An Introduction to Click Models for Web Search - (Morning ...clickmodels.weebly.com/.../5/2/2/...introduction-1.pdf · An Introduction to Click Models for Web Search (Morning block

Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap

An Introduction to Click Models for Web Search(Morning block 1)

Aleksandr Chuklin§,¶ Ilya Markov§ Maarten de Rijke§

[email protected] [email protected] [email protected]

§University of Amsterdam¶Google Switzerland

SIGIR 2015 Tutorial

AC–IM–MdR An Introduction to Click Models for Web Search 1


Who are we?

Aleksandr

Ilya Maarten

Software engineer

Postdoc at Professor at

at Google

U. Amsterdam U. Amsterdam



Who are we?

Aleksandr Ilya

Maarten

Software engineer Postdoc at

Professor at

at Google U. Amsterdam

U. Amsterdam



Who are we?

Aleksandr Ilya MaartenSoftware engineer Postdoc at Professor at

at Google U. Amsterdam U. Amsterdam



Aims

Describe existing click models in a unified way, so thatdifferent models can easily be related to each other

Compare commonly used click models

Provide ready-to-use formulas and implementations ofexisting click models and detail parameter estimationprocedures to facilitate the development of new ones

Summarize current efforts on click model evaluation –evaluation approaches, datasets and software packages

Provide an overview of click model applications anddirections for future development of click models



Aims








Aims








Aims








Aims








Structure of the tutorial

Two parts, with two blocks each

An Introduction to Click Models for Web SearchAdvanced Click Models and their applications to IR

Part I (this morning)

IntroductionBasic click modelsInference for click modelsBreakDemoEvaluationData and toolsResults on the basic click modelsRecap



Structure of the tutorial

Two parts, with two blocks each

An Introduction to Click Models for Web SearchAdvanced Click Models and their applications to IR

Part I (this morning)

IntroductionBasic click modelsInference for click modelsBreakDemoEvaluationData and toolsResults on the basic click modelsRecap



Materials

Complete draft of the book on which the tutorial is based:

Aleksandr Chuklin, Ilya Markov, Maarten de Rijke. ClickModels for Web Search. Synthesis Lectures on InformationConcepts, Retrieval, and Services. Morgan & Claypool, July,2015

Copy of the slides

Parts I and II

Code and data samples to follow live demos

See http://clickmodels.weebly.com for updates andadditional materials


http://clickmodels.weebly.com


Materials



Copy of the slides

Parts I and II






Materials



Copy of the slides

Parts I and II






Materials



Copy of the slides

Parts I and II






Morning block 1 – Outline

1 Introduction

2 Motivation

3 Basic Click Models

4 Click Probabilities

5 Parameter Estimation

6 Recap



What is a click model?



Notation

Expression Meaning

u A documentq A user’s queryr The rank of a documentur A document at rank rru The rank of a document u



Notation

Expression Meaning

c A placeholder for any concept associated with a SERP(e.g., query-document pair, rank, etc.)

S A set of user search sessionsSc A set of user search sessions containing a concept cXc An event X applied to a concept cxc The value that a random variable X takes,

when applied to a concept c

x(s)c The value that a random variable X takes,

when applied to a concept c in a particular session s




1 Introduction

2 Motivation




6 Recap



Why click models?

Understand users

Simulate users

Evaluate search

Improve search



Why click models?

Understand users

Simulate users

Evaluate search

Improve search



Why click models?

Understand users

Simulate users

Evaluate search

Improve search



Why click models?

Understand users

Simulate users

Evaluate search

Improve search



Why click models?

Understand users

Simulate users

Evaluate search

Improve search




1 Introduction

2 Motivation




6 Recap



Basic click models

Random click modelCTR modelsPosition-based modelCascade modelDependent click modelDynamic Bayesian network modelUser browsing model



Random click model

?



Random click model

P(Cu = 1) = ρ



CTR models

Rank-based CTR:

P(Cr = 1) = ρr

Document-based CTR:

P(Cu = 1) = ρuq



Position-based model

document u

Eu

Cu

Au

↵uq�ru



Position-based model

P(Cu = 1) = P(Eu = 1) · P(Au = 1)

P(Au = 1) = αuq

P(Eu = 1) = γru



Cascade model

document urdocument ur�1

Er�1

Cr�1

Ar�1

Er

Cr

Ar

......

