[PPT] - Graphical Models Henrik I. Christensen Robotics & Intelligent PowerPoint Presentation

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Introduction Bayes Nets Independence MRF Example P

Graphical Models

Henrik I. Christensen

Robotics & Intelligent Machines @ GT Georgia Institute of Technology, Atlanta, GA 30332-0280 hic@cc.gatech.edu

Henrik I. Christensen (RIM@GT) Graphical Models 1 / 55

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

Henrik I. Christensen (RIM@GT) Graphical Models 2 / 55

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Introduction Bayes Nets Independence MRF Example P

Introduction

Basically we can describe Bayesian inference through repeated use of the sum and product rules Using a graphical / diagrammatical representation is often useful

1

A way to visualize structure and consider relations

2

Provides insights into a model and possible independence

3

Allow us to leverage of the many graphical algorithms available

Will consider both directed and undirected models This is a very rich areas with numerous good references

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

Henrik I. Christensen (RIM@GT) Graphical Models 4 / 55

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Introduction Bayes Nets Independence MRF Example P

Bayesian Networks

Consider the joint probability p(a, b, c) Using product rule we can rewrite it as p(a, b, c) = p(c|a, b)p(a, b) Which again can be changed to p(a, b, c) = p(c|a, b)p(b|a)p(a) We can illustrate this as a b c

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Introduction Bayes Nets Independence MRF Example P

Bayesian Networks

Nodes represent variables Arcs/links represent conditional dependence We can use the decomposition for any joint distribution. The direct / brute-force application generates fully connected graphs We can represent much more general relations

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Introduction Bayes Nets Independence MRF Example P

Example with ”sparse” connections

We can represent relations such as p(x1)p(x2)p(x3)p(x4|x1, x2, x3)p(x5|x1, x3)p(x6|x4)p(x7|x4, x5) Which is shown below x1 x2 x3 x4 x5 x6 x7

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Introduction Bayes Nets Independence MRF Example P

The general case

We can think of this as coding the factors p(xk|pak) where pak is the set of parents to a variable xk The inference is then p(x) =

k

p(xk|pak) we will refer to this as factorization There can be no directed cycles in the graph The general form is termed a directed acyclic graph - DAG

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Introduction Bayes Nets Independence MRF Example P

Basic example

We have seen the polynomial regression before p(t, w) = p(w)

n

p(tn|w) Which can be visualized as

w t1 tN

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Introduction Bayes Nets Independence MRF Example P

Bayesian Regression

We can make the parameters and variables explicit p(t, w|x, α, σ2) = p(w|α)

n

p(tn|w, xn, σ2) as shown here

tn xn N w α σ2

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Introduction Bayes Nets Independence MRF Example P

Bayesian Regression - Learning

When entering data we can condition inference on it p(w|t) ∝ p(w)

n

p(tn|w)

tn xn N w α σ2

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Introduction Bayes Nets Independence MRF Example P

Generative Models - Example Image Synthesis

Image Object Orientation Position

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Introduction Bayes Nets Independence MRF Example P

Discrete Variables - 1

General joint distribution has K 2 − 1 parameters (for K possible

utcomes)

x1 x2 p(x1, x2|µ) =

K

i=1

K

j=1

µx1ix2j

ij

Independent joint distributions have 2(K − 1) parameters x1 x2 p(x1, x2|µ) =

K

i=1

µx1i

1i K

j=1

µx2j

2j

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Introduction Bayes Nets Independence MRF Example P

Discrete Variables - 2

General joint distribution over M variables will have K M − 1 parameters A Markov chain with M nodes will have K − 1 + (M − 1)K(K − 1) parameters x1 x2 xM

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Introduction Bayes Nets Independence MRF Example P

Discrete Variables - Bayesian Parms

x1 x2 xM µ1 µ2 µM

The parameters can be modelled explicitly p({xm, µm}) = p(x1|µ1)p(µ1)

M

m=2

p(xm|xm−1, µm)p(µm) It is assumed that p(µm) is a Dirachlet

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Introduction Bayes Nets Independence MRF Example P

Discrete Variables - Bayesian Parms (2)

x1 x2 xM µ1 µ

For shared paraemeters the situation is simpler p({xm}, µ1, µ) = p(x1|µ1)p(µ1)

M

m=2

p(xm|xm−1, µ)p(µ)

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Introduction Bayes Nets Independence MRF Example P

