Transcript
Page 1: Inference in Gaussian and Hybrid Bayesian Networks

Inference in Gaussian and Hybrid Bayesian Networks

ICS 275B

Page 2: Inference in Gaussian and Hybrid Bayesian Networks

Gaussian Distribution

2

1 p where

),(p, triplea asor ),N( as dRepresente

2

)(exp

2

1)(

2

xxP

Page 3: Inference in Gaussian and Hybrid Bayesian Networks

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-3 -2 -1 0 1 2 3

gaussian(x,0,1)gaussian(x,1,1)

2

)(exp

2

1)(

2xxP

N(, )

Page 4: Inference in Gaussian and Hybrid Bayesian Networks

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-3 -2 -1 0 1 2 3

gaussian(x,0,1)gaussian(x,0,2)

2

)(exp

2

1)(

2xxP

N(, )

Page 5: Inference in Gaussian and Hybrid Bayesian Networks

Multivariate Gaussian

Definition:

Let X1,…,Xn. Be a set of random variables. A multivariate Gaussian distribution over X1,…,Xn is a parameterized by an n-dimensional mean vector and an n x n positive definitive covariance matrix . It defines a joint density via:

)()(

2

1

||)2(

1)( 1

2/12/

xxXP T

n

Page 6: Inference in Gaussian and Hybrid Bayesian Networks

Multivariate Gaussian

)()(

2

1

||)2(

1)( 1

2/12/

xxXP T

n

Page 7: Inference in Gaussian and Hybrid Bayesian Networks

Linear Gaussian Distribution

Definition:

Let Y be a continuous node with continuous parents X1,…,Xk. We say that Y has a linear Gaussian model if it can be described using parameters 0, …,k and 2 such that:

P(y| x1,…,xk)=N (μy + 1x1 +…,kxk ; )

=N([μy,1,…,k] , )

Page 8: Inference in Gaussian and Hybrid Bayesian Networks

A B

),(~ aaNA

)],([~ bbawNB

A BA B

Page 9: Inference in Gaussian and Hybrid Bayesian Networks

kkYY 110

-10 -5 0 5 10X -10-5

05

10

Y

00.050.1

0.150.2

0.250.3

0.350.4

Page 10: Inference in Gaussian and Hybrid Bayesian Networks

Linear Gaussian Network

Definition

Linear Gaussian Bayesian network is a Bayesian network all of whose variables are continuous and where all of the CPTs are linear Gaussians.

Linear Gaussian BN Multivariate Gaussian

=>Linear Gaussian BN has a compact representation

Page 11: Inference in Gaussian and Hybrid Bayesian Networks

Inference in Continuous Networks

A B

abb

bba

baba

bab

a

A

babaa

σCMCKc

/σww

wM

MKC

CcNAPABP

APABPBP

wNNAP

*)/(

/

/

/ /1

/11-K where

11

),()(*)|(

)(*)|()(

)],,([P(B) ),()(

11

21

1

Page 12: Inference in Gaussian and Hybrid Bayesian Networks

Marginalization

answer. required theis )','(

' '

' ' quantitiesfour ofmatrix a is

' 'quantities twocontaining vector a is

),(

bb

bbba

abaa

a

ba

A

N

C

c

CcN

Page 13: Inference in Gaussian and Hybrid Bayesian Networks

Problems: When we Multiply two arbitrary Gaussians!

abb

bba

baba

bab

a

A

babaa

σCMCKc

/σww

wM

MKC

CcNAPABP

APABPBP

wNNAP

*)/(

/

/

/ /1

/11-K where

11

),()(*)|(

)(*)|()(

)],,([P(B) ),()(

11

21

1

Inverse of K and M is always well defined.

However, this inverse is not!

Page 14: Inference in Gaussian and Hybrid Bayesian Networks

Theoretical explanation: Why this is the case ? Inverse of a matrix

of size n x n exists when the matrix is of rank n.

If all sigmas and w’s are assumed to be 1.

(K-1+M-1) has rank 2 and so is not invertible.

