Click Models for Web Search - Lecture 2 · Click Models for Web Search Lecture 2 Aleksandr...
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Click probabilities Applications Parameter estimation
Click Models for Web SearchLecture 2
Aleksandr Chuklin§,¶ Ilya Markov§ Maarten de Rijke§
[email protected] [email protected] [email protected]
§University of Amsterdam¶Google Research Europe
AC–IM–MdR Click Models for Web Search 1
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Click probabilities Applications Parameter estimation
Course overview
Basic Click Models
Parameter Estimation Evaluation
Data and ToolsResultsApplications
Advanced Models
Recent Studies
Future Research
AC–IM–MdR Click Models for Web Search 2
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Click probabilities Applications Parameter estimation
Lecture 1
Basic Click Models
Parameter Estimation Evaluation
Data and ToolsResultsApplications
Advanced Models
Recent Studies
Future Research
AC–IM–MdR Click Models for Web Search 3
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Click probabilities Applications Parameter estimation
Lecture 1 recap
CTR models: counting clicks
Position-based model (PBM): examination and attractiveness
Cascade model (CM): previous examinations and clicks matter
Dynamic Bayesian network model (DBN): satisfactoriness
User browsing model (UBM): rank of previous click
AC–IM–MdR Click Models for Web Search 4
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Click probabilities Applications Parameter estimation
Lecture 2
Basic Click Models
Parameter Estimation Evaluation
Data and ToolsResultsApplications
Advanced Models
Recent Studies
Future Research
AC–IM–MdR Click Models for Web Search 5
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Click probabilities Applications Parameter estimation
Probability theory
Partitioned probability: A = A1 ∪ A2, A1 ∩ A2 = ∅
P(A) = P(A1,A2) = P(A1) + P(A2)
Bayes’ rule
P(A | B) · P(B) = P(B | A) · P(A)
B causes A: B → A
P(B) = P(B | A) · P(A)
AC–IM–MdR Click Models for Web Search 6
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Click probabilities Applications Parameter estimation
Probability theory (cont’d)
B → A, A = A1 ∪ A2, A1 ∩ A2 = ∅
P(B) = P(B | A) · P(A)
= P(B | A1,A2) · P(A1,A2)
= P(B | A1,A2) · (P(A1) + P(A2))
= P(B | A1,A2) · P(A1) + P(B | A1,A2) · P(A2)
= P(B | A1) · P(A1) + P(B | A2) · P(A2)
P(B) = P(B | A1) · P(A1) + P(B | A2) · P(A2)
AC–IM–MdR Click Models for Web Search 7
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Click probabilities Applications Parameter estimation
Lecture outline
1 Click probabilities
2 Applications
3 Parameter estimation
AC–IM–MdR Click Models for Web Search 8
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Click probabilities Applications Parameter estimation
Click probabilities
Full probability – probabilitythat a user clickson a document at rank r
P(Cr = 1)
Conditional probability –probability that a user clickson a document at rank rgiven previous clicks
P(Cr = 1 | C1, . . . ,Cr−1)
AC–IM–MdR Click Models for Web Search 9
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Click probabilities Applications Parameter estimation
Dependency between examination and clicks
document u
Eu
Cu
Au
↵uq�ru
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Click probabilities Applications Parameter estimation
Full click probability
P(Cr = 1) = +P(Cr = 1 | Er = 1) · P(Er = 1)
P(Cr = 1 | Er = 0) · P(Er = 0)
= P(Aur = 1) · P(Er = 1) + 0
= αurqεr
AC–IM–MdR Click Models for Web Search 11
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Click probabilities Applications Parameter estimation
Cascade models: dependency between examinations
document urdocument ur�1
Er�1
Cr�1
Ar�1
Er
Cr
Ar
......
↵ur�1q ↵urq
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Click probabilities Applications Parameter estimation
Full click probability
P(Cr = 1) = P(Aur = 1) · P(Er = 1) = αurqεr
εr+1 = P(Er+1 = 1)
= +P(Er = 1) · P(Er+1 = 1 | Er = 1)
P(Er = 0) · P(Er+1 = 1 | Er = 0)
= εr · P(Er+1 = 1 | Er = 1) + 0
= 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)
)
AC–IM–MdR Click Models for Web Search 13
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Click probabilities Applications Parameter estimation
Full click probability: Dynamic Bayesian network model
Dynamic Bayesian network model: satisfactoriness
document urdocument ur�1
Er�1
Cr�1
Ar�1
Er
Cr
Ar
......
