Probabilistic Graphical Models

Intelligent Agents:

Web-mining Agents

Probabilistic Graphical Models

Continuous Space

Probabilistic Graphical Models (PGMs)

1. Recap:

Propositional

modelling

• Factor model, Bayesian network, Markov network

• Semantics, inference tasks + algorithms + complexity

2. Probabilistic relational

models

(PRMs)

• Parameterised models, Markov logic networks

• Semantics, inference tasks

3. Lifted inference

• LVE, LJT, FOKC

• Theoretical analysis

4. Lifted learning

• Recap: propositional learning

• From ground to lifted models

• Direct lifted learning

5. Approximate Inference:

Sampling

• Importance sampling

• MCMC methods

6. Sequential models &

inference

• Dynamic PRMs

• Semantics, inference tasks + algorithms + complexity, learning

7. Decision making

• (Dynamic) Decision PRMs

• Semantics, inference tasks + algorithms, learning

8. Continuous Space

• Gaussian distributions

Models with Continuous Variables

•

Discretisation of continuous variables

•

_{Discrete model again}

•

Own set of problems

• Hard to find good discretisation

• High granularity might be necessary

➝ large ranges ➝ large factors

• Lose characteristics of variable

• Not each value necessarily associated with a probability

• Nearby values have similar probabilities ➝ hard to capture in a discrete distribution (no notion of closeness between range values)

Outline: 8. Continuous Space

A. Basics

•

_{Continuous variables, probability density function,}

cumulative probability distribution

•

Joint distribution, marginal density, conditional density

B. Gaussian models

•

_{(Multivariate) Gaussian distribution}

•

(Parameterised) Gaussian Bayesian networks

C. Probabilistic Soft Logic (PSL)

Probability Density Function

•

Continuous randvar

!

•

Range ℛ ! = 0,1

•

Function

' ∶ ℝ → ℝ

is a probability density function

(PDF) for

!

if it is a non-negative, integrable function

s.t.

+

ℛ ,

' - .- = 1

•

For any

/

(and

0

) in event space

1 ! ≤ / = +

34 5

' - .-

1 / ≤ ! ≤ 0 = +

5 6

' -

.-•

Function 1

is a cumulative distribution for !

•

Intuitively, value of ' -

at point -

is the incremental amount

that -

adds to the cumulative distribution during integration

PDFs: Uniform Distribution

•

Continuous randvar

!

has a uniform distribution

over

", $

, denoted

! ~ Unif ", $

, if it has the

PDF

* + = -

_{$ − "}

1

0

$ ≥ + ≥ "

otherwise

•

Density can be larger than 1 if

$ − " < 1

• Can be legal if the total area under the pdf is 1

-10 -5 0 5 10 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Distribution P(R ) -10 -5 0 5 10 0.00 0.02 0.04 0.06 0.08 0.10 PDF p(r)

Joint/Multivariate Distribution

•

Let

!

be a joint distribution over continuous

randvars

"

_#

, … , "

_&

•

Function

' (

_#

, … , (

_&

is a joint density function of

"

_#

, … , "

_&

if

•

' (

_#

, … , (

_&

≥ 0

for all values

(

_#

, … , (

_&

of

"

_#

, … , "

_&

•

'

is an integrable function

•

_{For any choice}

+

_#

, … , +

_&

and

,

_#

, … , ,

_&

,

! +

_#

≤ "

_#

≤ ,

_#

, … , +

_&

≤ "

_&

≤ ,

_&

= /

0₁ 2₁

… /

0₃ 2₃

' (

_#

, … , (

_&

4(

_#

… (

_&

Marginal Density

•

Given a joint density, integrate out the non-query

randvars

•

_{E.g., given}

! ", $

a joint density for randvars

%, &

, then

! " = (

)* +*

! ", $ ,$

•

_{Shorthand notations}

•

!

_-

= ! "

marginal density

•

!

_-,.