↵ur�1q ↵urq



Cascade model

Er = 1 and Ar = 1⇔ Cr = 1

P(Ar = 1) = αurq

P(E1 = 1) = 1

P(Er = 1 | Er−1 = 0) = 0

P(Er = 1 | Cr−1 = 1) = 0

P(Er = 1 | Er−1 = 1,Cr−1 = 0) = 1



Dependent click model


Er�1

Cr�1

Ar�1

Er

Cr

Ar

......

↵ur�1q ↵urq

Sr�1 Sr

�r�1 �r



Dependent click model

P(Er = 1 | Sr−1 = 1) = 0

P(Er = 1 | Er−1 = 1,Sr−1 = 0) = 1

P(Sr = 1 | Cr = 0) = 0

P(Sr = 1 | Cr = 1) = 1− λr



Dynamic Bayesian network model


Er�1

Cr�1

Ar�1

Er

Cr

Ar

......

↵ur�1q ↵urq

Sr�1 Sr

�

�ur�1q �urq



Dynamic Bayesian network model

P(Er = 1 | Sr−1 = 1) = 0

P(Er = 1 | Er−1 = 1, Sr−1 = 0) = γ

P(Sr = 1 | Cr = 1) = σurq



User browsing model

document ur

Er

Cr

Ar

...

↵urq

�rr0



User browsing model

P(Er = 1 | Cr ′ = 1,

Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′



Basic click models summary

Model P(Au = 1) P(Su = 1 | Cu = 1) P(Eu = 1 | E<ru ,S<ru ,C<ru )

RCM constant (ρ) N/A constant (1)RCTR constant (1) N/A rank (ρr )DCTR query-document (ρuq) N/A constant (1)PBM query-doc (αuq) N/A rank (γr )CM query-doc (αuq) constant (1) constant (1 or 0)DCM query-doc (αuq) rank (1− λr ) constant (1 or 0)DBN query-doc (αuq) query-doc (σuq) constant (γ)UBM query-doc (αuq) N/A other (γrr′ )




1 Introduction

2 Motivation




6 Recap



Click probabilities

Full probability:P(Cr = 1)

Conditional probability:

P(Cr = 1 | C1, . . .Cr−1)



Click probabilities



P(Cr = 1 | C1, . . .Cr−1)



Click probabilities



P(Cr = 1 | C1, . . .Cr−1)



Full click probability

P(Cu = 1) = P(Cu = 1 | Eru = 1) · P(Eru = 1) = αuqεru

εr+1 = P(Er+1 = 1)

= P(Er = 1) · P(Er+1 = 1 | Er = 1)

= εr ·(P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1) +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1))

DBN: εr+1 = εr ((1− σuq) γαuq + γ (1− αuq))





εr+1 = P(Er+1 = 1)

= P(Er = 1) · P(Er+1 = 1 | Er = 1)

= εr ·(P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1) +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1))






εr+1 = P(Er+1 = 1)

= P(Er = 1) · P(Er+1 = 1 | Er = 1)

= εr ·(P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1) +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1))






εr+1 = P(Er+1 = 1)

= P(Er = 1) · P(Er+1 = 1 | Er = 1)

= εr ·(P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1) +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1))






εr+1 = P(Er+1 = 1)

= P(Er = 1) · P(Er+1 = 1 | Er = 1)

= εr ·(P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1) +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1))




Conditional click probability

P(Cu = 1 | C<ru ) = P(Cu = 1 | Eru = 1,C<ru ) · P(Eru = 1 | C<ru )

= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,Cr = 1) · P(Er = 1 | Cr = 1,C<r ) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · P(Er = 1 | Cr = 0,C<r ) · (1− c(s)r )

= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)

1− αurqεr(1− c(s)r )

DCM: εr+1 = λrc(s)r +

(1− αurq)εr1− αurqεr

(1− c(s)r )





= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)



= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)




(1− c(s)r )





= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)



= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)




(1− c(s)r )





= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)



= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)




(1− c(s)r )





= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)



= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)




(1− c(s)r )





= αuqεru

εr+1 = P(Er+1 = 1 | C<r+1)

= P(Er+1 = 1 | Er = 1,C<r+1) · P(Er = 1 | C<r+1)



= P(Er+1 = 1 | Er = 1,Cr = 1) · c(s)r +

P(Er+1 = 1 | Er = 1,Cr = 0) · εr (1− αurq)




(1− c(s)r )




