Belief Propagation for Spatial Network Embeddings Andrew Frank - - PowerPoint PPT Presentation

Belief Propagation for Spatial Network Embeddings

Andrew Frank Alex Ihler Padhraic Smyth

Department of Computer Science UC Irvine

August 25, 2009

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Graphical Models Markov Random Fields

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Graphical Models Markov Random Fields

What Are Graphical Models?

Concise representations of probabilistic models Several types:

Bayesian networks (DAGs) Markov random fields (undirected graphs) Factor graphs (bipartite graphs) . . . and others!

Graphical Models Markov Random Fields

What Are Graphical Models?

Concise representations of probabilistic models Several types:

Bayesian networks (DAGs) Markov random fields (undirected graphs) Factor graphs (bipartite graphs) . . . and others!

A B C D E Nodes = random variables Edges = dependencies between variables

Graphical Models Markov Random Fields

What Are Graphical Models?

Concise representations of probabilistic models Several types:

Bayesian networks (DAGs) Markov random fields (undirected graphs) Factor graphs (bipartite graphs) . . . and others!

A B C D E suspects {innocent,guilty} Nodes = random variables Edges = dependencies between variables

Graphical Models Markov Random Fields

What Are Graphical Models?

Concise representations of probabilistic models Several types:

Bayesian networks (DAGs) Markov random fields (undirected graphs) Factor graphs (bipartite graphs) . . . and others!

A B C D E suspects {innocent,guilty} friends Nodes = random variables Edges = dependencies between variables

Graphical Models Markov Random Fields

Representing Conditional Independencies

Interpreting a Markov Random Field If all paths from X to Y pass through Z, then we can say X and Y are conditionally independent given Z. Graphically, with a Markov Random Field (MRF): A B C D E Textually, through enumeration: A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C . . .

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)p(D|C)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)p(D|C)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)p(D|C)p(E|C)

Graphical Models Markov Random Fields

Factorization

Conditional independence lets us factor a distribution: A B C D E A ⊥ D, E | C B ⊥ C, D, E | A C ⊥ B | A D ⊥ A, B, E | C E ⊥ A, B, D | C p(A, B, C, D, E) = p(A)p(B|A)p(C|A, B)p(D|A, B, C)p(E|A, B, C, D) = p(A)p(B|A)p(C|A)p(D|C)p(E|C) Largest factor involves 2 variables!

Graphical Models Markov Random Fields

Hammersley-Clifford Theorem

General factorization property of all MRFs: Hammersley-Clifford Theorem Every MRF factors as the product of potential functions defined

• ver cliques of the graph.

Potential functions are. . .

Strictly positive Unnormalized

Graphical Models Markov Random Fields

Hammersley-Clifford Theorem

General factorization property of all MRFs: Hammersley-Clifford Theorem Every MRF factors as the product of potential functions defined

• ver cliques of the graph.

Potential functions are. . .

Strictly positive Unnormalized

A B C D E p(·) ∝ fA(A)fB(B)fC(C)fD(D)fE(E)fAB(A, B)fAC(A, C)fCD(C, D)fCE(C, E)

Graphical Models Markov Random Fields

Specifying a Markov Random Field Model

Define the potential functions, e.g.: Let our domain be 0=innocent, 1=guilty. fB(B) =

• .4

B = 0 .6 B = 1 fAB(A, B) =

• 2

A = B 1 A = B

Graphical Models Markov Random Fields

Specifying a Markov Random Field Model

Define the potential functions, e.g.: Let our domain be 0=innocent, 1=guilty. fB(B) =

• .4

B = 0 .6 B = 1 Suspect B is acting suspicious. fAB(A, B) =

• 2

A = B 1 A = B

Graphical Models Markov Random Fields

Specifying a Markov Random Field Model

Define the potential functions, e.g.: Let our domain be 0=innocent, 1=guilty. fB(B) =

• .4

B = 0 .6 B = 1 Suspect B is acting suspicious. fAB(A, B) =

• 2

A = B 1 A = B Suspects A and B are friends.

Graphical Models Inference

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Graphical Models Inference

Marginalization with MRFs

Query p(A): p(A) =

• B,C,D,E

p(A, B, C, D, E) O(dn)

Graphical Models Inference

Marginalization with MRFs

Query p(A): p(A) =

• B,C,D,E

p(A, B, C, D, E) O(dn) A B C D E Use graph structure to compute p(A) in O(dn2).