= ! ", $

joint density

Conditional Density Function

•

Discrete case:

! "|$ = & =

' (,*+,_{' *+,}

•

Problem in continuous case:

! $ = & = 0

➝

! "|$ = &

undefined

•

To avoid problem, condition on event

& − / ≤ $ ≤ & + /

and consider limit when

/ ⟶ 0

! "|& = lim

6⟶7

! "|& − / ≤ $ ≤ & + /

•

If a continuous joint density

8 &, 9

exists, derive form

of this expression:

8 9|& =

8 &, 9

8 &

•

If

8 & = 0

, conditional density undefined

•

Chain rule and Bayes’ rule hold as well:

8 &, 9 = 8 & 8 9 &

8 9|& =

8 9 8 &|9

8 &

Outline: 8. Continuous Space

A. Basics

•

_{Continuous variables, probability density function,}

cumulative probability distribution

•

Joint distribution, marginal density, conditional density

B. Gaussian models

•

_{(Multivariate) Gaussian distribution}

•

(Parameterised) Gaussian Bayesian networks

C. Probabilistic Soft Logic (PSL)

Models with Continuous Variables

•

Problem: Space of possible parameterisation

essentially unbounded

•

Special case: (

Multivariate

)

Gaussian distributions

•

_{Two parameters per variable: mean, variance}

•

Strong assumptions, e.g.,

• Exponential decay away from its mean

• Linearity of interactions between randvars

➝ Assumptions often invalid but still work as a good approximation for many real-world distributions

•

_{Many generalisations exist which use Gaussians as a}

foundation

• Non-linear interactions

PDFs: Gaussian/Normal Distribution

•

Continuous randvar

!

has a Gaussian distribution

with mean

"

and variance

#

$

, denoted

! ~ & ", #

$

, if it has the PDF

( ) =

1

2-#

.

/ 0/1_$32 2

•

_{Expected value and variance of}

!

given by

"

and

#

$

• Standard deviation: # -10 -5 0 5 10 0.0 0.1 0.2 0.3 0.4 PDF R P(r) R~N(0,1) R~N(0,22) R~N(5,1.52) -10 -5 0 5 10 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Distribution R P(R ) R~N₍0,1) R~N(0, 22) R~N₍5,1.52) Standard Gaussian ! ~ & " = 0, #$ _{= 1 :} ( ) = 1 2-. / 0_$2

Multivariate Gaussian

•

Univariate Gaussian: two parameters

•

Mean !

and variance "

#

•

Multivariate Gaussian distribution over continuous

randvars

$

_%

, … , $

₍

characterised by

•

)-dimensional mean vector *

•

Symmetric )×)

covariance matrix Σ

•

I.e., - *; Σ

•

Density function defined as

/ 0 =

1

24

(

_Σ

exp −

1

2

0 − *

9

Σ

:%

0 − *

•

0 = ;

_%

, … , ;

₍ 9

•

Σ

determinant of Σ

•

To induce a well-defined density, Σ

must be

positive-definite

• For any 0 ∈ ℝ( s.t. 0 ≠ 0 ∶ 09Σ0 > 0

• Guaranteed to be non-singular ➝ non-zero determinant

Standard multivariate Gaussian $%, … , $( with

• * = B (all-zero vector)

Example

•

Joint Standard Gaussian

distribution over two

randvars

!

_"

, !

_$

, i.e.,

•

% = 0 0

(

, Σ = *

_$

•

_{Joint Gaussian distribution}

over three randvars

!

_"

, !

_$

, !

₊

•

_{Mean vector, covariance matrix:}

% =

−3

1

4

Σ =

4

2

2

5

−2

−5

−2 −5

8

•

Covariances

345 !

_"

; !

₊

and

345 !

_$

; !

₊

negative, i.e.,

!

₊

negatively correlated with

!

_"

(and

!