1 Introduction

2 Motivation




6 Recap



Click model

Set of events/random variables

Set of dependencies between these events

Correspondence between the model’s parameters and featuresof a query and results



Click model






Click model






Parameter estimation

Maximum likelihood estimationExpectation maximizationAlternative estimation methods



MLE for random click model

P(Cu = 1) = ρ

L =∏s∈S

∏u∈s

ρc(s)u (1− ρ)1−c

(s)u

LL =∑s∈S

∑u∈s

(c(s)u log(ρ) + (1− c

(s)u ) log(1− ρ)

)

ρ =

∑s∈S

∑u∈s c

(s)u∑

s∈S |s|




P(Cu = 1) = ρ

L =∏s∈S

∏u∈s

ρc(s)u (1− ρ)1−c

(s)u

LL =∑s∈S

∑u∈s

(c(s)u log(ρ) + (1− c

(s)u ) log(1− ρ)

)

ρ =

∑s∈S

∑u∈s c

(s)u∑

s∈S |s|




P(Cu = 1) = ρ

L =∏s∈S

∏u∈s

ρc(s)u (1− ρ)1−c

(s)u

LL =∑s∈S

∑u∈s

(c(s)u log(ρ) + (1− c

(s)u ) log(1− ρ)

)

ρ =

∑s∈S

∑u∈s c

(s)u∑

s∈S |s|




P(Cu = 1) = ρ

L =∏s∈S

∏u∈s

ρc(s)u (1− ρ)1−c

(s)u

LL =∑s∈S

∑u∈s

(c(s)u log(ρ) + (1− c

(s)u ) log(1− ρ)

)

ρ =

∑s∈S

∑u∈s c

(s)u∑

s∈S |s|



MLE for dependent click model

P(Au = 1) = αuq

P(Er = 1 | Er−1 = 1,Sr−1) = 1− Sr−1

P(Sr = 1 | Cr = 0) = 0

P(Sr = 1 | Cr = 1) = 1− λrCu = 1⇔ Eru = 1,Au = 1

Sr = 1 ⇐⇒ r = l ,

where l is the last-clicked rank




P(Au = 1) = αuq

P(Er = 1 | Er−1 = 1,Sr−1) = 1− Sr−1

P(Sr = 1 | Cr = 0) = 0

P(Sr = 1 | Cr = 1) = 1− λrCu = 1⇔ Eru = 1,Au = 1

Sr = 1 ⇐⇒ r = l ,

where l is the last-clicked rank



MLE for dependent click model: Attractiveness

P(Au = 1) = αuq

∀r ≤ l : Aur = 1 ⇐⇒ Cur = 1

L(αuq) =∏

s∈Suq

αI(A(s)

u =1)uq (1− αuq)1−I(A

(s)u =1)

LL(αuq) =∑s∈Suq

(I(. . . ) log(αuq) + (1− I(. . . )) log(1− αuq))

αuq =

∑s∈Suq I(A

(s)u = 1)

|Suq|=

∑s∈Suq I(C

(s)u = 1)

|Suq|=

∑s∈Suq c

(s)u

|Suq|




P(Au = 1) = αuq

∀r ≤ l : Aur = 1 ⇐⇒ Cur = 1

L(αuq) =∏

s∈Suq

αI(A(s)

u =1)uq (1− αuq)1−I(A

(s)u =1)


(I(. . . ) log(αuq) + (1− I(. . . )) log(1− αuq))

αuq =

∑s∈Suq I(A

(s)u = 1)

|Suq|=

∑s∈Suq I(C

(s)u = 1)

|Suq|=

∑s∈Suq c

(s)u

|Suq|




P(Au = 1) = αuq

∀r ≤ l : Aur = 1 ⇐⇒ Cur = 1

L(αuq) =∏

s∈Suq

αI(A(s)

u =1)uq (1− αuq)1−I(A

(s)u =1)


(I(. . . ) log(αuq) + (1− I(. . . )) log(1− αuq))

αuq =

∑s∈Suq I(A

(s)u = 1)

|Suq|=

∑s∈Suq I(C

(s)u = 1)

|Suq|=

∑s∈Suq c

(s)u

|Suq|



MLE for dependent click model: Continuation

P(Sr = 0) = λr

Sr = 1 ⇐⇒ Cr = 1, r = l

L(λr ) =∏s∈Sr

λI(S(s)

r =0)r (1− λr )1−I(S

(s)r =1)

