genome evolution:
DESCRIPTION
Genome evolution:. Lecture 8: Belief propagation. You are P(H|our data). I am P(H|all data). You are P(H|our data). Simple Tree: Inference as message passing. s. s. s. s. s. s. s. DATA. We need:. Understanding the tree model (and BNs): reversing edges. - PowerPoint PPT PresentationTRANSCRIPT
Genome Evolution. Amos Tanay 2009
Genome evolution:
Lecture 8: Belief propagation
Genome Evolution. Amos Tanay 2009
Simple Tree: Inference as message passing
s
s
s s
s
s
sYou are P(H|our data)
You are P(H|our data)
I am P(H|all data)
DATA
Genome Evolution. Amos Tanay 2009
Understanding the tree model (and BNs): reversing edges
The joint probability of the simple tree model:
),|Pr(),|Pr(),|Pr(),|Pr()|Pr(),Pr( 212213121 hxhxhxhhhxh
Can we change the position of the root and keep the joint probability as is?
)',|Pr()',|Pr()',|Pr()',|Pr()'|Pr(),Pr( 212213212 hxhxhxhhhxh
We need: )',|Pr()'|Pr(),|Pr()|Pr( 212121 hhhhhh
)'|Pr()',|Pr()'|Pr(),|Pr()|Pr( 2212121
11
hhhhhhhhh
)|Pr()'|Pr(/),|Pr()|Pr( 212121 hhhhhh
Genome Evolution. Amos Tanay 2009
Factor graphs
Defining the joint probability for a set of random variables given:
1) Any set of node subsets (hypergraph)
2) Functions on the node subsets (Potentials)
)(1
)Pr( aa xZ
x
)( ax
)|{, VaaAV
x
aa xZ )(
Joint distribution:
Partition function:
If the potentials are condition probabilities, what will be Z?
Things are difficult when there are several modes
Factor
R.V.
Not necessarily 1! (can you think of an example?)
Genome Evolution. Amos Tanay 2009
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
DBN PhyloHMM
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
hpaij hpai
j+1hpaij-1
hij hi
j+1hij-1
Converting directional models to factor graphs
(Loops!) Well defined
Z=1Z=1
)pa|Pr()( xxxa )pa|Pr()( xxxa
)pa|Pr()( xxxa Z!=1
Genome Evolution. Amos Tanay 2009
More definitions
The model: a
axZx )(log)log())log(Pr(
Potentials can be defined on discrete, real valued etc.it is also common to define general log-linear models directly:
))(logexp(1
)Pr( a
aa xwZ
x
Inference:
Dx a
aa xwZ
D ))(logexp(1
)|Pr(
)|Pr(/))(logexp(1
),|Pr(,|
DxwZ
DxDxx a
aai
i
Learning:
Find the factors parameterization: )|Pr(maxarg
D
Genome Evolution. Amos Tanay 2009
Inference in factor graphs: Algorithms
Directed models are sometimes more natural and easy to understand. Their popularity stems from their original role as expressing knowledge in AIThey are not very natural for modeling physical phenomena, except for time-dependent processes
Undirected models are analogous to well-developed models in statistical physics (e.g., spin glass models)We borrow computational ideas from physicists (the guys are big with approximations)
The models are convex which give them important algorithmic properties (Wainwright and Jordan 2003 and further development in recent time)
Dynamic programming:
Forward sampling (likelihood weighting):
Metropolis/Gibbs:
Mean field:
Structural variational inference:
No (also not in BN!)
No
Yes
Yes
Yes
Genome Evolution. Amos Tanay 2009
Belief propagation in a factor graph
)(1
)|( aaa xZ
xP
• Remember, a factor graph is defined given a set of random variables (use indices i,j,k.) and a set of factors on groups of variables (use indices a,b..)