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E

• B,C,D,E

f(A)f(B)f(C)f(D)f(E)f(A, B)f(A, C)f(C, D)f(C, E)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E

• B,C,D,E

f(A)f(B)f(C)f(D)f(E)f(A, B)f(A, C)f(C, D)f(C, E)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E

• B,C,D

f(A)f(B)f(C)f(D)f(A, B)f(A, C)f(C, D)

• E

f(E)f(C, E)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C)

• B,C,D

f(A)f(B)f(C)f(D)f(A, B)f(A, C)f(C, D)mEC(C)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C)

• B,C,D

f(A)f(B)f(C)f(D)f(A, B)f(A, C)f(C, D)mEC(C)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C)

• B,C

f(A)f(B)f(C)f(A, B)f(A, C)mEC(C)

• D

f(D)f(C, D)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C)

• B,C

f(A)f(B)f(C)f(A, B)f(A, C)mEC(C)mDC(C)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C)

• B,C

f(A)f(B)f(C)f(A, B)f(A, C)mEC(C)mDC(C)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C)

• B

f(A)f(B)f(A, B)

• C

f(C)f(A, C)mEC(C)mDC(C)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A)

• B

f(A)f(B)f(A, B)mCA(A)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A)

• B

f(A)f(B)f(A, B)mCA(A)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A) f(A)mCA(A)

• B

f(B)f(A, B)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A) mBA(A) f(A)mCA(A)mBA(A)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A) mBA(A) f(A)mCA(A)mBA(A)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

View marginalization as a “message-passing” algorithm

Variables are computational nodes. Intermediate results are “messages” between nodes.

A B C D E mEC(C) mDC(C) mCA(A) mBA(A) ∝ p(A)

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

Message update equation for pairwise MRFs: mst(xt) =

• xs

 f(xs)f(xs, xt)

• xu∈N(xs)\xt

mus(xs)  

Exact for tree-structured graphs.

Graphical Models Inference

Belief Propagation (Sum-Product Algorithm)

Message update equation for pairwise MRFs: mst(xt) =

• xs

 f(xs)f(xs, xt)

• xu∈N(xs)\xt

mus(xs)  

Exact for tree-structured graphs.

What about on graphs with loops?

Use the same equation! (“Loopy” BP) No longer exact May not converge Often does quite well

Graphical Models Inference

Related Algorithms

−1 1 2 0.5 1 1.5 Exact −1 1 2 0.5 1 1.5 BP, Easy −1 1 2 1 2 3 Mean Field, Easy −1 1 2 0.5 1 1.5 TRW−BP, Easy −1 1 2 0.5 1 1.5 Exact −1 1 2 1 2 3 BP, Hard −1 1 2 1 2 3 Mean Field, Hard −1 1 2 0.5 1 1.5 TRW−BP, Hard

Self-Localization Problem Description

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {} 2 {} 3 {} 4 {} 5 {} Nodes that are “close enough” can estimate distance between them.

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {(2,3)} 2 {(1,3)} 3 {} 4 {} 5 {} Nodes that are “close enough” can estimate distance between them.

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {(2,3)(3,2)} 2 {(1,3)} 3 {(1,2)} 4 {} 5 {} Nodes that are “close enough” can estimate distance between them.

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {(2,3)(3,2)} 2 {(1,3)} 3 {(1,2)} 4 {(5,1)} 5 {(4,1)} Nodes that are “close enough” can estimate distance between them.

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {(2,3)(3,2)(4,4) (5,3)} 2 {(1,3)(3,1) (5,3)} 3 {(1,2)(2,1)} 4 {(5,1)(1,4)} 5 {(4,1)(1,3) (2,3)} Nodes that are “close enough” can estimate distance between them.

Self-Localization Problem Description

Localization Scenario

Nodes distributed throughout a planar region.

(people, mobile sensors, . . . )

1 2 3 4 5 Local measurements: node {(neighbor, distance)} 1 {(2,3)(3,2)(4,4) (5,3)} 2 {(1,3)(3,1) (5,3)} 3 {(1,2)(2,1)} 4 {(5,1)(1,4)} 5 {(4,1)(1,3) (2,3)} Nodes that are “close enough” can estimate distance between them. Task: recover node locations.

Self-Localization Model Formulation

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Self-Localization Model Formulation

Local Detection Model

Variables:

xs, location in R2 of node s

• st, indicates whether nodes s and t detect each other

dst, noisy observation of ||xs − xt||

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 ||xs − xt|| p(ost| ||xs−xt||) Detection Noise 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 dst p(dst| ||xs−xt|| = 2) Distance Sensor Noise

Self-Localization Model Formulation

Joint Model

p(x, o, d) =

• (s,t)

p(ost | xs, xt)

• (s,t):ost=1

p(dst | xs, xt)

• s

p(xs) x1 x2 x3 x4 x5

• st = 1
• st = 0

fst(xs, xt) =

• p(ost = 1|xs, xt)p(dst|xs, xt)

if ost = 1 1 − p(ost = 1|xs, xt) if ost = 0

Self-Localization Model Formulation

Handling Continuous Variables

Variable domains are continuous! (locations in R2) = ⇒ replace sums with integrals? mst(xt) =

• xs

 f(xs)f(xs, xt)

• xu∈N(xs)\xt

mus(xs)  

Self-Localization Model Formulation

Handling Continuous Variables

Variable domains are continuous! (locations in R2) = ⇒ replace sums with integrals? mst(xt) =

• xs

 f(xs)f(xs, xt)

• xu∈N(xs)\xt

mus(xs)   Theory holds, but now we must compute the integrals.

