Population Markov Chain Monte Carlo and Genetic Networks Fujun Ye - - PowerPoint PPT Presentation

Population Markov Chain Monte Carlo and Genetic Networks

Fujun Ye

MSc in Artificial Intelligence Supervised by Dirk Husmeier

Outline

Introduction MCMCMC MCMCMC for missing values Result Evaluation (complete data) Result Evaluation (missing values) Summary

Introduction

Genetic Network Clustering and Differential equation Bayesian Network MCMC

Genetic Network

+ A F B b ab f2 a f eq

• +
• +

Clustering

Differential Equation

Advantage

provide detailed understanding of the biological systems

Shortcoming

short of data noisy data

Inferring Bayesian Network From Expression Data

∏

=

=

n i i G i n

X Pa X P X X X P

1 2 1

)) ( | ( ) ,..., , (

Bayesian Network

A C B D E

) ( ) | ( ) | ( ) , | ( ) | ( ) , , , , ( a P a b P a c P c b d P d e P e d c b a P =

Problems

The number of different network structures grows

super exponentially with the number of nodes

M M’ P(M|D) Where the data set is large, the optimal structure M’ is well defined Where the data set is small, there are many networks which can explain the data fairly well. M P(M|D) M’

MCMC

MCMC samples networks from its posterior

distribution

Calculate the posterior probability of a

feature

) | ( D f P

=

∑ ∑

i i i i i

D M P M f P D M P ) | ( ) | ( ) | (

∑

=

i i i k k k

M P M D P M P M D P D M P ) ( ) | ( ) ( ) | ( ) | (

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Coincidence dependence

P (M|D) M

Escape from local optima

using traditional MCMC

Small step size versus big step size

P (M|D) M

Problems

Huge search space and coincidence

dependence — Prescreening is important!

Local optima — Traversal operator is

important!

Fixed step size —

Varied step size is more reasonable

MCMCMC

Metropolis-coupled Markov Chain Monte Carlo

(MCMCMC)

Pre-processing method Traversal operators Algorithm MCMCMC for missing values

MCMCMC

1 2

T T >

1

1 =

T

2 3

T T >

For each chain, move a step based on Chain swap ) ) | ( ) | ( ) ( ) | ( ) ( ) | ( , 1 min( ) , (

' ' 1 ' ' '

M M Q M M Q M P M D P M P M D P M M A

T

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

M k i M k i a

M M M M M T T T T T S ... ... ... ... ... ...

2 1 2 1

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

M i k M k i b

M M M M M T T T T T S ... ... ... ... ... ...

2 1 2 1

Acceptance Probability

[ ] [ ] [ ] [ ]

} ) ( ) | ( ) ( ) | ( ) ( ) | ( ) ( ) | ( , 1 min

1 1 1 1

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ =

k i i k

T k k T i i T k k T i i a

M P M D P M P M D P M P M D P M P M D P P

Pre-processing method

∫

= θ θ θ d M P M D P M D P ) | ( ) , | ( ) | (

Penalize complex model | | * | |

n n nv

v

n n

π α α

π =

n n n v

n v nv π π

α α =

∑

∏∏ ∏

Γ + Γ + Γ Γ =

n v nv nv nv n n n

n n n n n n n n n n n

n n M D p

π π π π π π π

α α α α ) ( ) ( ) ( ) ( ) | (

The log likelihood is

∑

=

n n D

n score M D p ) , , ( )) | ( log( π where

∑

+ + Γ − Γ =

n n n n n

n nv n n

n D n score

π π π π

α α π ))] ( log( )) ( [log( ) , , ( ∑∑ Γ − + Γ

n n n n n n n n

nv v nv nv

n

π π π π

α α ))] ( log( ) ) ( [log(

Use some max fan in Find all possible parents-configurations for each node and delete low

score parents-configurations

Keep C parents-configurations for each node and cardinality

Threshold is set as:

= θ

m scoresl m scoresl scoresh / / ) ( * + − λ

π π’ score(n,π,D) When data is quite sparse and noisy score(n,π,D) π π’ Using pre-screening method

Traversal operators

Importance sampling---

Sample a parents-configuration for a node

= ) | ( nodei p

j

π

C n D i score C D i score

n

• ld

k k k k j

) 1 ( ) , , ( ) , , (

_ , 1

− + +

∑

≠ =

π π

= ) | ( ) | ( Mold Mnew Q Mnew Mold Q

∑ ∑

≠ = ≠ =

− + + − + +

n new k k k k new k n

• ld

k k k k

• ld

k

C n D i score C D i score C n D i score C D i score

_ , 1 _ _ , 1 _

) 1 ( ) , , ( ) , , ( ) 1 ( ) , , ( ) , , ( π π π π

= ) | ( ) | ( Mold D P Mnew D P

)) , , ( ) , , ( exp(

_ _

D i score D i score

• ld

k new k

π π −

∑

=

=

n i

Ki Ki nodei P

1

) (

DIN sampling --- If the new network is loopy

Step 1 2 3 1 2 1 3 The old model 2 3 1 Step 2 2 3 1 The new model

) ) | ( * ) ( * ) | ( ) | ( * ) ( * ) | ( , 1 ( ) | ( Mold Mnew Q Mold P Mold D P Mnew Mold Q Mnew P Mnew D P Min Mold Mnew A =