1 1- 0

1- 1 0

0 0 0

/ 0

/ /1 0

0 0 0

0 0 0

0 1- 1-

0 1- 1

0 0 0

0 /

0 / /11-K where

1 1- 0

1- 2 1-

0 1- 111

)|,()|(*)|(

2

1

2

1

1

bxbx

bxb

abab

aba

/σww

wM

/σww

w

MKC

XBAPXBPBAP

Page 15: Inference in Gaussian and Hybrid Bayesian Networks

Density vs conditional

However, Theorem: If the product of the gaussians

represents a multi-variate gaussian density, then the inverse always exists. For example, For P(A|B)*P(B)=P(A,B) = N(c,C) then

inverse of C always exists. P(A,B) is a multi-variate gaussian (density).

But P(A|B)*P(B|X)=P(A,B|X) = N(c,C) then inverse of C may not exist. P(A,B|X) is a conditional gaussian.

Page 16: Inference in Gaussian and Hybrid Bayesian Networks

Inference: A general algorithm Computing marginal of a given variable, say Z.

1 w'-

w- '1

1

)2log(2

1

2

],...,[Let w

),,(),,...,,(),...,|(

2

2

2

2

2

1

11

wwK

wh

g

ww

khgwwNXXyP

y

y

y

y

y

y

k

ykyk

Step 1:

Convert all conditional gaussians to canonical form

Page 17: Inference in Gaussian and Hybrid Bayesian Networks

Inference: A general algorithm Computing marginal of a given variable, say Z.

Step 2: Extend all g’s,h’s and

k’s to the same domain by adding 0’s.

0 /1

0 0 tochanged is K'

same theremainsK domain, same theK to and K' Extending

]/1['

)',','()(

/

/ /1

),,()|(

2

b

b

abab

aba

σ

σK

khgBP

/σww

wK

khgBAP

Page 18: Inference in Gaussian and Hybrid Bayesian Networks

Inference: A general algorithm Computing marginal of a given variable, say Z. Step 3: Add all g’s, all h’s and all k’s. Step 4: Let the variables involved in the

computation be: P(X1,X2,…,Xk,Z)= N(μ,∑)

Page 19: Inference in Gaussian and Hybrid Bayesian Networks

Inference: A general algorithm Computing marginal of a given variable, say Z.

z

k

ZZ

ZZZkZ

Zk

zzNZP

....

...........

..............................

),()(

1

z

1

1111

Step 5:

Extract the marginal

Page 20: Inference in Gaussian and Hybrid Bayesian Networks

Inference: Computing marginal of a given variable

For a continuous Gaussian Bayesian Network, inference is polynomial O(N3). Complexity of matrix inversion

So algorithms like belief propagation are not generally used when all variables are Gaussian.

Can we do better than N^3? Use Bucket elimination.

Page 21: Inference in Gaussian and Hybrid Bayesian Networks

Bucket elimination Algorithm elim-bel (Dechter 1996)

b

Multiplication operator

P(a|e=0)

W*=4”induced width” (max clique size)

bucket B:

P(a)

P(c|a)

P(b|a) P(d|b,a) P(e|b,c)

bucket C:

bucket D:

bucket E:

bucket A:

e=0

B

C

D

E

A

e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

Marginalization operator

Page 22: Inference in Gaussian and Hybrid Bayesian Networks

Multiplication Operator

Convert all functions to canonical form if necessary.

Extend all functions to the same variables (g1,h1,k1)*(g2,h2,k2) =(g1+g2,h1+h2,k1+k2)

Page 23: Inference in Gaussian and Hybrid Bayesian Networks

Again our problem!

b

Multiplication operator

P(a)

W*=4”induced width” (max clique size)

bucket B:

P(a)

P(c|a)

P(b|a) P(d|b,a) P(e|b,c)

bucket C:

bucket D:

bucket E:

bucket A:

P(e)

B

C

D

E

A

e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

Marginalization operator

h(a,d,c,e) does not represent a density and so cannot be computed in our usual form N(μ,σ)

Page 24: Inference in Gaussian and Hybrid Bayesian Networks

Solution: Marginalize in canonical form Although intermediate functions computed in bucket

elimination are conditional, we can marginalize in canonical form, so we can eliminate the problem of non-existence of inverse completely.