Self-Localization Model Formulation

Particle Belief Propagation (PBP)

Draw weighted particles from each variable’s domain. Run (importance-corrected) discrete BP over these particles. mst(xt) =

• xs

 f(xs, xt)f(xs)

• xu∈N(xs)\xt

mus(xs)  

Self-Localization Model Formulation

Particle Belief Propagation (PBP)

Draw weighted particles from each variable’s domain. Run (importance-corrected) discrete BP over these particles. mst(xt) =

• xs

 f(xs, xt)f(xs)

• xu∈N(xs)\xt

mus(xs)   = E

xs∼W(xs)

 f(xs, xt) f(xs) W(xs)

• xu∈N(xs)\xt

mus(xs)  

Self-Localization Model Formulation

Particle Belief Propagation (PBP)

Draw weighted particles from each variable’s domain. Run (importance-corrected) discrete BP over these particles. mst(xt) =

• xs

 f(xs, xt)f(xs)

• xu∈N(xs)\xt

mus(xs)   = E

xs∼W(xs)

 f(xs, xt) f(xs) W(xs)

• xu∈N(xs)\xt

mus(xs)   ˆ m(i)

st ≈ 1

n

n

• k=1

 f

• x(k)

s

, x(i)

t

f

• x(k)

s

• W
• x(k)

s

• xu∈N(xs)\xt

mus

• x(k)

s

• 



Self-Localization Experimental Results

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Self-Localization Experimental Results

Results

Synthetic example:

Anchor Mobile Target

“Exact”

Anchor Mobile Target

Particle BP

Self-Localization Experimental Results

Results

Synthetic example:

Anchor Mobile Target

“Exact”

Anchor Mobile Target Anchor Mobile Target

Particle BP

Self-Localization Experimental Results

Results

Synthetic example:

Anchor Mobile Target

“Exact”

Anchor Mobile Target Anchor Mobile Target

Particle BP

Anchor Mobile Target

TRW Particle BP

Latent Space Embeddings of Social Networks Problem Description

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Latent Space Embeddings of Social Networks Problem Description

Main Idea

Intuition

Actors live in a latent, d−dimensional “social space”. Proximity in social space increases the likelihood of a link.

Hoff, Raftery, Handcock. Latent space approaches to social network analysis. JASA 2002.

Latent Space Embeddings of Social Networks Problem Description

Connection to Localization

Localization

Geographic space Detection ⇒ physical proximity Location is the end goal

Latent space embedding

Social space Network link ⇒ proximity in latent space Latent location indirectly useful

Latent Space Embeddings of Social Networks Problem Description

Connection to Localization

Localization

Geographic space Detection ⇒ physical proximity Location is the end goal

Latent space embedding

Social space Network link ⇒ proximity in latent space Latent location indirectly useful

Latent Space Embeddings of Social Networks Problem Description

Connection to Localization

Localization

Geographic space Detection ⇒ physical proximity Location is the end goal

Latent space embedding

Social space Network link ⇒ proximity in latent space Latent location indirectly useful

Latent Space Embeddings of Social Networks Model Formulation

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Latent Space Embeddings of Social Networks Model Formulation

Local Model

Variables:

zs, location in latent space of node s yst, social network link indicator

p(yst | zs, zt) = σ(α − ||zs − zt||)

0.2 0.4 0.6 0.8 1 Link Probability ||zs − zt|| p(yst | ||zs − zt||)

Latent Space Embeddings of Social Networks Model Formulation

Joint Model

Social Network: 1 2 3 4 5 MRF model: z1 z2 z3 z4 z5 yst = 1 yst = 0 fst(zs, zt) = p(yst|zs, zt)

Latent Space Embeddings of Social Networks Preliminary Results

Outline

1

Graphical Models Markov Random Fields Inference

2

Self-Localization Problem Description Model Formulation Experimental Results

3

Latent Space Embeddings of Social Networks Problem Description Model Formulation Preliminary Results

Latent Space Embeddings of Social Networks Preliminary Results

Test Data Set

Sampson’s monk data

18 monks living in a monestary Links indicate a “liking” relation Well studied data set

MLE (Hoff, Raftery, Handcock ’02)

Latent Space Embeddings of Social Networks Preliminary Results

PBP Embedding of Monk Data

−10 −5 5 −6 −4 −2 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Monk Embedding, Marginal Modes

Latent Space Embeddings of Social Networks Preliminary Results

PBP Embedding of Monk Data

−10 −5 5 −6 −4 −2 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Monk Embedding, Marginal Modes −10 −5 5 −6 −4 −2 2 4 Monk 16 Marginal

Conclusions

Conclusions

BP is a generic inference method for computing marginals. PBP can estimate marginals in the self-localization problem. Could BP be useful for latent space network modeling?

Conclusions

Conclusions

BP is a generic inference method for computing marginals. PBP can estimate marginals in the self-localization problem. Could BP be useful for latent space network modeling?

Thank you!