↵ur�1q ↵urq
Sr�1 Sr
�
�ur�1q �urq
P(Cr+1 = 1) = αur+1qεr ·
(+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)
)
P(Cr+1 = 1) = αur+1qεr ·
(+
(1− σurq)γ · αurq
γ · (1− αurq)
)AC–IM–MdR Click Models for Web Search 14
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Click probabilities Applications Parameter estimation
Conditional click probability
P(Cr = 1 | C1, . . . ,Cr−1) = P(Cr = 1 | C<r )
= +P(Cr = 1 | Er = 1,C<r ) · P(Er = 1 | C<r )
P(Cr = 1 | Er = 0,C<r ) · P(Er = 0 | C<r )
= P(Aur = 1) · P(Er = 1 | C<r ) + 0
= αurqεr
εr+1 = +
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 )
c(s)r – a click on rank r in query session s
AC–IM–MdR Click Models for Web Search 15
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Click probabilities Applications Parameter estimation
Click probabilities summary
Full probability
P(Cr+1 = 1) =
αur+1qεr ·
(+P(Er+1 = 1 | Er = 1,Cr = 1) · P(Cr = 1 | Er = 1)
P(Er+1 = 1 | Er = 1,Cr = 0) · P(Cr = 0 | Er = 1)
)
Conditional probability
P(Cr+1 = 1 | C1, . . . ,Cr )
= αur+1q ·
+
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 )
AC–IM–MdR Click Models for Web Search 16
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Click probabilities Applications Parameter estimation
What do click models give us?
General:
Understanding of user behavior
Specific:
Conditional click probabilities
Full click probabilities
Attractiveness and satisfactoriness for query-document pairs
AC–IM–MdR Click Models for Web Search 17
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Click probabilities Applications Parameter estimation
Lecture outline
1 Click probabilities
2 ApplicationsUser interaction analysisSimulating usersModel-based metricsApproximating document relevance
3 Parameter estimation
AC–IM–MdR Click Models for Web Search 18
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Click probabilities Applications Parameter estimation
Lecture outline
2 ApplicationsUser interaction analysisSimulating usersModel-based metricsApproximating document relevance
AC–IM–MdR Click Models for Web Search 19
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Click probabilities Applications Parameter estimation
User interaction analysis
Random click model (global CTR): ρ = 0.122
Rank-based CTR:ρ1 = 0.429, ρ2 = 0.190, ρ3 = 0.136, . . . , ρ10 = 0.048
Position-based model:γ1 = 0.998, γ2 = 0.939, γ3 = 0.759, . . . , γ10 = 0.260
Dynamic Bayesian network model: γ = 0.9997
Click models are trained on the first 10K sessions of the WSCD 2012 dataset.
AC–IM–MdR Click Models for Web Search 20
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Click probabilities Applications Parameter estimation
Lecture outline
2 ApplicationsUser interaction analysisSimulating usersModel-based metricsApproximating document relevance
AC–IM–MdR Click Models for Web Search 21
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Click probabilities Applications Parameter estimation
Simulating users
Algorithm Simulating user clicks
Input: click model M, query session sOutput: vector of simulated clicks (c1, . . . , cn)
1: for r ← 1 to |s| do2: Pr ← PM(Cr = 1 | C1 = c1, . . . ,Cr−1 = cr−1)︸ ︷︷ ︸
conditional click probability
3: Generate cr from Bernoulli(Pr )4: end for
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Click probabilities Applications Parameter estimation
Lecture outline
2 ApplicationsUser interaction analysisSimulating usersModel-based metricsApproximating document relevance
AC–IM–MdR Click Models for Web Search 23
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Click probabilities Applications Parameter estimation
Model-based metrics
Utility-based metrics
uMetric =n∑
r=1
P(Cr = 1)·Ur
Effort-based metrics
eMetric =n∑
r=1
P(Sr = 1) ·Fr
AC–IM–MdR Click Models for Web Search 24
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Click probabilities Applications Parameter estimation
Expected reciprocal rank
RR =1
r, where Sr = 1
ERR =∑r
1
r· P(Sr = 1)
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Click probabilities Applications Parameter estimation
Dynamic Bayesian network model (DBN)
P(Ar = 1) = αurq
P(E1 = 1) = 1
P(Er = 1 | Sr−1 = 1) = 0
P(Er = 1 | Sr−1 = 0) = γ
P(Sr = 1 | Cr = 0) = 0
P(Sr = 1 | Cr = 1) = σurq
P(Sr = 1) =?