λr =

∑s∈Sr I(S

(s)r = 0)

|Sr |=

∑s∈Sr (1− I(r = l))

|Sr |

=

∑s∈Sr I(r 6= l)

|Sr |




P(Sr = 0) = λr

Sr = 1 ⇐⇒ Cr = 1, r = l

L(λr ) =∏s∈Sr

λI(S(s)

r =0)r (1− λr )1−I(S

(s)r =1)

λr =

∑s∈Sr I(S

(s)r = 0)

|Sr |=

∑s∈Sr (1− I(r = l))

|Sr |

=

∑s∈Sr I(r 6= l)

|Sr |




P(Sr = 0) = λr

Sr = 1 ⇐⇒ Cr = 1, r = l

L(λr ) =∏s∈Sr

λI(S(s)

r =0)r (1− λr )1−I(S

(s)r =1)

λr =

∑s∈Sr I(S

(s)r = 0)

|Sr |

=

∑s∈Sr (1− I(r = l))

|Sr |

=

∑s∈Sr I(r 6= l)

|Sr |




P(Sr = 0) = λr

Sr = 1 ⇐⇒ Cr = 1, r = l

L(λr ) =∏s∈Sr

λI(S(s)

r =0)r (1− λr )1−I(S

(s)r =1)

λr =

∑s∈Sr I(S

(s)r = 0)

|Sr |=

∑s∈Sr (1− I(r = l))

|Sr |

=

∑s∈Sr I(r 6= l)

|Sr |




P(Sr = 0) = λr

Sr = 1 ⇐⇒ Cr = 1, r = l

L(λr ) =∏s∈Sr

λI(S(s)

r =0)r (1− λr )1−I(S

(s)r =1)

λr =

∑s∈Sr I(S

(s)r = 0)

|Sr |=

∑s∈Sr (1− I(r = l))

|Sr |

=

∑s∈Sr I(r 6= l)

|Sr |




αuq =1

|Suq|∑s∈Suq

c(s)u

λr =1

|Sr |∑s∈Sr

I(r 6= l)



Expectation maximization

LL =∑s∈S

log

(∑X

P(

X,C(s) | Ψ))

Q =∑s∈S

EX|C(s),Ψ

[logP

(X,C(s) | Ψ

)]



Expectation maximization

LL =∑s∈S

log

(∑X

P(

X,C(s) | Ψ))

Q =∑s∈S

EX|C(s),Ψ

[logP

(X,C(s) | Ψ

)]



Expectation maximization: E-step (grouping)

Each node X depends only on itsparents P(X )

P(C ) = {A,B}

P(A) = ∅

A

C

B

E

D





P(C ) = {A,B}

P(A) = ∅

A

C

B

E

D





P(C ) = {A,B}

P(A) = ∅

A

C

B

E

D




Grouping Q around θc :

Q(θc) =∑s∈S

EX|C(s),Ψ

[logP

(X,C(s) | Ψ

)]

=∑s∈S

EX|C(s),Ψ

[∑ci∈s

(I(X

(s)ci = 1,P(X

(s)ci ) = p

)log(θc) +

I(X

(s)ci = 0,P(X

(s)ci ) = p

)log(1− θc)

)+ Z

]

=∑s∈S

∑ci∈s

(P(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)log(θc) +

P(X

(s)ci = 0,P(X

(s)ci ) = p | C(s),Ψ

)log(1− θc)

)+ Z,





Q(θc) =∑s∈S

EX|C(s),Ψ

[logP

(X,C(s) | Ψ

)]=∑s∈S

EX|C(s),Ψ

[∑ci∈s

(I(X

(s)ci = 1,P(X

(s)ci ) = p

)log(θc) +

I(X

(s)ci = 0,P(X

(s)ci ) = p

)log(1− θc)

)+ Z

]

=∑s∈S

∑ci∈s

(P(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)log(θc) +

P(X

(s)ci = 0,P(X

(s)ci ) = p | C(s),Ψ

)log(1− θc)

)+ Z,





Q(θc) =∑s∈S

EX|C(s),Ψ

[logP

(X,C(s) | Ψ

)]=∑s∈S

EX|C(s),Ψ

[∑ci∈s

(I(X

(s)ci = 1,P(X

(s)ci ) = p

)log(θc) +

I(X

(s)ci = 0,P(X

(s)ci ) = p

)log(1− θc)

)+ Z

]

=∑s∈S

∑ci∈s

(P(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)log(θc) +

P(X

(s)ci = 0,P(X

(s)ci ) = p | C(s),Ψ

)log(1− θc)

)+ Z,



Expectation maximization: M-step

ESS(x) =∑s∈S

∑ci∈s

P(X

(s)ci = x ,P(X

(s)ci ) = p | C(s),Ψ

),

∂Q(θc)

∂θc

=∑s∈S

∑ci∈s

(P(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)θc

−

P(X

(s)ci = 0,P(X

(s)ci ) = p | C(s),Ψ

)1− θc

)

= 0.