)( iia xm
• Think of messages as transmitting beliefs:
a->i : given my other inputs variables, and ignoring your message, you are x
i->a : given my other inputs factors and my potential, and ignoring your message, you are x
• xa refers to an assignment of values to the inputs of the factor a
• Z is the partition function (which is hard to compute)
• The BP algorithm is constructed by computing and updating messages:
• Messages from factors to variables:
• Messages from variables to factors: )( iai xm
(any value attainable by xi)->real values
Genome Evolution. Amos Tanay 2009
Messages update rules:
)()(\)(
iicaiNc
iai xmxm
ia xx
jajiaNj
aaiia xmxxm )()()(\)(
Messages from variables to factors:
Messages from factors to variables:
a
i aiN \)(
a
iiaN \)(
Genome Evolution. Amos Tanay 2009
The algorithm proceeds by updating messages:
• Define the beliefs as approximating single variables posterios (p(hi|s)):
)()()(
iiaiNa
ii xmxb
Algorithm:
Initialize all messages to uniformIterate until no message change:
Update factors to variables messagesUpdate variables to factors messages
• Why this is different than the mean field algorithm?
)()( ii hqhq
Genome Evolution. Amos Tanay 2009
Beliefs on factor inputs
This is far from mean field, since for example:
)()(
)()()()(
\)()(
)()(
jjcajNcaNj
a
jjaNj
jajaNj
aaa
xmx
xbxmxxb
The update rules can be viewed as derived from constraints on the beliefs:
1.requirement on the variables beliefs (bi)
2.requirement on the factor beliefs (ba)
3.Marginalization requirement:
a
i aiN \)(
a
iiaN \)(
ia xxjjc
ajNcaNjaiiiid
iNdxmxxbxm )()()()(
\)()()(
ia xxjjc
ajNciaNjaiia xmxxm )()()(
\)(\)(
ia xx
aaii xbxb\
)()(
)()()(
iiaiNa
ii xmxb
)()()(\)()(
jjcajNcaNj
aaa xmxxb
Genome Evolution. Amos Tanay 2009
BP on Tree = Up-Down
s4 s3
h2
h3e
s2 s1
h1
b a
c
d
)|Pr()|Pr()( 12111hshsxup ih
111)( 1 hbhach mmhm
)()()(
)()()(
2\
1
1\
1
11
11
smxhm
smxhm
bshx
bhb
ashx
aha
ib
ia
32
32
1
,313232 )|Pr()|Pr()()(
)(
hhhh
ih
hhhhhdownhup
xdown
3 2
2
3
33
1
31
)(),()(),(
)()(),(
)()()(
323313
3313
\31
h hehedc
hhehdc
hxchcchc
hmhhhhh
hmhmhh
hmxhmc
2 1
3
Genome Evolution. Amos Tanay 2009
Loopy BP is not guaranteed to converge
X Y
Y
x
01
10
Y
x
01
10
1 1
0 0
This is not a hypothetical scenario – it frequently happens when there is too much symmetryFor example, most mutational effects are double stranded and so symmetric which can result in loops.
Genome Evolution. Amos Tanay 2009
The Bethe Free Energy
H. Bethe
• LBP was introduced in several domains (BNs, Coding), and is consider very practical in many cases.
• ..but unlike the variational approaches we studied before, it is not clear how it approximate the likelihood/partition function, even when it converges..
hh
hqhqshphqF )(log)()|,(log)( • Compare to the variational free energy:
Theorem: beliefs are LBP fixed points if and only if they are locally optimal for the Bethe free energy
iiii
aaabethe
aaabethe
BetheBetheBethe
xbxbdxbxbH
xxbU
HUF
)(log)()1()(log)(
)(log)(
• In the early 2000, Yedidia, Freeman and Weiss discovered a connection between the LBP algorithm and the Bethe free energy developed by Hans Bethe to approximate the free energy in crystal field theory back in the 40’s/50’s.