Extension to Linear Gaussian Models

The model can be extended to have each node as a Gaussian process/variable that is a linear function of its parents p(xi|pai) = N  xi

j∈pai

wijxj + bi, vi  

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence

Considerations of independence is important as part of the analysis and setup of a system As an example a is independent of b given c p(a|b, c) = p(a|c) Or equivalently p(a, b|c) = p(a|b, c)p(b|c) = p(a|c)p(b|c) Frequent notation in statistics a ⊥ ⊥ b|c

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 1

c a b

p(a, b, c) = p(a|c)p(b|c)p(c) p(a, b) =

c

p(a|c)p(b|c)p(c) a / ⊥ ⊥b | ∅

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 1

c a b

p(a, b|c) = p(a, b, c) p(c) = p(a|c)p(b|c) a ⊥ ⊥ b | c

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 2

a c b

p(a, b, c) = p(a)p(c|a)p(b|c) p(a, b) = p(a)

c

p(c|a)p(b|c) = p(a)p(b|a) a / ⊥ ⊥b|∅

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 2

a c b

p(a, b|c) = p(a, b, c) p(c) = p(a)p(c|a)p(b|c) p(c) = p(a|c)p(b|c) a⊥ ⊥b|c

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 3

c a b

p(a, b, c) = p(a)p(b)p(c|a, b) p(a, b) = p(a)p(b) a ⊥ ⊥ b | ∅ This is the opposite of Case 1 - when c unobserved

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Introduction Bayes Nets Independence MRF Example P

Conditional Independence - Case 3

c a b

p(a, b|c) = p(a, b, c) p(c) = p(a)p(b)p(c|a, b) p(c) a / ⊥ ⊥b | c This is the opposite of Case 1 - when c observed

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Introduction Bayes Nets Independence MRF Example P

Diagnostics - Out of fuel?

G B F

B = Battery F = Fuel Tank G = Fuel Gauge p(G = 1|B = 1, F = 1) = 0.8 p(G = 1|B = 1, F = 0) = 0.2 p(G = 1|B = 0, F = 1) = 0.2 p(G = 1|B = 0, F = 0) = 0.1 p(B = 1) = 0.9 p(F = 1) = 0.9 ⇒ p(F = 0) = 0.1

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Introduction Bayes Nets Independence MRF Example P

Diagnostics - Out of fuel?

G B F

p(F = 0|G = 0) = p(G = 0|F = 0)p(F = 0) p(G = 0) ≈ 0.257 Observing G=0 increased the probability of an empty tank

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Introduction Bayes Nets Independence MRF Example P

Diagnostics - Out of fuel?

G B F

p(F = 0|G = 0, B = 0) = p(G = 0|B = 0, F = 0)p(F = 0) p(G = 0|B = 0, F)p(F) ≈ 0.111 Observing B=0 implies less likely empty tank

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Introduction Bayes Nets Independence MRF Example P

D-separation

Consider non-intersecting subsets A, B, and C in a directed graph A path between subsets A and B is considered blocked if it contains a node such that:

1

the arcs are head-to-tail or tail-to-tail and in the set C

2

the arcs meet head-to-head and neither the node or its descendents are in the set C.

If all paths between A and B are blocked then A is d-separated from B by C. If there is d-separation then all the variables in the graph satisfies A ⊥ ⊥ B|C

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Introduction Bayes Nets Independence MRF Example P

D-Separation - Discussion

f e b a c f e b a c

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

Henrik I. Christensen (RIM@GT) Graphical Models 31 / 55

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Introduction Bayes Nets Independence MRF Example P

Markov Random Fields

The causal dependency is not always clear. Markov Random Fields (MRF) considers undirected graphs Dependency is here basically consideration of how “cliques” of nodes connect to other “cliques” Consider connectivity and trying to identify possible “connected” groups of nodes

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Introduction Bayes Nets Independence MRF Example P

Undirected graph example

A C B

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Introduction Bayes Nets Independence MRF Example P

Cliques and Maximum Cliques

x1 x2 x3 x4

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Introduction Bayes Nets Independence MRF Example P

Joint Distribution for Cliques

p(x) = 1 Z

C

ψC(xC) where ψc(xC) is the potential over the clique C and the normalization factor is defined by Z =

x
C

ψC(xC) The potential is defined by the Boltzmann distribution ψC(xC) = e−E(xc) and E is an energy

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Introduction Bayes Nets Independence MRF Example P

Image De-noising

See details in the book for formulation & results

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

Henrik I. Christensen (RIM@GT) Graphical Models 37 / 55

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Introduction Bayes Nets Independence MRF Example P

Outline of SLAM problem

Prediction of pose & map features Update of map & pose estimate World Model

Pose estimate Corrected pose est

Ensure the robot does not get lost!