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Click probabilities Applications Parameter estimation
DBN: Satisfaction
P(Sr = 1) = P(Sr = 1 | Cr = 1) · P(Cr = 1)
= σurq · P(Cr = 1)
= σurq · αurq · P(Er = 1)
= σurq · αurq ·r−1∏i=1
(γ · (1− σuiq · αuiq)
)= Rurq ·
r−1∏i=1
(γ · (1− Ruiq)
)
AC–IM–MdR Click Models for Web Search 27
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Click probabilities Applications Parameter estimation
Expected reciprocal rank
ERR =∑r
1
r· P(Sr = 1)
=∑r
1
r· Rurq ·
r−1∏i=1
(γ · (1− Ruiq)
)
AC–IM–MdR Click Models for Web Search 28
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Click probabilities Applications Parameter estimation
Model-based metrics
Model-based metric
Click model Utility-based Effort-based
DBN uSDBN ERRDBN EBU rrDBNUBM uUBM –
AC–IM–MdR Click Models for Web Search 29
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Click probabilities Applications Parameter estimation
Lecture outline
2 ApplicationsUser interaction analysisSimulating usersModel-based metricsApproximating document relevance
AC–IM–MdR Click Models for Web Search 30
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Click probabilities Applications Parameter estimation
Approximating document relevance
αu1q σu1q
αu2q σu2q
αu3q σu3q
αu4q σu4q
αu5q σu5q
PBM, UBM DBN
AC–IM–MdR Click Models for Web Search 31
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Click probabilities Applications Parameter estimation
Approximating document relevance
Clicks are affected by rank =⇒ do not represent documentrelevance directly
Attractiveness and satisfactoriness do not depend on rank =⇒can be used as indicators of document relevance
They are used by search engines as retrieval features
Documents can simply be ranked by αuq, σuq, or αuqσuq
AC–IM–MdR Click Models for Web Search 32
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Click probabilities Applications Parameter estimation
Applications summary
Click model’s output Application
Understanding of user behavior User interaction analysisConditional click probabilities User simulationFull click probabilities Model-based metricsParameter values Ranking
AC–IM–MdR Click Models for Web Search 33
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Click probabilities Applications Parameter estimation
Lecture outline
1 Click probabilities
2 Applications
3 Parameter estimationMaximum likelihood estimationExpectation maximizationExpectation maximization examplesAlternative estimation methods
AC–IM–MdR Click Models for Web Search 34
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Click probabilities Applications Parameter estimation
Lecture outline
3 Parameter estimationMaximum likelihood estimationExpectation maximizationExpectation maximization examplesAlternative estimation methods
AC–IM–MdR Click Models for Web Search 35
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Click probabilities Applications Parameter estimation
MLE for random click model
P(Cu = 1) = ρ
L =∏s∈S
∏u∈s
ρc(s)u (1− ρ)1−c(s)
u
︸ ︷︷ ︸likelihood of Bernoulli random variable
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|=
# clicks
# shown docs
AC–IM–MdR Click Models for Web Search 36
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Click probabilities Applications Parameter estimation
Lecture outline
3 Parameter estimationMaximum likelihood estimationExpectation maximizationExpectation maximization examplesAlternative estimation methods
AC–IM–MdR Click Models for Web Search 37
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Click probabilities Applications Parameter estimation
Expectation maximization
1 Set parameters to some initial values2 Repeat until convergence
E-step: derive the expectation of the likelihood functionM-step: maximize this expectation
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Click probabilities Applications Parameter estimation
EM terminology
θc – parameter of a click model
Xc – random variablecorresponding to θc
P(Xc) – parents of Xc
Examples:
P(C ) = {A,B}P(A) = ∅
A
C
B
E
DA
C
B
E
DA
C
B
E
D
AC–IM–MdR Click Models for Web Search 39
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Click probabilities Applications Parameter estimation