Page 25: Inference in Gaussian and Hybrid Bayesian Networks

Algorithm

In each bucket, convert all functions in canonical form if necessary, multiply them and marginalize out the variable in the bucket as shown in the previous slide.

Theorem: P(A) is a density and is correct. Complexity: Time and space: O((w+1)^3)

where w is the width of the ordering used.

Page 26: Inference in Gaussian and Hybrid Bayesian Networks

Continuous Node, Discrete ParentsDefinition:

Let X be a continuous node, and let U={U1,U2,…,Un} be its discrete parents and Y={Y1,Y2,…,Yk} be its continuous parents. We say that X has a conditional linear Gaussian (CLG) CPT if, for every value uD(U), we have a a set of (k+1) coefficients au,0, au,1, …, au,k+1 and a variance u

2 such that:

),(),|(1

2,0,

k

iuiiuu yaaNyuXp

Page 27: Inference in Gaussian and Hybrid Bayesian Networks

CLG Network

Definition:

A Bayesian network is called a CLG network if every discrete node has only discrete parents, and every continuous node has a CLG CPT.

Page 28: Inference in Gaussian and Hybrid Bayesian Networks

Inference in CLGs

Can we use the same algorithm? Yes, but the algorithm is unbounded if we are not

careful. Reason:

Marginalizing out discrete variables from any arbitrary function in CLGs is not bounded. If we marginalize out y and k from f(x,y,i,k) , the result is

a mixture of 4 gaussians instead of 2. X and y are continuous variables I and k are discrete binary variables.

Page 29: Inference in Gaussian and Hybrid Bayesian Networks

Solution: Approximate the mixture of Gaussians by a single gaussian

Page 30: Inference in Gaussian and Hybrid Bayesian Networks

Multiplication and Marginalization Convert all functions to

canonical form if necessary.

Extend all functions to the same variables

(g1,h1,k1)*(g2,h2,k2) =(g1+g2,h1+h2,k1+k2)

MultiplicationStrong marginal when marginalizing continuous variables

Weak marginal when marginalizing discrete variables

Page 31: Inference in Gaussian and Hybrid Bayesian Networks

Problem while using this marginalization in bucket elimination Requires computing ∑ and μ which is not possible

due to non-existence of inverse. Solution: Use an ordering such that you never have

to marginalize out discrete variables from a function that has both discrete and continuous gaussian variables.

Special case: Compute marginal at a discrete node Homework: Derive a bucket elimination algorithm

for computing marginal of a continuous variable.

Page 32: Inference in Gaussian and Hybrid Bayesian Networks

b

Multiplication operator

P(a)

W*=4”induced width” (max clique size)

bucket B:

P(a)

P(c|a)

P(b|a,e) P(d|b,a) P(d|b,c)

bucket C:

bucket D:

bucket E:

bucket A:

P(e) e)(a,hD

(a)hE

e)d,(a,hC

Marginalization operator

Special Case: A marginal on a discrete variable in a CLG is to be computed.B,C and D are continuous variables and A and E is discrete

e)c,d,(a,hB

Page 33: Inference in Gaussian and Hybrid Bayesian Networks

Complexity of the special case Discrete-width (wd): Maximum number of

discrete variables in a clique Continuous-width (wc): Maximum number of

continuous variables in a clique Time: O(exp(wd)+wc^3) Space: O(exp(wd)+wc^3)

Page 34: Inference in Gaussian and Hybrid Bayesian Networks

Algorithm for the general case:Computing Belief at a continuous node of a CLG Convert all functions to canonical form. Create a special tree-decomposition Assign functions to appropriate cliques

(Same as assigning functions to buckets) Select a Strong Root Perform message passing

Page 35: Inference in Gaussian and Hybrid Bayesian Networks

Creating a Special-tree decomposition Moralize the Bayesian Network. Select an ordering such that all continuous

variables are ordered before discrete variables (Increases induced width).

Page 36: Inference in Gaussian and Hybrid Bayesian Networks

Elimination order

w

y

x

z

Strong elimination order:• First eliminate continuous variables• Eliminate discrete variable when no

available continuous variables

Moralized graph has this edge

W and X are discrete variables and Y and Z are continuous.

