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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§
a.chuklin@uva.nl i.markov@uva.nl derijke@uva.nl
§University of Amsterdam¶Google Switzerland
SIGIR 2015 Tutorial
AC–IM–MdR An Introduction to Click Models for Web Search 1
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Who are we?
Aleksandr
Ilya Maarten
Software engineer
Postdoc at Professor at
at Google
U. Amsterdam U. Amsterdam
AC–IM–MdR An Introduction to Click Models for Web Search 2
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Who are we?
Aleksandr Ilya
Maarten
Software engineer Postdoc at
Professor at
at Google U. Amsterdam
U. Amsterdam
AC–IM–MdR An Introduction to Click Models for Web Search 2
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Who are we?
Aleksandr Ilya MaartenSoftware engineer Postdoc at Professor at
at Google U. Amsterdam U. Amsterdam
AC–IM–MdR An Introduction to Click Models for Web Search 2
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 3
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 3
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 3
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 3
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 3
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 4
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 4
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 5
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 5
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 5
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 5
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 6
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
What is a click model?
AC–IM–MdR An Introduction to Click Models for Web Search 7
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 8
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 9
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 10
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Why click models?
Understand users
Simulate users
Evaluate search
Improve search
AC–IM–MdR An Introduction to Click Models for Web Search 11
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Why click models?
Understand users
Simulate users
Evaluate search
Improve search
AC–IM–MdR An Introduction to Click Models for Web Search 11
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Why click models?
Understand users
Simulate users
Evaluate search
Improve search
AC–IM–MdR An Introduction to Click Models for Web Search 11
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Why click models?
Understand users
Simulate users
Evaluate search
Improve search
AC–IM–MdR An Introduction to Click Models for Web Search 11
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Why click models?
Understand users
Simulate users
Evaluate search
Improve search
AC–IM–MdR An Introduction to Click Models for Web Search 11
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 12
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Basic click models
Random click modelCTR modelsPosition-based modelCascade modelDependent click modelDynamic Bayesian network modelUser browsing model
AC–IM–MdR An Introduction to Click Models for Web Search 13
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Random click model
?
AC–IM–MdR An Introduction to Click Models for Web Search 14
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Random click model
P(Cu = 1) = ρ
AC–IM–MdR An Introduction to Click Models for Web Search 14
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
CTR models
Rank-based CTR:
P(Cr = 1) = ρr
Document-based CTR:
P(Cu = 1) = ρuq
AC–IM–MdR An Introduction to Click Models for Web Search 15
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Position-based model
document u
Eu
Cu
Au
↵uq�ru
AC–IM–MdR An Introduction to Click Models for Web Search 16
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Position-based model
P(Cu = 1) = P(Eu = 1) · P(Au = 1)
P(Au = 1) = αuq
P(Eu = 1) = γru
AC–IM–MdR An Introduction to Click Models for Web Search 17
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Cascade model
document urdocument ur�1
Er�1
Cr�1
Ar�1
Er
Cr
Ar
......
↵ur�1q ↵urq
AC–IM–MdR An Introduction to Click Models for Web Search 18
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 19
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Dependent click model
document urdocument ur�1
Er�1
Cr�1
Ar�1
Er
Cr
Ar
......
↵ur�1q ↵urq
Sr�1 Sr
�r�1 �r
AC–IM–MdR An Introduction to Click Models for Web Search 20
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 21
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Dynamic Bayesian network model
document urdocument ur�1
Er�1
Cr�1
Ar�1
Er
Cr
Ar
......
↵ur�1q ↵urq
Sr�1 Sr
�
�ur�1q �urq
AC–IM–MdR An Introduction to Click Models for Web Search 22
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 23
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
User browsing model
document ur
Er
Cr
Ar
...
↵urq
�rr0
AC–IM–MdR An Introduction to Click Models for Web Search 24
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
User browsing model
P(Er = 1 | Cr ′ = 1,
Cr ′+1 = 0, . . . ,Cr−1 = 0) = γrr ′
AC–IM–MdR An Introduction to Click Models for Web Search 25
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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′ )
AC–IM–MdR An Introduction to Click Models for Web Search 26
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 27
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Click probabilities
Full probability:P(Cr = 1)
Conditional probability:
P(Cr = 1 | C1, . . .Cr−1)
AC–IM–MdR An Introduction to Click Models for Web Search 28
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Click probabilities
Full probability:P(Cr = 1)
Conditional probability:
P(Cr = 1 | C1, . . .Cr−1)
AC–IM–MdR An Introduction to Click Models for Web Search 28
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Click probabilities
Full probability:P(Cr = 1)
Conditional probability:
P(Cr = 1 | C1, . . .Cr−1)
AC–IM–MdR An Introduction to Click Models for Web Search 28
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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))
AC–IM–MdR An Introduction to Click Models for Web Search 29
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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))
AC–IM–MdR An Introduction to Click Models for Web Search 29
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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))
AC–IM–MdR An Introduction to Click Models for Web Search 29
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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))
AC–IM–MdR An Introduction to Click Models for Web Search 29
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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))
AC–IM–MdR An Introduction to Click Models for Web Search 29
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 )
AC–IM–MdR An Introduction to Click Models for Web Search 30
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 31
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation 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
AC–IM–MdR An Introduction to Click Models for Web Search 32
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation 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
AC–IM–MdR An Introduction to Click Models for Web Search 32
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation 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
AC–IM–MdR An Introduction to Click Models for Web Search 32
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Parameter estimation
Maximum likelihood estimationExpectation maximizationAlternative estimation methods
AC–IM–MdR An Introduction to Click Models for Web Search 33
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 34
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 34
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 34