_$

)

• When !_" (!_$) goes up, !₊ goes down

7_" 7$

Marginalisation

•

Trivial with covariance matrix:

•

Compute pairwise covariances, i.e., generating the

distribution in its covariance form

•

Given covariance form

Σ

: Read off from

", Σ

•

Assume a joint Gaussian distribution over

$, %

where

$ ∈ ℝ

(

and

% ∈ ℝ

)

•

One can decompose mean and covariance:

* +, , = .

"

_"

$ %

;

Σ

_$$

Σ

_$%

Σ

_%$

Σ

_%%

•

where

• "_$ ∈ ℝ(, "_% ∈ ℝ), • Σ_$$ an 0×0 matrix, Σ_$% an 0×2 matrix, Σ_%$ = Σ_$%3 an 2×0 matrix, Σ_%% a 2×2 matrix

•

Then, marginal distribution over

%

given by Gaussian

Example

•

Given joint Gaussian distribution over three

randvars

!

_"

, !

_$

, !

_%

•

_{Mean vector, covariance matrix:}

& =

−3

1

4

Σ =

4

2

2

5

−2

−5

−2 −5

8

•

0 !

_"

, !

_$

given by Gaussian distribution with

& = 1

Dual: Information/Precision Form

•

Rewrite

exp −

%

&

' − (

)

_Σ

+%

_{' − (}

_{by setting}

Γ = Σ

+%

and multiplying out:

−

1

2

' − (

)

Γ

' − ( = −

1

2

'

)

Γ' − 2'

)

Γ( + (

)

Γ(

•

(

)

Σ

+%

(

is constant over the different

'

, therefore,

•

Γ(

called potential vector

= ')_Γ( = ')_Γ( )) _▹1)) _{= 1} = Γ( )_')) ) _▹(13))_{= 3})₁) = Γ( )_' ) _▹1)) _{= 1} = Γ( )_{' ▹5}) _{= 5, 5 a scalar}

6 ' ∝ exp −

1

2

'

)

Γ' − 2'

)

Γ(

= exp −

1

2

'

)

Γ' + '

)

Γ(

= exp −

1

2

'

)

Γ' + Γ(

)

'

Dual: Information/Precision Form

•

For a decomposition

!, #

where

! ∈ ℝ

&

and

# ∈ ℝ

'

:

Γ = Σ

+,

=

Σ

_Σ

!!

Σ

!# #!

Σ

##

+-=

Γ

_Γ

!!

Γ

!# #!

Γ

##

•

Getting to

Σ

• Σ_!! = Γ_!! − Γ_!#Γ_##+-Γ_#! +-• Σ_## = Γ_## − Γ_#!Γ_!!+-Γ_!# +-• Σ_!# = −Γ_!!+-Γ_!# Γ_## − Γ_#!Γ_!!+-Γ_!# +- = Σ_#!/ • Σ_#! = −Γ_##+-Γ_#! Γ_!! − Γ_!#Γ_##+-Γ_#! +- = Σ_!#/

•

Getting to

Γ

• Γ_!! = Σ_!! − Σ_!#Σ_##+-Σ_#! +-• Γ_## = Σ_## − Σ_#!Σ_!!+-Σ_!# +-• Γ_!# = −Σ_!!+-Σ_!# Σ_## − Σ_#!Σ_!!+-Σ_!# +- = Γ_#!/ • Γ_#! = −Σ_##+-Σ_#! Σ_!! − Σ_!#Σ_##+-Σ_#! +- = Γ_!#/

Conditioning

•

Conditioning a Gaussian on observations

! = #

easy to perform in the information form by setting

!

to

#

in one of the following

$ % ∝ exp −

1

2

% − -

.

Γ % −

-∝ exp −

1

2

%

.

Γ% + Γ-

.

%

•

Assuming a decomposition into

1

and

!

$ %, # = 3

-

_-

1

!

;

Σ

₁

Σ

_1!