Page 37: Inference in Gaussian and Hybrid Bayesian Networks

Elimination order (1)

w

y

x

z

dim: 2 dim: 2

dim: 2

1

Page 38: Inference in Gaussian and Hybrid Bayesian Networks

Elimination order (2)

w

y

x

z

dim: 2 dim: 2

2

1

Page 39: Inference in Gaussian and Hybrid Bayesian Networks

Elimination order (3)

w

y

x

z

3 dim: 2

2

1

Page 40: Inference in Gaussian and Hybrid Bayesian Networks

Elimination order (4)

w

y

x

z

3 4

2

1w

y

z

3

2

1

w

y

x3 4

2w

y

3

2

Cliques 1

Cliques 2

separator

Page 41: Inference in Gaussian and Hybrid Bayesian Networks

Bucket tree or Junction tree (1)

w

y

z

w

y

x

w

y

Cliques 1

Cliques 2: root

separator

Page 42: Inference in Gaussian and Hybrid Bayesian Networks

Algorithm for the general case:Computing Belief at a continuous node of a CLG

Convert all functions to canonical form. Create a special tree-decomposition Assign functions to appropriate cliques

(Same as assigning functions to buckets) Select a Strong Root Perform message passing

Page 43: Inference in Gaussian and Hybrid Bayesian Networks

Assigning Functions to cliques Select a function and place it in an arbitrary

clique that mentions all variables in the function.

Page 44: Inference in Gaussian and Hybrid Bayesian Networks

Algorithm for the general case:Computing Belief at a continuous node of a CLG

Convert all functions to canonical form. Create a special tree-decomposition Assign functions to appropriate cliques

(Same as assigning functions to buckets) Select a Strong Root Perform message passing

Page 45: Inference in Gaussian and Hybrid Bayesian Networks

Strong Root

We define a strong root as any node R in the bucket-tree which satisfies the following property: for any pair (V,W) which are neighbors on the tree with W closer to R than V, we have

variablesdiscrete ofset theis

variablescontinuous ofset theis

W Vor \

WV

Page 46: Inference in Gaussian and Hybrid Bayesian Networks

Example Strong rootStrong Root

Page 47: Inference in Gaussian and Hybrid Bayesian Networks

Algorithm for the general case:Computing Belief at a continuous node of a CLG

Create a special tree-decomposition Assign functions to appropriate cliques

(Same as assigning functions to buckets) Select a Strong Root Perform message passing

Page 48: Inference in Gaussian and Hybrid Bayesian Networks

Message passing at a typical node x2

oNode “a” contains functions assigned to it according to the tree-decomposition scheme denoted by pj(a)

)()),(()),((),(

j

j

basepa biaiba apaisephbaseph

)),(( axseph naxn

a

b

x1

)),(( 11 axseph ax

Page 49: Inference in Gaussian and Hybrid Bayesian Networks

Message Passing

rootroot

Collect

rootroot

Distribute

Figure from P. Green

Two pass algorithm: Bucket-tree propagation

Page 50: Inference in Gaussian and Hybrid Bayesian Networks

Lets look at the messagesCollect Evidence

∫C

∫L

∫Mout

∫Min∫D

∫D

Strong Root

Page 51: Inference in Gaussian and Hybrid Bayesian Networks

Distribute Evidence

∫E∑W,B

∫E∑W,B

∑W

∫E∑B

∑F

Strong Root

Page 52: Inference in Gaussian and Hybrid Bayesian Networks

Lauritzens theorem

When you perform message passing such that collect evidence contains only strong marginals and distribute evidence may contain weak marginals, the junction-tree algorithm in exact in the sense that: The first (mean) and second moments (variance)

computed are true moments

Page 53: Inference in Gaussian and Hybrid Bayesian Networks

Complexity

Polynomial in #of continuous variables in a clique (n3)

Exponential in #of discrete variables in a clique Possible options for approximation

Ignore the strong root assumption and use approximation like MBTE, IJGP, Sampling

Respect the strong root assumption and use approximation like MBTE, IJGP, Sampling Inaccuracies only due to discrete variables if done in one

pass of MBTE.

