SLIDE 1
Probabilistic & Unsupervised Learning Belief Propagation
Maneesh Sahani
maneesh@gatsby.ucl.ac.uk
Gatsby Computational Neuroscience Unit, and MSc ML/CSML, Dept Computer Science University College London Term 1, Autumn 2014
SLIDE 2 Recall: Belief Propagation on undirected trees
Joint distribution of undirected tree: p(X) = 1 Z
fi(Xi)
fij(Xi, Xj) Xj Xi Messages computed recursively: Mj→i(Xi) :=
fij(Xi, Xj)fj(Xj)
Ml→j(Xj) Marginal distributions: p(Xi) ∝ fi(Xi)
Mk→i(Xi) p(Xi, Xj) ∝ fij(Xi, Xj)fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
SLIDE 3 Loopy Belief Propagation
Joint distribution of undirected graph: p(X) = 1 Z
fi(Xi)
fij(Xi, Xj) Xj Xi Messages computed recursively (with few guarantees of convergence): Mj→i(Xi) :=
fij(Xi, Xj)fj(Xj)
Ml→j(Xj) Marginal distributions are approximate in general: p(Xi) ≈ bi(Xi) ∝ fi(Xi)
Mk→i(Xi) p(Xi, Xj) ≈ bij(Xi, Xj) ∝ fij(Xi, Xj)fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
SLIDE 4 Dealing with loops
◮ Accuracy: BP posterior marginals are approximate on all non-trees, but converged
approximations are frequently found to be good.
◮ Convergence: no general guarantee, but BP does converge in some cases:
◮ Trees. ◮ Graphs with a single loop. ◮ Distributions with sufficiently weak interactions. ◮ Graphs with long (and weak) loops ◮ Gaussian networks: means correct, variances may also converge.
◮ Damping: Common approach to encourage convergence (cf EP)
M new
i→j (Xj) := (1 − α)M old i→j(Xj) + α
fij(Xi, Xj)fi(Xi)
Mk→i(Xi)
◮ Grouping variables: Variables can be grouped into cliques to improve accuracy.
◮ Region graph approximations. ◮ Cluster variational method. ◮ Junction graph.
SLIDE 5
Different Interpretations of Loopy Belief Propagation
Loopy BP can be interpreted as a fixed point algorithm from a few different perspectives:
◮ Expectation propagation. ◮ Tree-based reparametrization. ◮ Bethe free energy.
SLIDE 6
Different Interpretations of Loopy Belief Propagation
Loopy BP can be interpreted as a fixed point algorithm from a few different perspectives:
◮ Expectation propagation. ◮ Tree-based reparametrization. ◮ Bethe free energy.
SLIDE 7 Loopy BP as message-based Expectation Propagation
⇒
Approximate pairwise factors fij by product of messages: fij(Xi, Xj) ≈ ˜ fij(Xi, Xj) = Mi→j(Xj)Mj→i(Xi) Thus, the full joint is approximated by a factorised distribution: p(X) ≈ 1 Z
fi(Xi)
˜
fij(Xi, Xj) = 1 Z
Mj→i(Xi)
bi(Xi) but with multiple factors for most Xi.
SLIDE 8
Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
SLIDE 9 Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
◮ Deletion:
q¬ij(X)
= fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
fs(Xs)
Mt→s(Xs)
SLIDE 10 Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
◮ Deletion:
q¬ij(Xi, Xj) = fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
fs(Xs)
Mt→s(Xs)
SLIDE 11 Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
◮ Deletion:
q¬ij(Xi, Xj) = fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
fs(Xs)
Mt→s(Xs)
◮ Projection:
{Mnew
i→j, Mnew j→i} = argmin KL[fij(Xi, Xj)q¬ij(Xi, Xj)Mj→i(Xi)Mi→j(Xj)q¬ij(Xi, Xj)]
SLIDE 12 Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
◮ Deletion:
q¬ij(Xi, Xj) = fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
fs(Xs)
Mt→s(Xs)
◮ Projection:
{Mnew
i→j, Mnew j→i} = argmin KL[fij(Xi, Xj)q¬ij(Xi, Xj)Mj→i(Xi)Mi→j(Xj)q¬ij(Xi, Xj)]
Now, q¬ij() factors ⇒ rhs factors ⇒ min is achieved by marginals of fij()q¬ij()
SLIDE 13 Loopy BP as message-based EP
Xj Xi Then the EP updates to the messages are:
◮ Deletion:
q¬ij(Xi, Xj) = fi(Xi)fj(Xj)
Mk→i(Xi)
Ml→j(Xj)
fs(Xs)
Mt→s(Xs)
◮ Projection:
{Mnew
i→j, Mnew j→i} = argmin KL[fij(Xi, Xj)q¬ij(Xi, Xj)Mj→i(Xi)Mi→j(Xj)q¬ij(Xi, Xj)]
Now, q¬ij() factors ⇒ rhs factors ⇒ min is achieved by marginals of fij()q¬ij() Mnew
j→i(Xi)q¬ij(Xi) =
- Xj
- fij(Xi, Xj)fj(Xj)
- l∈ne(j)\i
Ml→j(Xj)
Mk→i(Xi)
⇒ Mnew
j→i(Xi) =
- Xj
- fij(Xi, Xj)fj(Xj)
- l∈ne(j)\i
Ml→j(Xj)
SLIDE 14
Different Interpretations of Loopy Belief Propagation
Loopy BP can be interpreted as a fixed point algorithm from a few different perspectives:
◮ Expectation propagation. ◮ Tree-based reparametrization. ◮ Bethe free energy.