Conditioning

•

For a decomposition into

!, #

$ % ∝ exp −1 2 % − - .Σ01 % − -= exp −1 2 % − - .Γ % − -= −1 2 % # − -₄ -₅ . Γ44 Γ45 Γ₅₄ Γ₅₅ %# − -₄ -₅ = −1 2 % − -₄ # − -₅ . _Γ 44 Γ45 Γ₅₄ Γ₅₅ % − -₄ # − -₅ = −1 2 % − -4 .Γ44 % − -4 − 1 22 % − -4 .Γ45 # − -5 − 1 2 # − -5 .Γ55 # − -5 ∝ −1 2 % − -4 .Γ44 % − -4 − % − -4 .Γ45 # − -5 = −1 2 % − -4 .Γ44 % − -4 − % − -4 .Γ45 # − -5 − 6 + 6 = exp −1 2 % − -4 + Γ4401Γ54 # − -5 . Γ₄₄ % − -₄ + Γ₄₄01Γ₄₅ # − -₅ exp 6 ∝ exp −1 2 % − -4 + Γ4401Γ54 # − -5 . Γ₄₄ % − -₄ + Γ₄₄01Γ₅₄ # − -₅

Does not depend on %

6 = 1

Conditioning

•

_{For a decomposition into}

!, #

•

Result:

$|& = # ~ ) *

∗

, Σ

∗

•

In covariance form:

• *∗ = *_- + Σ_-&Σ_&&/0 # − *_&

• Σ∗ = Σ_-- − Σ_-&Σ_&&/0Σ

&-2 3 ∝ exp −1 2 3 −: ;Σ/0 3 − : = exp −1 2 3 − : ;Γ 3 −: 2 !|& = # = exp −1 2 3 # − : -:_& ; Γ-- Γ-& Γ_&- Γ_&& # −3 : -:_& ∝ exp −1 2 3 − :- + Γ--/0Γ&- # − :& ; Γ_-- 3 − :_- + Γ_--/0Γ_&- # − :_& *∗ = *_- − Γ_-/0Γ_&- # − *_& Σ∗ = Γ

--Query Answering: Summary

•

For marginalisation, read off parameters in

covariance form

•

_{Marginal query for}

!

:

" #

_!

; Σ

_!!

•

_{For conditioning, one needs to invert the}

covariance matrix to obtain the information form

•

Conditioning on

& = (

:

!|& = ( ~ " +

∗

, Σ

∗

• In covariance form

• +∗ = +. + Σ.&Σ&&01 ( − +&

• Σ∗ = Σ_.. − Σ_.&Σ_&&01Σ_&.

Independencies in Gaussians

•

Let randvars

!

_"

, … , !

_%

have a joint distribution

& '; Σ

•

_Then,

!

_*

, !

₊

independent iff

Σ

_*+

= 0

•

Joint distribution needs to be Gaussian for this

equivalence to hold

• If the distribution is not Gaussian, Σ_*+ = 0 might be the case

and there still might be a dependence between !_*, !₊

•

Conditional independence can be read of in the

inverse of the covariance matrix,

Σ

."

•

_{Given a Gaussian distribution}

/ 0

_"

, … , 0

_%

= & '; Σ

Example

•

_{Joint Standard Gaussian}

distribution over two

randvars

!

_"

, !

_$

, i.e.,

•

_{% = 0 0}

(

, Σ = *

_$

•

!

_"

, !

_$

independent

as

Σ

_+,

= Σ

_,+

= 0

•

Gaussian for

!

_"

, !

_$

, !

_-

from before

•

Covariance and inverse covariance matrix:

Σ =

4

2

2

5

−2

−5

−2 −5

8

Σ

3"

=

−0.125 0.5833 0.3333

0.3125 −0.125

0

0

0.3333 0.3333

•

!

_"

, !

_-

conditionally independent given

!

_$

•

Σ

_"-3"

= 0

iff

7 ⊨ !

_"

⊥ !

_-

| !

_"

, !

_$

, !

_-

∖ !

_"

, !

_-

=

!

_"

⊥ !

_-

|!

_$

<_" <$ 7 <_", <_$