Page 54: Inference in Gaussian and Hybrid Bayesian Networks

W=0 W=1

X=0 X=1

Initialization (1)

w

y

x

z

dim: 2 dim: 2

dim: 2

dim: 2

w=0

0.5

w=1

0.5

x=0

0.4

x=1

0.6

)100

010,

2.1

2.0;(

yN )

20

02,

0.1

0.1;(

yN

)90

09,

5.07.0

3.09.0

5.0

5.0;(

yzN )

30

02,

5.07.0

2.03.0

5.0

2.0;(

yzN

Page 55: Inference in Gaussian and Hybrid Bayesian Networks

Initialization (2)

wyz wxywy

Cliques 1 Cliques 2 (root)

w=0g=log(0.5),h=[],K

=[]

w=1g=log(0.5),h=[],K

=[]

x=0g=log(0.4),h=[],K

=[]

x=1g=log(0.6),h=[],K

=[]X=0 X=1

g = -4.1245

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

g = -3.0310

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

W=0 W=1

g = -4.0629

h = [0.0889 -0.0111 -0.0556 0.0556]

K =

g = -2.7854

h = [0.0867 -0.0633 -0.1000 -0.1667]

K = 0.1444 - 0.0089 - 0.1 0.0778

- 0.0089 0.0378 - 0.0333 - 0.0556- 0.1 - 0.0333 0.1111 0

0.0778 - 0.0556 0 0.1111

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

Page 56: Inference in Gaussian and Hybrid Bayesian Networks

W=0 W=1

g = -4.7560

h =

K =

g = -3.4786

h =

K =

Initialization (3)

wyz wxywy

Cliques 1 Cliques 2 (root)

0.0889 - 0.0111 - 0.0556 0.0556

0.1444 - 0.0089 - 0.1 0.0778- 0.0089 0.0378 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.0867 - 0.0633 - 0.1 - 0.1667

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

wx=00 wx=10

g = -5.1308

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

g = -5.1308

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

wx=01 wx=11

g = -3.5418

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

g = -3.5418

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

empty

Page 57: Inference in Gaussian and Hybrid Bayesian Networks

Message Passing

wyz wxywy

Cliques 1 Cliques 2 (root)

empty

Collect evidencewywyzwy )()(*

)(

)()()(

**

wy

wywxywxy

Distribute evidencewywxywy )()( ***

)(

)()()(

*

****

wy

wywyzwyz

Page 58: Inference in Gaussian and Hybrid Bayesian Networks

Collect evidence (1)

wyz wxywy

Cliques 1 Cliques 2 (root)

empty

2221

1211

2

1

2

1 , ,KK

KKKh

h

h

y

yy

121

112122

11

11212

11

1111

11

ˆ

ˆ

)||log)2log((2

1

KKKKK

KK

KKpgg T

hhh

hh

)ˆ,ˆ,ˆ;(][ 2121 KgdTTT hyyyy y2y3

y1y2

y2

(y1,y2)(y2)

Page 59: Inference in Gaussian and Hybrid Bayesian Networks

Collect evidence (2)

wyz wxywy

Cliques 1 Cliques 2 (root)

empty

W=0 W=1

g = -4.7560

h =

K =

g = -3.4786

h =

K =

0.0889 - 0.0111 - 0.0556 0.0556

0.1444 - 0.0089 - 0.1 0.0778- 0.0089 0.0378 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.0867 - 0.0633 - 0.1 - 0.1667

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

W=0 W=1

g = -0.6931

h = [0.1388 0]’ *1.0e-16

K = [0.2776 -0.0694;0.0347 0]*1.0e-16

g = -0.6931

h = [0 0]’

K = [0 0 0 0]

marginalization

Page 60: Inference in Gaussian and Hybrid Bayesian Networks

Collect evidence (3)

wyz wxywy

Cliques 1 Cliques 2 (root)

empty

W=0 W=1

g = -0.6931

h = [0.1388 0]’ *1.0e-16

K = [0.2776 -0.0694;0.0347 0]*1.0e-16

g = -0.6931

h = [0 0]’