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 34
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 35
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 35
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 36
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 36
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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|
AC–IM–MdR An Introduction to Click Models for Web Search 36
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 |
AC–IM–MdR An Introduction to Click Models for Web Search 37
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 |
AC–IM–MdR An Introduction to Click Models for Web Search 37
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 |
AC–IM–MdR An Introduction to Click Models for Web Search 37
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 |
AC–IM–MdR An Introduction to Click Models for Web Search 37
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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 |
AC–IM–MdR An Introduction to Click Models for Web Search 37
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
MLE for dependent click model
αuq =1
|Suq|∑s∈Suq
c(s)u
λr =1
|Sr |∑s∈Sr
I(r 6= l)
AC–IM–MdR An Introduction to Click Models for Web Search 38
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization
LL =∑s∈S
log
(∑X
P(
X,C(s) | Ψ))
Q =∑s∈S
EX|C(s),Ψ
[logP
(X,C(s) | Ψ
)]
AC–IM–MdR An Introduction to Click Models for Web Search 39
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization
LL =∑s∈S
log
(∑X
P(
X,C(s) | Ψ))
Q =∑s∈S
EX|C(s),Ψ
[logP
(X,C(s) | Ψ
)]
AC–IM–MdR An Introduction to Click Models for Web Search 39
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 40
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 40
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 40
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: E-step (grouping)
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,
AC–IM–MdR An Introduction to Click Models for Web Search 41
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: E-step (grouping)
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,
AC–IM–MdR An Introduction to Click Models for Web Search 41
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: E-step (grouping)
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,
AC–IM–MdR An Introduction to Click Models for Web Search 41
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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.
AC–IM–MdR An Introduction to Click Models for Web Search 42
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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.
AC–IM–MdR An Introduction to Click Models for Web Search 42
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: M-step
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)
AC–IM–MdR An Introduction to Click Models for Web Search 43
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: M-step
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)
AC–IM–MdR An Introduction to Click Models for Web Search 43
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Expectation maximization: M-step
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)
AC–IM–MdR An Introduction to Click Models for Web Search 43
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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}
AC–IM–MdR An Introduction to Click Models for Web Search 44
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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}
AC–IM–MdR An Introduction to Click Models for Web Search 44
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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}
AC–IM–MdR An Introduction to Click Models for Web Search 44
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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)
AC–IM–MdR An Introduction to Click Models for Web Search 45
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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)
AC–IM–MdR An Introduction to Click Models for Web Search 45
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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)
AC–IM–MdR An Introduction to Click Models for Web Search 45
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Attractiveness
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
AC–IM–MdR An Introduction to Click Models for Web Search 46
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Attractiveness
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
AC–IM–MdR An Introduction to Click Models for Web Search 46
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Attractiveness
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
AC–IM–MdR An Introduction to Click Models for Web Search 46
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Attractiveness
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
AC–IM–MdR An Introduction to Click Models for Web Search 46
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Attractiveness
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
)
AC–IM–MdR An Introduction to Click Models for Web Search 47
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 48
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 48
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 48
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
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
AC–IM–MdR An Introduction to Click Models for Web Search 48
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Examination
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
AC–IM–MdR An Introduction to Click Models for Web Search 49
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Examination
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
AC–IM–MdR An Introduction to Click Models for Web Search 49
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model: Examination
γ(t+1)rr ′ =
1
|Srr ′ |∑s∈Srr′
(c(s)u + (1− c
(s)u )
γ(t)rr ′ (1− α(t)
uq )
1− γ(t)rr ′ α(t)uq
)
AC–IM–MdR An Introduction to Click Models for Web Search 50
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
EM for user browsing model
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
)
γ(t+1)rr ′ =
1
|Srr ′ |∑s∈Srr′
(c(s)u + (1− c
(s)u )
γ(t)rr ′ (1− α(t)
uq )
1− γ(t)rr ′ α(t)uq
)
AC–IM–MdR An Introduction to Click Models for Web Search 51
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Alternative estimation methods
Bayesian inference
Probit link
Matrix factorization
AC–IM–MdR An Introduction to Click Models for Web Search 52
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Alternative estimation methods
Bayesian inference
Probit link
Matrix factorization
AC–IM–MdR An Introduction to Click Models for Web Search 52
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Alternative estimation methods
Bayesian inference
Probit link
Matrix factorization
AC–IM–MdR An Introduction to Click Models for Web Search 52
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Morning block 1 – Outline
1 Introduction
2 Motivation
3 Basic Click Models
4 Click Probabilities
5 Parameter Estimation
6 Recap
AC–IM–MdR An Introduction to Click Models for Web Search 53
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation 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
AC–IM–MdR An Introduction to Click Models for Web Search 54
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation 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
AC–IM–MdR An Introduction to Click Models for Web Search 54
Introduction Motivation Basic Click Models Click Probabilities Parameter Estimation Recap
Acknowledgments
All content represents the opinion of the authors which is not necessarily shared orendorsed by their respective employers and/or sponsors.
AC–IM–MdR An Introduction to Click Models for Web Search 55
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