SLIDE 15 Loopy BP as tree-based reparametrisation
Tree-structured distributions can be parametrised in many ways: p(X) = 1 Z
fi(Xi)
fij(Xi, Xj) undirected tree (1)
= p(Xr)
p(Xi|Xpa(i)) directed (rooted) tree (2)
=
p(Xi)
p(Xi, Xj) p(Xi)p(Xj) pairwise marginals (3) where (3) requires that
Xj p(Xi, Xj) = p(Xi).
The undirected tree representation is not unique—multiplying a factor fij(Xi, Xj) by g(Xi) and dividing fi(Xi) by the same g(Xi) does not change the distribution. BP can be seen as an iterative replacement of fi(Xi) by the local marginal of pij(Xi, Xj), along with the corresponding reparametrisation of fij(Xi, Xj). Cf. Hugin propagation. Converged BP on a tree finds p(Xi) and p(Xi, Xj), allowing us to transform (1) to (3).
SLIDE 16 Reparametrisation on trees
Xd Xe Xf Xg Xa Xb Xc p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
SLIDE 17 Reparametrisation on trees
Xd Xe Xf Xg Xa Xb Xc fde fdf fdg
1·fab·1
fac fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
SLIDE 18 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→a
Xb Xc fde fdf fdg
1·fab·1
fac fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
SLIDE 19 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→a
Xb Xc fde fdf fdg
fab Mb→a
fac fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 20 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→a
Xb Xc fde fdf fdg
fab Mb→a 1·fac·Mb→a
fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 21 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→aMc→a
Xb Xc fde fdf fdg
fab Mb→a 1·fac·Mb→a
fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 22 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→aMc→a
Xb Xc fde fdf fdg
fab Mb→a fac Mc→a
fad p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 23 Reparametrisation on trees
Xd Xe Xf Xg Xa
Mb→aMc→a
Xb Xc fde fdf fdg
fab Mb→a fac Mc→a Mb→aMc→afad ·1
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 24 Reparametrisation on trees
Xd
Ma→d
Xe Xf Xg Xa
Mb→aMc→a
Xb Xc fde fdf fdg
fab Mb→a fac Mc→a Mb→aMc→afad ·1
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 25 Reparametrisation on trees
Xd
Ma→d
Xe Xf Xg Xa
Mb→aMc→a
Xb Xc fde fdf fdg
fab Mb→a fac Mc→a fad Ma→d
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 26 Reparametrisation on trees
Xd
Ma→d
Xe Xf Xg Xa
Mb→aMc→a
Xb Xc
fab Mb→a fac Mc→a fad Ma→d
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 27 Reparametrisation on trees
Xd
M→d
Xe Xf Xg Xa
Mb→aMc→a
Xb Xc
fab Mb→a fac Mc→a fad Ma→d
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 28 Reparametrisation on trees
Xd
M→d
Xe Xf Xg Xa
Mb→aMc→a
Xb Xc
fab Mb→a fac Mc→a Mb→aMc→a
fad Ma→d Ma→d
M→d
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 29 Reparametrisation on trees
Xd
M→d
Xe Xf Xg Xa
M→a
Xb Xc