K = [0 0 0 0]

wx=00 wx=10

g = -5.1308

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

g = -5.1308

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

wx=01 wx=11

g = -3.5418

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

g = -3.5418

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

multiplication

wx=00 wx=10

g = -5.8329

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

g = -5.8329

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

wx=01 wx=11

g = -4.2350

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

g = -4.2350

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

Page 61: Inference in Gaussian and Hybrid Bayesian Networks

Distribute evidence (1)

wyz wxywy

Cliques 1 Cliques 2 (root)

W=0 W=1

g = -4.7560

h =

K =

g = -3.4786

h =

K =

0.0889 - 0.0111 - 0.0556 0.0556

0.1444 - 0.0089 - 0.1 0.0778- 0.0089 0.0378 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.0867 - 0.0633 - 0.1 - 0.1667

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

W=0 W=1

g = -0.6931

h = [0.1388 0]’ *1.0e-16

K = [0.2776 -0.0694;0.0347 0]*1.0e-16

g = -0.6931

h = [0 0]’

K = [0 0 0 0]

division

Page 62: Inference in Gaussian and Hybrid Bayesian Networks

Distribute evidence (2)

wyz wxywy

Cliques 1 Cliques 2 (root)

W=0 W=1

g = -4.0629

h =

K =

g = -2.7854

h =

K =

0.0889 - 0.0111 - 0.0556 0.0556

0.1444 - 0.0089 - 0.1 0.0778- 0.0089 0.0378 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.0867 - 0.0633 - 0.1 - 0.1667

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

Page 63: Inference in Gaussian and Hybrid Bayesian Networks

Distribute evidence (3)

wyz wxywy

Cliques 1 Cliques 2 (root)

wx=00 wx=10

g = -5.8329

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

g = -5.8329

h = [-0.02 0.12]’

K = [0.1 0; 0 0.1]

wx=01 wx=11

g = -4.2350

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

g = -4.2350

h = [0.5 -0.5]’

K = [0.5 0.5;0.5 0.5]

Marginalize over x

w=0 w=1

logp = -0.6931

mu = [0.52 -0.12]’

Sigma =

logp = -0.6931

mu = [0.52 -0.12]’

Sigma =5.5456 - 0.6336

- 0.6336 6.36165.5456 - 0.6336

- 0.6336 6.3616

Page 64: Inference in Gaussian and Hybrid Bayesian Networks

Distribute evidence (4)

wyz wxywy

Cliques 1 Cliques 2 (root)

W=0 W=1

g = -4.0629

h =

K =

g = -2.7854

h =

K =

0.0889 - 0.0111 - 0.0556 0.0556

0.1444 - 0.0089 - 0.1 0.0778- 0.0089 0.0378 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.0867 - 0.0633 - 0.1 - 0.1667

0.2083 - 0.1467 0.15 - 0.2333- 0.1467 0.1033 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

w=0 w=1

logp = -0.6931

mu = [0.52 -0.12]’

Sigma =

logp = -0.6931

mu = [0.52 -0.12]’

Sigma =5.5456 - 0.6336

- 0.6336 6.36165.5456 - 0.6336

- 0.6336 6.3616

multiplication

w=0 w=1

g = -4.3316

h = [0.0927 -0.0096]’

K =

g = -0.6931

h = [0.0927 -0.0096]’

K =0.1824 0.01820.0182 0.159

0.1824 0.01820.0182 0.159

Canonical form

Page 65: Inference in Gaussian and Hybrid Bayesian Networks

Distribute evidence (5)

wyz wxywy

Cliques 1 Cliques 2 (root)

W=0 W=1

g = -8.3935

h =

K =

g = -7.1170

h =

K =

0.1816 - 0.0207 - 0.0556 0.05560.3268 0.0093 - 0.1 0.07780.0093 0.1968 - 0.0333 - 0.0556

- 0.1 - 0.0333 0.1111 00.0778 - 0.0556 0 0.1111

0.1793 - 0.073 - 0.1 - 0.1667

0.3907 - 0.1285 0.15 - 0.2333- 0.1285 0.2623 - 0.1 0.1667

0.15 - 0.1 0.5 0- 0.2333 0.1667 0 0.3333

Page 66: Inference in Gaussian and Hybrid Bayesian Networks

After Message Passing

p(wyz) p(wxy)p(wy)

Cliques 1 Cliques 2 (root)

Local marginal distributions


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