ESS(x) =∑s∈S

∑ci∈s

P(X

(s)ci = x ,P(X

(s)ci ) = p | C(s),Ψ

),

∂Q(θc)

∂θc

=∑s∈S

∑ci∈s

(P(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)θc

−

P(X

(s)ci = 0,P(X

(s)ci ) = p | C(s),Ψ

)1− θc

)

= 0.




θ(t+1)c =

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s∑x=1

x=0 P(X

(s)ci = x ,P(X

(s)ci ) = p | C(s),Ψ

)

=

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s P(P(X

(s)ci ) = p | C(s),Ψ

)=

ESS (t)(1)

ESS (t)(1) + ESS (t)(0)




θ(t+1)c =

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s∑x=1

x=0 P(X

(s)ci = x ,P(X

(s)ci ) = p | C(s),Ψ

)

=

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s P(P(X

(s)ci ) = p | C(s),Ψ

)

=ESS (t)(1)

ESS (t)(1) + ESS (t)(0)




θ(t+1)c =

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s∑x=1

x=0 P(X

(s)ci = x ,P(X

(s)ci ) = p | C(s),Ψ

)

=

∑s∈S

∑ci∈s P

(X

(s)ci = 1,P(X

(s)ci ) = p | C(s),Ψ

)∑

s∈S∑

ci∈s P(P(X

(s)ci ) = p | C(s),Ψ

)=

ESS (t)(1)

ESS (t)(1) + ESS (t)(0)



EM for user browsing model

document ur

Er

Cr

Ar

...

↵urq

�rr0

P(Au = 1) = αuq

P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Au) = ∅P(Er ) = {C1, . . . ,Cr−1}




document ur

Er

Cr

Ar

...

↵urq

�rr0

P(Au = 1) = αuq

P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Au) = ∅

P(Er ) = {C1, . . . ,Cr−1}




document ur

Er

Cr

Ar

...

↵urq

�rr0

P(Au = 1) = αuq

P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Au) = ∅P(Er ) = {C1, . . . ,Cr−1}



EM for user browsing model: Attractiveness

P(Au = 1) = αuq

P(Au = 1,P(Au) = p | C) = P(Au = 1 | C)

P(P(Au) = p | C) = 1

α(t+1)uq =

∑s∈Suq P(Au = 1 | C)∑

s∈Suq 1=

1

|Suq|∑s∈Suq

P(Au = 1 | C)




P(Au = 1) = αuq

P(Au = 1,P(Au) = p | C) = P(Au = 1 | C)

P(P(Au) = p | C) = 1

α(t+1)uq =

∑s∈Suq P(Au = 1 | C)∑

s∈Suq 1=

1

|Suq|∑s∈Suq

P(Au = 1 | C)




P(Au = 1) = αuq

P(Au = 1,P(Au) = p | C) = P(Au = 1 | C)

P(P(Au) = p | C) = 1

α(t+1)uq =

∑s∈Suq P(Au = 1 | C)∑

s∈Suq 1=

1

|Suq|∑s∈Suq

P(Au = 1 | C)




P(Au = 1 | C) = P(Au = 1 | Cu)

= I(Cu = 1)P(Au = 1 | Cu = 1) +

I(Cu = 0)P(Au = 1 | Cu = 0)

= cu + (1− cu)P(Cu = 0 | Au = 1)P(Au = 1)

P(Cu = 0)

= cu + (1− cu)(1− γrr ′)αuq

1− γrr ′αuq




P(Au = 1 | C) = P(Au = 1 | Cu)

= I(Cu = 1)P(Au = 1 | Cu = 1) +

I(Cu = 0)P(Au = 1 | Cu = 0)

= cu + (1− cu)P(Cu = 0 | Au = 1)P(Au = 1)

P(Cu = 0)

= cu + (1− cu)(1− γrr ′)αuq

1− γrr ′αuq




P(Au = 1 | C) = P(Au = 1 | Cu)

