Network determination based on birth-death MCMC inference A. - - PowerPoint PPT Presentation

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Network determination based on birth-death MCMC inference A. - - PowerPoint PPT Presentation

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N ETWORK B IRTH - DEATH MCMC METHOD S PECIFIC ELEMENT OF BDMCMC METHOD E XAMPLES Network determination based on birth-death MCMC inference A. Mohammadi and E. Wit February 4, 2013 N ETWORK B IRTH - DEATH MCMC METHOD S PECIFIC ELEMENT OF BDMCMC

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

Network determination based on birth-death MCMC inference

  • A. Mohammadi and E. Wit

February 4, 2013

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

WHAT IS TALK ABOUT ?

Problem

◮ High-dimensional cases: p(p − 1)/2 ≫ n

◮ Bayesian approches : Not fast ◮ glasso : Sensitivity to tuning parameters

Solution

◮ We proposed Bayesian method which is fast and accurate ◮ Implement to R package: BDgraph

Trans-dimensional MCMC

◮ Reversible-jump MCMC ◮ Birth-Death MCMC

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

Network

Gaussian graphical model with 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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NETWORK BIRTH-DEATH MCMC METHOD SPECIFIC ELEMENT OF BDMCMC METHOD EXAMPLES

BIRTH-DEATH PROCESS

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

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NETWORK BIRTH-DEATH MCMC METHOD 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 time and deleting edge in death

time

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

SIMPLE CASE

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

SIMPLE CASE

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

SIMPLE CASE

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

SIMPLE CASE

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

SIMPLE CASE

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

SIMPLE CASE

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

BALANCE CONDITION

Preston (1976): Backward Kolmogorov

If balance conditions are hold, process converges to unique stationary distribution.

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

PROPOSED BIRTH-DEATH MCMC ALGORITHM

Proposal 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)

Proposal birth-death MCMC algorithm

Starting 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 death Step 2: Sampling from new precision matrix: K+

ξ or K− ξ

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

SAMPLING BIRTH-DEATH MCMC ALGORITHM

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

◮ Birth-death MCMC algorithm for general case ...

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

SPECIFIC ELEMENT OF BDMCMC METHOD

◮ Prior distributions ◮ Computing death rates ◮ Sampling from precision matrix

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

G-WISHART DISTRIBUTION

Sampling from G-Wishart distribution

◮ Accept-reject algorithm ◮ Metropolis-Hastings algorithm ◮ Block Gibbs sampler

◮ According to maximum cliques ◮ Edgewise block Gibbs sampler

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

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))

  • γbbξ(

Limitation

Algorithm is very slow !!

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NETWORK BIRTH-DEATH MCMC METHOD 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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NETWORK BIRTH-DEATH MCMC METHOD 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)

R package

We compile our method into BDgraph package which is available from CRAN web site

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

SOME RESULT

Effect of Sample size

Number of data 20 30 40 60 80 100 150 p(true graph | data) 0.018 0.067 0.121 0.2 0.22 0.35 0.43 false discovery 1 false negative

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

SOME RESULT

ˆ pξ =

             

1 1 0.03 0.06 0.02 0.02 0.03 1 1 1 0.04 0.03 0.02 0.03 0.03 1 1 0.06 0.04 0.06 0.03 1 1 0.05 0.04 0.03 1 1 0.05 0.13 1 1 0.13 1 1 1

             

. ˆ K =

             

1.3 0.6 0.5 1.4 0.5 1 0.5 1.2 0.6 1.3 0.4 0.9 0.5 0.9 0.4 1

             

.

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

SIMULATION: 120 NODES

MG =

  • N120(0, Σ)|K = Σ−1 ∈ PG
  • ,

◮ n = 1000 ≪ 7260 ◮ Priors: K ∼ WG(3, I120) and G ∼ TU(all possible graphs) ◮ 10000 iterations and 5000 iterations as burn-in

Result

◮ Time 190 minutes ◮ p(true graph | data) = 0.41 which is most probable graph

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

CELL SIGNALING DATA

Flow 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

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

CONCLUSION

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

Thanks for your attention

References

MOHAMMADI, A. AND E. C. WIT (2012) Efficient birth-death MCMC inference for Gaussian graphical models. arXiv preprint arXiv:1210.5371 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