fab Mb→a fac Mc→a Mb→aMc→a
fad Ma→d Ma→d
M→d
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 30 Reparametrisation on trees
Xd
M→d
Xe Xf Xg Xa
M→a
Xb Xc
fab Mb→a fac Mc→a fad Ma→d Md→a
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 31 Reparametrisation on trees
Xd
M→d
Xe Xf Xg Xa
M→a
Xb Xc
fab Mb→a fac Mc→a fad Ma→d Md→a
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi)
SLIDE 32 Reparametrisation on trees
Xd
pd
Xe
pe
Xf
pf
Xg
pg
Xa
pa
Xb
pb
Xc
pc pde pd pe pdf pd pf pdg pd pg pab papb pac papc pad papd
p(X) =
fij(Xi, Xj)
⇓
p(X) =
p(Xi)
p(Xi, Xj) p(Xi)p(Xk) Define f 0
ij = fij, f 0 i = p0 i = 1. Iterate over edges (ij):
pn(Xi, Xj) = f n−1
i
(Xi)f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
i (Xi) = pn(Xi) =
pn(Xi, Xj) = f n−1
i
(Xi)
f n−1
ij
(Xi, Xj)f n−1
j
(Xj)
f n
ij (Xi, Xj) =
f n−1
ij
(Xi, Xj)
Mj→i(Xi) After all messages have propagated: f ∞
i
(Xi) =
Mj→i(Xi) = p(Xi) f ∞
ij (Xi, Xj) =
fij(Xi, Xj) Mj→i(Xi)Mi→j(Xj) =
Mk→i(Xi)fij(Xi,Xj)
l∈ne(j)\i
Ml→j(Xj)
Mk→i(Xi)Mj→i(Xi)Mi→j(Xj)
l∈ne(j)\i
Ml→j(Xj) =
p(Xi, Xj) p(Xi)p(Xj)
SLIDE 33 Reparametrisation on non-trees
◮ If BP converges on a non-tree, it will have successfully reparametrised the distribution to
have locally consistent beliefs: p(X) =
b(Xi)
b(Xi, Xj) b(Xi)b(Xj) with
b(Xi, Xj) = b(Xi) etc.
◮ However, the marginals will not usually be correct or globally consistent. That is
i
b(Xi)
b(Xi, Xj) b(Xi)b(Xj)
◮ What can be said about these pseudomarginals? ◮ Consider the following (theoretical) message scheduling scheme:
◮ Identify all the spanning trees of the graph. ◮ Pass messages along edges of each spanning tree in turn. ◮ Iterate over spanning trees to convergence
SLIDE 34 Loopy BP as tree-based reparametrisation
graph spanning tree 1 spanning tree 2
p(X) = 1 Z
f 0
i (Xi)
f 0
ij (Xi, Xj)
= 1
Z
f 0
i (Xi)
f 0
ij (Xi, Xj)
f 0
ij (Xi, Xj)
= 1
Z
f 1
i (Xi)
f 1
ij (Xi, Xj)
f 1
ij (Xi, Xj)
where f 1
i (Xi) = pT1(Xi), f 1 ij (Xi, Xj) = pT1 (Xi,Xj) pT1 (Xi)pT1 (Xj), f 1 ij = f 0 ij .
= 1
Z
f 1
i (Xi)
f 1
ij (Xi, Xj)
f 1
ij (Xi, Xj)
. . .
SLIDE 35 Loopy BP as tree-based reparametrisation
At convergence, loopy BP has reparametrised the joint distribution as: p(X) = 1 Z
f ∞
i
(Xi)
f ∞
ij (Xi, Xj)
where for any tree T embedded in the graph, f ∞
i
(Xi) = pT(Xi)
f ∞
ij (Xi, Xj) =
pT(Xi, Xj) pT(Xi)pT(Xj) Thus, the local marginals of all subtrees are locally consistent with each other, and the pseudomarginals represent valid beliefs for any of the subtrees. p(X) = 1 Z
bi(Xi)
bij(Xi, Xj) bi(Xi)bj(Xj)
SLIDE 36
Different Interpretations of Loopy Belief Propagation
Loopy BP can be interpreted as a fixed point algorithm from a few different perspectives:
◮ Expectation propagation. ◮ Tree-based reparametrization. ◮ Bethe free energy.
