Inference in DBNs with non-disjoint clustersperso.crans.org/~genest/CFF.pdf · 2015. 10. 1. ·...

Inference in DBNs with non-disjoint clusters

Matthieu Pichené

Introduction

Apoptosis pathway

Mcl1

Mcl1

Method

simulations Analysis

MATHEMATICAL FORMALISM

BIOLOGICAL SYSTEM

Method

Method

Method(Approximate)  abstrac1on    

of  the  low  level    biochemical  model  

DBNs

ES

S

E

P

S + E <—> ES —> P + E

t0 t1 t2 t3

k+1

k-1

k+2

{1 2 3 4 5

{1 2 3 4 5

{1 2 3 4 5

{1 2 3 4 5

Every specie at time point t is a random

variable over a discrete

number of values.

Number of configurations at each time point: ValuesSpecies

DBNs

ES

S

E

P

t0 t1 t2 t3

+CPT

S ES

E P

k+1

k-1

k+2S + E <—> ES —> P + E

CPTS + E <—> ES —> P + Ek+1

k-1

k+2

S S E ES Pr1 1 1 1 0.11 2 1 2 0.22 2 3 3 0.1…

SES S E ES P Pr112…

E S E ES Pr112…

P ES P Pr112…

ES

E P

DBNs

ES

S

E

P

t0 t1 t2 t3

+CPT

S ES

E P

k+1

k-1

k+2S + E <—> ES —> P + E

Complexity of exact inference: at least ValuesSpecies

DBNs

• We need an approximation. Express configurations as product of probabilities

• Simplest idea : Consider all species independent ( Factored Frontier )

Factored Frontier

ES

S

E

P

t0 t1 t2 t3

k+1

k-1

k+2

Hypothesis : Independent

S + E <—> ES —> P + E

complexity of FF inference: Species x ValuesNbPar+1

Pt2(P=h)= f(Pt1(P),Pt1(ES),CPT)

Low accuracy

Clustered Factored Frontier

• Use of clusters containing the species that have the most mutual information

• Clusters may vary over time

• All sets of states for species in a clusters are calculated (that limits the length of clusters)

Clustered Factored Frontier

• Use information theory (Eric) to obtain the important relations

• We (Eric) chose the tree to minimize distance

• Tree implies cluster of size 2

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136 238 5 61439 337 4 7404335464515 8113029345612132528272657492217191820162421514832333150554447525354 923104142 0

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Mutual information on the whole graph

Mutual Information on the Tree Approximation

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0.5

1

1.5

2

2.5

3

Species correlations (Eric)

Hypothesis :

Pr(St=h,ESt=l,Et=m,Pt=h) =

Pr(St=h,ESt=l) Pr(ESt=l, Et=m) Pr(ESt=l,Pt=h)

Pr2(ESt=l)

S ES E

P

Clustered Factored Frontierwe assume that relations not in tree are irrelevant

Apoptosis pathway

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Apoptosis pathway

Clustered Factored Frontier

ES

S

E

P

t0 t1 t2 t3

+CPT

S ES

E P

k+1

k-1

k+2S + E <—> ES —> P + E

Clustered Factored Frontier

ES

S

E

P

t0 t1 t2 t3

+CPT

S ES

E P

k+1

k-1

k+2S + E <—> ES —> P + E

Pt1(s’,es’)=Σs,es,e (Pt0(s,es,e)CPT(s,es,e,s’)CPT(s,es,e,es’))

How our algorithm work

Hypothesis :

Pr(St=h,ESt=l,Et=m,Pt=h) =

Pr(St=h,ESt=l) Pr(ESt=l, Et=m) Pr(ESt=l,Pt=h)

Pr2(ESt=l)

S ES E

P

How to compute P(parents(Cluster))

Proposition : P(Xp = vp, XL = VL, XR =VR) = P(Xp = vp, XL = VL) x P(Xp = vp, XR =VR)

P(Xp = vp)

p

L R

How to compute P(parents(Cluster))

Parent_Cluster= set of nodes necessary to use the CPTs.

How to compute P(parents(Cluster))

How to compute P(parents(Cluster))

How to compute P(parents(Cluster))

How to compute P(parents(Cluster))

How to compute P(parents(Cluster))

Independence between trees Complexity : Species x Values Parents_Cluster+1

Algorithm comparison

FF ClusteredFF Exact computation

Complexity Species x ValuesNbParents

Species x ValuesParents_Cluster+1 > ValuesSpecies

Accuracy Low ? but better than FF Exact

Conclusion

• Our program is currently still being written. Results will tell if the accuracy is good or not.

• After the first results are obtained we will upgrade it to accept bigger clusters and non-tree graphs

How our algorithm work

How our algorithm work

How our algorithm work

How our algorithm work

How our algorithm work

Order S x N

How our algorithm work

• For each time T groups of clusters are found

• Most efficient path is found to calculate each cluster

• Calculate probability using CPTs

• Results are saved, cluster probabilities are kept in memory

Clustered Factored Frontier

A

A*

A <—> A* CPT:

96.04% A = h , A* = l 0.04% A = l , A* = h 1.96% A = h , A* = h 1.96% A = l , A* = l 0.04% A = h , A* = l 96.04% A = l , A* = h 1.96% A = h , A* = h 1.96% A = l , A* = l

98% : A = h A* = l —> A = h 2% : A = h A* = l —> A = l 2% : A = l A* = h —> A = h 98% : A = l A* = h —> A = l 2% : A = h A* = l —> A* = h 98% : A = h A* = l —> A* = l 98% : A = l A* = h —> A* = h 2% : A = l A* = h —> A* = l

50% A = h A* = l 50% A = l A* = h :

Clustered Factored Frontier

A

A*

A <—> A* CPT:

53.04% A = h , A* = l 53.04% A = l , A* = h 1.96% A = h , A* = h 1.96% A = l , A* = l

98% : A = h A* = l —> A = h 2% : A = h A* = l —> A = l 2% : A = l A* = h —> A = h 98% : A = l A* = h —> A = l 2% : A = h A* = l —> A* = h 98% : A = h A* = l —> A* = l 98% : A = l A* = h —> A* = h 2% : A = l A* = h —> A* = l

50% A = h A* = l 50% A = l A* = h :