Genome Evolution. Amos Tanay 2009
Generalization: Regions-based free energy
RR Aa
RXi
R caci 11
• Start with a factor graph (X,A)
• Introduce regions (XR,AR) and multipliers cR
• We require that:
• We will work with valid regions graphs:
)()()(
)(log)()(
)()()(
)(log)(
RRRRRR
xRRRRRR
xRRRRR
AaaRR
bHbUbF
xbxbbH
xExbbU
xxE
R
R
R
RR XaNAa )(
Region-based average energy
Region average energy
Region Entropy
Region Free energy
})({})({})({
)(})({
)(})({
R
R
R
RRRRR
RRRRR
RRRR
bHbUbF
bHcbH
bUcbU
Region-based entropy
Region-based free energy
Genome Evolution. Amos Tanay 2009
Bethe regions are the factors neighbors sets and single variables regions:
a
c
b
111 ccbac ccc
We compensate for the multiple counting of variables using the multiplicity constant
We can add larger regions
As long as we update the multipliers:
11 iia dcc
Ra
Rac
Rbc
RR
RR cc'
'1
Genome Evolution. Amos Tanay 2009
Multipliers compensate on average, not on entropy
Claim: For valid regions, if the regions’ beliefs are exact:
a x
aaaa
c
R x RaaaRRR
RRRRRR
a
RaR
R
xxbxxbcbUcbU )(log)()(log)()(})({)1(
We cannot guarantee much on the region-based entropy:
Claim: the region-based entropy is exact when the model is a uniform distributionProof: exercise. This means that the entropy count the correct number of degrees of
freedom – e.g. for binary variables, H=Nlog2
Definition: a region based free energy approximation is said to be max-ent normal if its region-based entropy is maximized when the beliefs are uniform.
An non max-ent approximation can minimize the region free energy by selecting erroneously high entropy beliefs!
Rx
RRRRRR xbxbbH )(log)()(
)()( RRRR xpxb
x
RR xExpbU )()(})({then the average region-based energy is exact:
a x
aaaax a
aax a
xxpxxpxExpU )(log)()(log)()()(
Genome Evolution. Amos Tanay 2009
Bethe’s region are max-ent normal
Claim: The Bethe regions gives a max-ent normal approximation (i.e. it maximize the region-based entropy on the uniform distribution)
a x aNi xiiiiaaaa
i xiiiiBethe
a ii
xbxbxbxbxbxbH)(
)(log)()(ln)()(log)(
Entropy Information
(maximal on uniform) (nonnegative, and 0 on uniform)
iiii
aaabethe
BetheBetheBethe
xbxbdxbxbH
HUF
)(log)()1()(log)(
)( abI)( ibH
Genome Evolution. Amos Tanay 2009
Start with a complete graph and binary factors
Add all variable triplets, pairs and singleton as regions
Generate multipliers:triplets = 1 (20 overall)pairs = -3 (15 overall)singletons = 6 (6 overall) ( guarantee consistency)
Example: A Non max-ent approximation
Look at the consistent beliefs:
The Region entropy (for any region) = ln2. The total region entropy is:
otherwise
xxxxxxb
otherwise
xxxxbxb kji
kjiji
jii 0
2/1),,(
0
2/1),(;5.0)0(
2ln112ln362ln452ln20 R
RRR HcH
We claimed before the entropy of the uniform distribution will be exact: 6ln2
RR
RR cc'
'1
Genome Evolution. Amos Tanay 2009
We want to solve a variational problem:
While enforcing constraints on the regions’ beliefs:
Inference as minimization of region-based free energy
})({min RR bF
1)( Rx
RR xb
)()( ''\ '
RRxx
RR xbxbRR
Unlike the structured variational approximation we discussed before, and although the beliefs are (regionally) compatible, we can have cases with optimal beliefs that are not representing a true global posterior distribution
C
BA
Y
x
4.01.0
1.04.0
1.04.0
4.01.0,
4.01.0
1.04.0CBA bbb
Y
x
1.04.0
4.01.0
Optimal region beliefs are identical to the factors:
5.0
5.0ib
Y
x
4.01.0
1.04.0
It can be shown that this cannot be the result of any joint distribution on the three variables
(note the negative feedback loop here)
Genome Evolution. Amos Tanay 2009
Claim: When it converges, LBP finds a minimum of the Bethe free energy.
Proof idea: we have an optimization problem (minimum energy) with constraints (beliefs are consistent and adds up to 1). We write down a Lagrangian that expresses both minimization goal and constraints, and show that it is minimized when the LBP update rules are holding.
Inference as minimization of region-based free energy
i iNa x xxaaiiiai
i xiii
a xaaaBethe
i ia
ia
xbxbx
xbxbFL
)( \
)]()()[(
1)(1)(
Important technical point: we shall assume that in the fixed point all beliefs are non zero. This can be shown to hold if all factors are “soft” (do not contain zero values for any assignment).
