

|K|

(b∗−2)/2

exp

  • −1

2tr(D∗(K−ξ − K))

  • γb
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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

Ratio of normalizing constant

IG(b, D) IG−ξ(b, D) = 2√πtiitjj Γ

b+νi

2

  • Γ

b+νi−1

2

  • EG [fT(ψν)]

EG−ξ [fT(ψν)]

50 100 150 200 250 300 1.00 1.05 1.10 1.15 1.20

Plot for ratio of normalizing constants

number of nodes ratio of expectation

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

Death rates for high-dimensional cases

δξ(K) = 2√πtiitjj Γ

b+νi

2

  • Γ

b+νi−1

2

  • |K−ξ|

|K|

(b∗−2)/2

× exp

  • −1

2tr(D∗(K−ξ − K))

  • γbbξ(kξ)

BDgraph package

◮ bdmcmc.high : for high-dimensional graphs ◮ bdmcmc.low : for low-dimensional graphs

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SIMULATION: 8 NODES

MG =

  • N8(0, Σ)|K = Σ−1 ∈ PG
  • K =

             

1 .5 .4 1 .5 1 .5 1 .5 1 .5 1 .5 1 .5 1

             

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SOME 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 false negative 1

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GRAPH SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

SIMULATION: 120 NODES

MG =

  • N120(0, Σ)|K = Σ−1 ∈ PG
  • ,

◮ n = 2000 ≪ 7260 ◮ Priors: K ∼ WG(3, I120) 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

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SUMMARY

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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:1210.5371v4 WANG, H. AND S. LI (2012) Efficient Gaussian graphical model determination under G-Wishart prior distributions. Electronic Journal of Statistics, 6:168-198 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):317-335

PRESTON, C. J. (1976) Special birth-and-death processes. Bull. Inst. Internat. Statist.,

34:1436-1462