SLIDE 37 Loopy BP and Bethe free energy
In the reparametrisation view, BP solves for marginal beliefs bij(Xi, Xj) and bi(Xi) =
Xj bij(Xi, Xj) such that
p(X) = 1 Z
fi(Xi)
fij(Xi, Xj) =
bi(Xi)
bij(Xi, Xj) bi(Xi)bj(Xj)
SLIDE 38 Loopy BP and Bethe free energy
In the reparametrisation view, BP solves for marginal beliefs bij(Xi, Xj) and bi(Xi) =
Xj bij(Xi, Xj) such that
p(X) = 1 Z
fi(Xi)
fij(Xi, Xj) =
bi(Xi)
bij(Xi, Xj) bi(Xi)bj(Xj) Another view of loopy BP is as a set of fixed point equations for finding stationary points of an
- bjective function called the Bethe free energy, which is defined in terms of the locally
consistent beliefs (or pseudomarginals) bi ≥ 0 and bij ≥ 0:
bi(xi) = 1
∀i
bij(xi, xj) = bi(xi)
∀i, j ∈ ne(i), xi
SLIDE 39
Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q]
SLIDE 40
Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q] We define the (negative) Bethe free energy: Fbethe(b) = Ebethe(b) + Hbethe(b)
SLIDE 41 Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q] We define the (negative) Bethe free energy: Fbethe(b) = Ebethe(b) + Hbethe(b)
◮ The Bethe average energy is the expected log-joint evaluated as though the
pseudomarginals were correct:
Ebethe(b) =
bi(xi) log fi(xi) +
bij(xi, xj) log fij(xi, xj)
SLIDE 42 Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q] We define the (negative) Bethe free energy: Fbethe(b) = Ebethe(b) + Hbethe(b)
◮ The Bethe average energy is the expected log-joint evaluated as though the
pseudomarginals were correct:
Ebethe(b) =
bi(xi) log fi(xi) +
bij(xi, xj) log fij(xi, xj)
◮ The Bethe entropy is approximate: it is the sum of the pseudomarginal entropies
corrected for pairwise (pseudo)interactions, but neglecting higher
Hbethe(b) =
H[bi] −
KL[bijbibj]
= −
bi(xi) log bi(xi) −
bij(xi, xj) log bij(xi, xj) bi(xi)bj(xj)
SLIDE 43 Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q] We define the (negative) Bethe free energy: Fbethe(b) = Ebethe(b) + Hbethe(b)
◮ The Bethe average energy is the expected log-joint evaluated as though the
pseudomarginals were correct:
Ebethe(b) =
bi(xi) log fi(xi) +
bij(xi, xj) log fij(xi, xj)
◮ The Bethe entropy is approximate: it is the sum of the pseudomarginal entropies
corrected for pairwise (pseudo)interactions, but neglecting higher
Hbethe(b) =
H[bi] −
KL[bijbibj]
= −
bi(xi) log bi(xi) −
bij(xi, xj) log bij(xi, xj) bi(xi)bj(xj)
◮ On a tree, both the beliefs and the Bethe entropy expression are correct, so Fbethe = F.
SLIDE 44 Loopy BP and Bethe free energy
Recall that the variational free energy is: F(q) = log P(X)q + H[q] We define the (negative) Bethe free energy: Fbethe(b) = Ebethe(b) + Hbethe(b)
◮ The Bethe average energy is the expected log-joint evaluated as though the
pseudomarginals were correct:
Ebethe(b) =
bi(xi) log fi(xi) +
bij(xi, xj) log fij(xi, xj)
◮ The Bethe entropy is approximate: it is the sum of the pseudomarginal entropies
corrected for pairwise (pseudo)interactions, but neglecting higher
Hbethe(b) =
H[bi] −
KL[bijbibj]
= −
bi(xi) log bi(xi) −
bij(xi, xj) log bij(xi, xj) bi(xi)bj(xj)
◮ On a tree, both the beliefs and the Bethe entropy expression are correct, so Fbethe = F. ◮ Message updates in loopy BP can now be derived by finding the stationary points of a
Lagrangian with local consistency and normalisation constraints. The BP messages are related to the Lagrange multipliers.