Genome Evolution. Amos Tanay 2009
The Bethe Lagrangian
i iNa x xxaaiiiai
i xiii
a xaaaBethe
i ia
ia
xbxbx
xbxbFL
)( \
)]()()[(
1)(1)(
i x
iiiiia x
aaaaa x
aaaaBethe
iaa
xbxbdxbxbxxbF )(log)()1()(log)()(log)(
i iNa x xxaaiiiai
i xiii
a xaaa
i ia
i
a
xbxbx
xb
xb
)( \
)]()()[(
1)(
1)(
Large region beliefs are normalized
Variable region beliefs are normalized
Marginalization
Genome Evolution. Amos Tanay 2009
The Bethe lagrangian
Take the derivatives with respect to each ba and bi:
))(1exp()()(
)(1)(log)(log)(
)(
)(
aNiiaiaaaaa
aNiiaiaaaaa
aia
xxxb
xxbxxb
L
i iNa x xxaaiiiai
i xiii
a xaaaBethe
i ia
ia
xbxbx
xbxbFL
)( \
)]()()[(
1)(1)(
i x
iiiiia x
aaaaa x
aaaaBethe
iaa
xbxbdxbxbxxbF )(log)()1()(log)()(log)(
)))((1
11exp()(
)()1)()(log1()(
)(
)(
aNiiaii
iii
iNaiaiiiii
ii
xd
xb
xxbdxb
L
Genome Evolution. Amos Tanay 2009
Bethe minima are LBP fixed points
))(exp()())(1exp()()()(
)(iai
aNia
aNiiaiaaaaa xxxxxb
)1
)(exp()))((
1
11exp()(
)()(
i
iai
iNaaNiiaii
iii d
xx
dxb
)(log)(log)(\)(
iicaiNc
iaiiai xmxmx
So here are the conditions:
And we can solve them if:
)()()(\)()(
iicaiNcaNi
aaa xmxxb
)()(1
1)(
)(\)()(iia
iNaiic
aiNci
iNaii xmxm
dxb
Giving us:
We saw before these conditions, with the marginalization constraint, are generating the update rules! So L minimum -> LBP fixed point is proven.The other direction quite direct – see Exercise
LBP is in fact computing the lagrange multipliers – a very powerful observation
Genome Evolution. Amos Tanay 2009
Generalizing LBP for region graphs
)()()()( '})(\{)P(')()P(
DDPRRDDPRDD
RRPRP
aaAa
RR xmxmxxbR
Parent-to-child beliefs:
A region graph is graph on subsets of nodes in the factor graph, with valid multipliers (as defined above)
RD(R) – Decedents of R
P(R)
RR Aa
RXi
R caci 11
• regions (XR,AR) and multipliers cR
• We require that:
• We will work with valid regions graphs:
RR XaNAa )(
RR
RR cc'
'1
P(D(R))\D(R)P(R) – Parents of R
D(R)
Genome Evolution. Amos Tanay 2009
Generalizing LBP for region graphs
)()()()( '})(\{)P(')()P(
DDPRRDDPRDD
RRPRP
aaAa
RR xmxmxxbR
Parent-to-child algorithm:
RP
RP
x JJIRPDJI
JJIRPNJIaFaRRP xm
xmxxm
\
\
)(
)()()(
),(),(
),(),(
I
J
D(P)+P
Not D(P)+P
D(R) – Decedents of R
P(R) – Parents of R
P
RD(R)+R
IJ
D(P)+PP
RD(R)+R
N(I,J) = I not in D(P)+P J in D(P)+P but not D(R)+R
D(I,J) = I in D(P)+P but not D(R)+R J in D(R)+R
Genome Evolution. Amos Tanay 2009
GLBP in practice
LBP is very attractive for users: really simple to implement, very fast
LBP performance is limited by the size of region assignments Xa which can grow rapidly with the factor’s degrees or the size of large regions
GLBP will be powerful when large regions can capture significant dependencies that are not captured by individual factors – think small positive loop or other symmetric effects
LBP messages can be computed synchronously (factors->variables->factors…), other scheduling options may boost up performance considerably
LBP is just one (quite indirect) way by which Bethe energies can be minimized. Other approaches are possible – which can be guaranteed to converge
The Bethe/Region energy minimization can be further constraint to force beliefs are realizable. This gives rise to the concept of Wainwright-Jordan marginal polytope and convex algorithms on it.