)) ), ( , ( ) ), ( , ( exp( )) ), ( , ( ) ), ( , ( exp( ) | ( ) | (

1 1

D n

• n

score D n

• n

score D n n n score D n n n score Mold D P Mnew D P

i i n j j j i i n j j j

π π π π + + =

∑ ∑

= =

) | ( ) | ( Mold Mnew Q Mnew Mold Q = I simply us an approximation since it is quite time consuming to calculate the proposal probability

∑ ∑

≠ = ≠ =

− + + − + +

n new k k k k new k n

• ld

k k k k

• ld

k

C n D i score C D i score C n D i score C D i score

_ , 1 _ _ , 1 _

) 1 ( ) , , ( ) , , ( ) 1 ( ) , , ( ) , , ( π π π π

DIN proposal Traditional MCMC

Algorithm

Initialization Each iteration

• Move a step for every chain
• Chain swap

Keep the first chain

T>=1 Chain Swap DIN Sampling Illegal Legal M1 M2 … Mm Importance sampling M1’ Importance sampling Mi’ A (Mi’, Mi) Mi’ S( m Chains) S’ Pa (S’, S) T=1

MCMCMC for missing values

I4 I3 I10 I6 2 2 1 1 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 1 2 2 2 1 1 2 1 2 2 1 2

X1 X2 X3 X4 X5 1 ? 2 1 1 2 2 2 1 2 ? 1 1 ? 1 1 2 ? 2 ? ? 1 1 1 2 1 1 2 ? 1 1 ? 2 1 1 2 1 1 2 ? 2 2 ? 1 2 1 2 2 ? ? I1 I2 I3 I4 I5 I6 … 1 3 1 5 2 1 2 7 3 4 3 9 …

T=1 T>=1 Chain Swap

DIN Sampling

Illegal Legal M1, D1 M2, D2 … Mm, Dm Importance sampling M1’ Importance sampling Mi’ A (Mi’, Mi| Di) Mi’ S( m Chains) S’ Pa (S’, S) T=1 Dmi Dmi’ Di’ Observed data A (Di’, Di| Gi)

Proposal method for before

burn in

) , , | ( m n M v Q

n

=∑

+ +

n m n m n

v v v v v

N N ) 1 ( 1

π π

) , , | , ( m n M v v Q

mis n π

= ∑

+ +

mis n nm mis n nm mis n

v v v v v v v

N N

π

π π π π ,

) 1 ( 1

) , | ( n M v Q

n

=∑

+ +

n n n

v v v

N N ) 1 ( 1

X1 X2 X3 X4 X5 1 ? 2 1 1 2 2 2 1 2 ? 1 1 2 1 1 2 ? ? ? ? 1 1 1 2 1 1 2 ? 1 1 ? 2 1 1 2 1 1 2 ? 2 2 ? 1 2 1 2 2 ? ?

1 2 5 4 3

Acceptance probability ) , (

'

MissVal MissVal Accept = ) ) | ( ) | ( ) | ( ) | ( , 1 min(

' ' '

M D P MissVal MissVal Q M D P MissVal MissVal Q

After burn in

Acceptance probability

) , (

'

MissVal MissVal Accept

=

) ) | ( ) | ( ) | ( ) | ( , 1 min(

' ' '

M D P MissVal MissVal Q M D P MissVal MissVal Q

) | (

' MissVal

MissVal Q = ∏ ∑

Ω ∈

+ +

) ( _

) 1 ( 1

cmis i j ij new i

N N ) | (

'

MissVal MissVal Q = ∏ ∑

Ω ∈

+ +

) ( _

) 1 ' ( 1

cmis i j ij

• ld

i

N N

Result Evaluation (complete data)

fn tp tp sensitivty + = fp tn tn y specificit + = ary complement fp tn tn y specificit + − = 1 = fp tn fp + fn tp

2 1 3 2 1 3

fp

4 4

tn tn

ROC curve

tp is the number of true positive edges. fn is the number of false negative edges. fp is the number of false positive edges. tn is the number of true negative edges.

Model Genetic Network

MCMCMC against order MCMC MCMCMC against structure MCMC MCMCMC against Population MCMC

• Temperatures= [1, 1, 3, 9, 30]
• Keep at most 10 parents-configurations for each node and cardinality.
• With 60000 iterations: 30000 burn in and keep the last 30000 samples.

Alarm Network

Arabidopsis data

Result Evaluation (missing values)

Model Genetic Network

Before burn in(30000 burn in, 30000 iterations After burn in 40000 iterations Temp=[1,1,3,9,12]

• The ROC curve for noise=0.2 data=200 with different missing rate
• Temp = [1, 1, 3, 9, 12]
• Use 30000 burn in and 30000 iterations.

Every 10 steps keep one sample. (before burn in algorithm)

B cell Lymphoma data

Summary

MCMCMC Order MCMC Structure MCMC Population MCMC

Problems with MCMCMC

T=1 T=50 iterations

logP(D|M)+ logP(M)