SLIDE 45 Bethe fixed point equations
The Bethe free-energy Lagrangian is:
L =
bi(xi) log fi(xi) +
bij(xi, xj) log fij(xi, xj)
[Ebethe] −
bi(xi) log bi(xi) −
bij(xi, xj) log bij(xi, xj) bi(xi)bj(xj)
[Hbethe] +
ξi
xi
bi(xi) − 1
+
xi
ξij(xi)
xj
bij(xi, xj) − bi(xi)
ξji(xj)
xi
bij(xi, xj) − bj(xj)
Setting derivatives wrt beliefs to 0 gives
∂L ∂bi(xi) = log fi(xi) − log bi(xi) +
bij(xi, xj) bi(xi)
+ξi −
ξij(xi) + const = 0 ⇒bi(xi) ∝ fi(xi)
e−ξij(xi)
∂L ∂bij(xi, xj) = log fij(xi, xj) − log bij(xi, xj) + log bi(xi)bj(xj) + ξij(xi) + ξji(xj) + const = 0 ⇒bij(xi, xj) ∝ fij(xi, xj)bi(xi)bj(xj)eξij(xi)eξji(xj)
SLIDE 46 Bethe fixed point messages
The Bethe Lagrangian fixed point equations are: bi(xi) ∝ fi(xi)
e−ξij(xi) bij(xi, xj) ∝ fij(xi, xj)bi(xi)bj(xj)eξij(xi)eξji(xj) Comparison with BP suggests that messages should have the form Mj→i(xi) = e−ξij(xi). Indeed, solving for ξij(xi) by enforcing the constraint
xj bij(xi, xj) = bi(xi) we have:
bij(xi, xj) ∝
fij(xi, xj)bi(xi)bj(xj)eξij(xi)eξji(xj)
⇒ bi(xi) ∝ bi(xi)eξij(xi)
xj
fij(xi, xj)bj(xj)eξji(xj)
⇒ e−ξij(xi) ∝
fij(xi, xj)bj(xj)eξji(xj)
=
fij(xi, xj)fj(xj)
e−ξjl(xj) thus recovering the BP message passing rules.
SLIDE 47
Loopy BP and Bethe free energy
◮ Fixed points of loopy BP are exactly the stationary points of the Bethe free energy. ◮ Stable fixed points of loopy BP are local maxima of Bethe free enegy (note the negative
definition of free energy for consistency with the variational free energy).
◮ For binary attractive networks, Bethe free energy at fixed points of loopy BP provides an
upper bound on the log partition function log Z—this is useful for learning undirected graphical models as it leads to a lower bound on the log likelihood.
SLIDE 48
Loopy BP vs variational approximation
◮ Beliefs bi and bij in loopy BP are only locally consistent pseudomarginals, not
necessarily consistent marginals of the implied joint distribution.
◮ Bethe free energy accounts for interactions between different sites, while variational free
energy assumes independence.
◮ The loop series or Plefka expansion of the log partition function Z: the variational free
energy forms the first order terms, while Bethe free energy contains higher order terms (involving generalized loops).
◮ Loopy BP tends to be signficantly more accurate whenever it converges.
SLIDE 49 Extensions and variations
◮ Generalized BP: group variables together to treat their
interactions exactly.
◮ Convergent alternatives: Fixed points of loopy BP are stationary
points of the Bethe free enegy. We can also derive algorithms that increase the Bethe free energy at every step, and are thus are guaranteed to converge.
◮ Convex alternatives: We can derive convex cousins of the negative of the Bethe free
- energy. These give rise to algorithms that will converge to a unique global maximum.
◮ We have considered sum-product loopy BP to compute marginals. The treatment of
loopy Viterbi or max-product algorithms is different.
SLIDE 50 References
◮ Probabilistic Reasoning in Intelligent Systems. J. Pearl. Morgan Kaufman, 1988. ◮ Turbo decoding as an instance of Pearl’s belief propagation algorithm. R. J. McEliece, D.
. Cheng. IEEE Journal on Selected Areas in Communication, 1998, 16(2):140-152.
◮ Iterative decoding of compound codes by probability propagation in graphical models. F
. Kschischang and B. Frey. IEEE Journal on Selected Areas in Communication, 1998, 16(2):219-230.
◮ A family of algorithms for approximate Bayesian inference. T. Minka. PhD Thesis, 2001. ◮ Tree-based reparameterization framework for analysis of sum-product and related
- algorithms. M. J. Wainwright, T. S. Jaakkola and A. S. Willsky. IEEE Transactions on
Information Theory, 2004, 49(5).
◮ Constructing free energy approximations and generalized belief propagation algorithms.
- J. S. Yedidia, W. T. Freeman and Y. Weiss. IEEE Transactions on Information Theory,
2005, 51:2282-2313.