= I(Cu = 1)P(Au = 1 | Cu = 1) +

I(Cu = 0)P(Au = 1 | Cu = 0)

= cu + (1− cu)P(Cu = 0 | Au = 1)P(Au = 1)

P(Cu = 0)

= cu + (1− cu)(1− γrr ′)αuq

1− γrr ′αuq




P(Au = 1 | C) = P(Au = 1 | Cu)

= I(Cu = 1)P(Au = 1 | Cu = 1) +

I(Cu = 0)P(Au = 1 | Cu = 0)

= cu + (1− cu)P(Cu = 0 | Au = 1)P(Au = 1)

P(Cu = 0)

= cu + (1− cu)(1− γrr ′)αuq

1− γrr ′αuq




α(t+1)uq =

1

|Suq|∑s∈Suq

(c(s)u + (1− c

(s)u )

(1− γ(t)rr ′ )α(t)uq

1− γ(t)rr ′ α(t)uq

)



EM for user browsing model: Examination

P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Er ) = {C1, . . . ,Cr−1}p = [c1, . . . , cr ′−1, 1, 0, . . . , 0]

Srr ′ = {s : cr ′ = 1, cr ′+1 = 0, . . . , cr−1 = 0}

P(Er = x ,P(Er ) = p | C) = P(Er = x | C)

P(P(Er ) = p | C) = 1




P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Er ) = {C1, . . . ,Cr−1}p = [c1, . . . , cr ′−1, 1, 0, . . . , 0]

Srr ′ = {s : cr ′ = 1, cr ′+1 = 0, . . . , cr−1 = 0}

P(Er = x ,P(Er ) = p | C) = P(Er = x | C)

P(P(Er ) = p | C) = 1




P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Er ) = {C1, . . . ,Cr−1}p = [c1, . . . , cr ′−1, 1, 0, . . . , 0]

Srr ′ = {s : cr ′ = 1, cr ′+1 = 0, . . . , cr−1 = 0}

P(Er = x ,P(Er ) = p | C) = P(Er = x | C)

P(P(Er ) = p | C) = 1




P(Er = 1 | Cr ′ = 1,Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′

P(Er ) = {C1, . . . ,Cr−1}p = [c1, . . . , cr ′−1, 1, 0, . . . , 0]

Srr ′ = {s : cr ′ = 1, cr ′+1 = 0, . . . , cr−1 = 0}

P(Er = x ,P(Er ) = p | C) = P(Er = x | C)

P(P(Er ) = p | C) = 1




γ(t+1)rr ′ =

∑s∈Srr′

P(Er = 1 | C)∑s∈Srr′

1=

1

|Srr ′ |∑s∈Srr′

P(Er = 1 | C)

P(Er = 1 | C) = cu + (1− cu)γrr ′(1− αuq)

1− γrr ′αuq




γ(t+1)rr ′ =

∑s∈Srr′

P(Er = 1 | C)∑s∈Srr′

1=

1


P(Er = 1 | C)

P(Er = 1 | C) = cu + (1− cu)γrr ′(1− αuq)

1− γrr ′αuq




γ(t+1)rr ′ =

1


(c(s)u + (1− c

(s)u )

γ(t)rr ′ (1− α(t)

uq )


)




α(t+1)uq =

1

|Suq|∑s∈Suq

(c(s)u + (1− c

(s)u )

(1− γ(t)rr ′ )α(t)uq


)

γ(t+1)rr ′ =

1


(c(s)u + (1− c

(s)u )

γ(t)rr ′ (1− α(t)

uq )


)



Alternative estimation methods

Bayesian inference

Probit link

Matrix factorization




Bayesian inference

Probit link





Bayesian inference

Probit link





1 Introduction

2 Motivation




6 Recap



Morning block 1 – Recap

Basic click models as probabilistic graphical models

MLE: Precise parameter estimation method for simple cases

EM: Approximate parameter estimation for complex cases

After the break

DemoEvaluationData and toolsExperimental results



Morning block 1 – Recap

Basic click models as probabilistic graphical models

MLE: Precise parameter estimation method for simple cases

EM: Approximate parameter estimation for complex cases

After the break

DemoEvaluationData and toolsExperimental results



Acknowledgments

All content represents the opinion of the authors which is not necessarily shared orendorsed by their respective employers and/or sponsors.


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An Introduction to Click Models for Web Search - (Morning ...clickmodels.weebly.com/.../5/2/2/...introduction-1.pdf · An Introduction to Click Models for Web Search (Morning block