an introduction to click models for web search - (morning...
TRANSCRIPT
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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§
[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
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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
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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
![Page 4: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/10.jpg)
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
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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
![Page 12: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/12.jpg)
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
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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
![Page 14: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/16.jpg)
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
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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
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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
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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
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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
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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
![Page 22: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader035.vdocuments.us/reader035/viewer/2022070811/5f09c4427e708231d4286827/html5/thumbnails/26.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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)
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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))
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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
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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
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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
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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
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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 )
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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 )
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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 )
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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
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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 )
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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 )
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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|
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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