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Introduction Bayes Nets Independence MRF Example P

The basics for SLAM

State of robot is modelled as the pose

xR =
x

y θ T Map features can be represented as points or lines, i.e.:

xi =
xi

yi T

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Introduction Bayes Nets Independence MRF Example P

Estimation as a Kalman Problem

Prediction by odometric modelling Updating as a Kalman process, with the state

xstate =

    

xR
x1

. . .

xn

    

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Introduction Bayes Nets Independence MRF Example P

Why is SLAM difficult?

The number of map hypotheses is very large Often the signal to noise ratio for features is ≈ 1 Robust discriminative features are not common The “process” is often approximated

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Introduction Bayes Nets Independence MRF Example P

Problems?

1 Flexible inclusion/exclusion of measurements? 2 Handling of linearization? 3 Dealing with topological constraints?

Loop closing etc.

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Introduction Bayes Nets Independence MRF Example P

Data handling

Easy inclusion and/or exclusion of data at any time in the process. How to avoid too early a commitment to a particular map hypothesis. Design of a representation that allow any-time inclusion/exclusion of data?

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Introduction Bayes Nets Independence MRF Example P

Linearizations?

Linearization might cause divergence in the data. Reported by several, e.g. Julier et al (2001) and Castellanos et al (2004) Consistent handling of non-linearities

Start by exact handling of non-linearities As data matures a linearization is permitted Identification of major non-linearities to include them

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Introduction Bayes Nets Independence MRF Example P

Topological constraints?

Consistent inclusion of topological constraints Two step strategy:

1

Close approximation of system in a trivial way

2

Fine tune full model by adding “smaller” corrections

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Introduction Bayes Nets Independence MRF Example P

Problem statement

{xi} the robot path (set of poses), (i ∈ {1 . . . Np}) {zj} feature coordinates (j ∈ {1..Nm}) {di} dead reckoning measurements, between feature measurements {fk} feature measurements, (k ∈ {1..Nf }) Λ the f ↔ z association P(x, z, d, f , Λ) = P(d, f |x, z, Λ)P(x, z, Λ)

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Introduction Bayes Nets Independence MRF Example P

Probabilistic model

P(x, z, d, f , Λ) ∝ P(d, f |x, z, Λ)P(x, z, Λ) P(d, f |x, z, Λ) ∝ P(d|x)P(f |x, z, Λ) P(x, z, Λ) ∝ P(λ) = P(Nf ) ∝ e−λNf P(x, z, d, f , Λ) ∝ P(d|x)P(f |x, z, Λ)e−λNf

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Introduction Bayes Nets Independence MRF Example P

An energy model

Definition of energy/entropy of the model: E(x, z, d, f , Λ) = − log(P(d|x)) − log(P(f |x, z, Λ)) + λNf Or E(x, z, d, f , Λ) = Ed + Ef + EΛ Or: . . .

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Introduction Bayes Nets Independence MRF Example P

An energy model

E(x, z, d, f , Λ) = Ed(x) + Ef (x, z) + EΛ(nj) (1) Ed = −

Np

i=1

log(P(di|xi−1, xi)) = 1 2

Np

i=1

ξT

i kiξi (2)

Ef = − log(P(f |x, z, Λ)) = 1 2

Nm

k=1

ηT

k kkηk

(3) EΛ = −

Nf

j=1

λ(nj − 1) (4) ξi = T(xi|xi−1) − di ηk = h(T(zj|xi)) − fk

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Introduction Bayes Nets Independence MRF Example P

Organizing a model

Graph representation Two types of nodes:

1

State notes

Poses (xi) Features (zj)

2

Energy Nodes (Computation of Eqn (1))

Connected to the state nodes needed for computation Movement (di, ki)

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Introduction Bayes Nets Independence MRF Example P

Starting a model

Ef Ef Ed X1 Z1 Z0 X0

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Introduction Bayes Nets Independence MRF Example P

Entering more data

X0 Z0 Z1 X1 X2 X3 X4

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Introduction Bayes Nets Independence MRF Example P

A graphical model example

Z4 Z2 Z3 X10 X9 X8 X7 X6 X5 X4 X3 X2 X1 Z1 Z0 X0

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Introduction Bayes Nets Independence MRF Example P

Map updating

Optimal solution to Eq. (1): (argminE) is not realistic. Relaxation techniques allow iterative updating In a time step:

1

Add a new state node (pose)

2

Any new features/measurements?

3

Update the rest of the map – minimize energy

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Introduction Bayes Nets Independence MRF Example P

Outline

1

Introduction

2

Bayesian Networks

3

Conditional Independence

4

Markov Random Fields

5

Small Example

6

Summary

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