EM objective
Find the value of parameter θcthat optimizes log-likelihood LL of the model
given observed query sessions S
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Click probabilities Applications Parameter estimation
E-step
LL =∑s∈S
log
(∑X
P(
X,C(s) | Ψ))
X – all random variables
Ψ – all parameters
C(s) – clicks in a query session s
Q =∑s∈S
EX|C(s),Ψ
[logP
(X,C(s) | Ψ
)]
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Click probabilities Applications Parameter estimation
E-step (grouping)
Q(θc) =∑s∈S
EX|C(s),Ψ
[logP
(X,C(s) | Ψ
)]=∑s∈S
EX|C(s),Ψ
[logP
(X (s)c ,P(X (s)
c ) = p)
+ Z]
=∑s∈S
EX|C(s),Ψ
[∑ci∈s
(I(X (s)ci = 1,P
)log(θc) +
I(X (s)ci = 0,P
)log(1− θc)
)+ Z
]
=∑s∈S
∑ci∈s
(P(X (s)ci = 1,P | C(s),Ψ
)log(θc) +
P(X (s)ci = 0,P | C(s),Ψ
)log(1− θc)
)+ Z
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Click probabilities Applications Parameter estimation
M-step
∂Q(θc)
∂θc=∑s∈S
∑ci∈s
(P(Y
(s)ci = 1)
θc− P(Y
(s)ci = 0)
1− θc
)= 0
θ(t+1)c =
∑s∈S
∑ci∈s P(Y
(s)ci = 1)∑
s∈S∑
ci∈s∑x=1
x=0 P(Y(s)ci = x)
=
∑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
)
Probabilities are computed using parameter values θ(t)c
calculated on previous iteration t
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Click probabilities Applications Parameter estimation
EM summary
Q(θc ) =∑s∈S
EX|C(s),Ψ
[log P
(X, C(s) | Ψ
)]
=∑s∈S
EX|C(s),Ψ
[ ∑ci∈s
(I(X (s)ci
= 1,P(X (s)ci
) = p)
log(θc ) + I(X (s)ci
= 0,P(X (s)ci
) = p)
log(1− θc )
)+ Z
]
=∑s∈S
∑ci∈s
(P(X (s)ci
= 1,P(X (s)ci
) = p | C(s),Ψ)
log(θc ) + P(X (s)ci
= 0,P(X (s)ci
) = p | C(s),Ψ)
log(1− θc )
)+ Z
∂Q(θc )
∂θc=∑s∈S
∑ci∈s
(P(X
(s)ci
= 1,P(X(s)ci
) = p | C(s),Ψ)
θc−
P(X
(s)ci
= 0,P(X(s)ci
) = p | C(s),Ψ)
1− θc
)= 0
θ(t+1)c =
∑s∈S
∑ci∈s P
(X
(s)ci
= 1,P(X(s)ci
) = p | C(s),Ψ)
∑s∈S
∑ci∈s
∑x=1x=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)
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Click probabilities Applications Parameter estimation
Lecture outline
3 Parameter estimationMaximum likelihood estimationExpectation maximizationExpectation maximization examplesAlternative estimation methods
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Click probabilities Applications Parameter estimation
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}
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Click probabilities Applications Parameter estimation
EM for User Browsing Model: Attractiveness
P(Au = 1) = αuq, P(Au) = ∅
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)
Suq – sessions initiated by query q and containing document uamong the results
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Click probabilities Applications Parameter estimation
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
cu – a click on document u
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Click probabilities Applications Parameter estimation
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
)
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Click probabilities Applications Parameter estimation
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
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Click probabilities Applications Parameter estimation
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
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Click probabilities Applications Parameter estimation
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
)
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Click probabilities Applications Parameter estimation
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
)
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Click probabilities Applications Parameter estimation
Parameter estimation summary
Maximum likelihood estimation
Parameters estimated directly from dataParameters do not depend on each otherSingle pass over a click logVery efficient but not very effective
Expectation maximization
Parameters depend on each otherIterative estimationEffective but not efficient
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Click probabilities Applications Parameter estimation
Lecture outline
3 Parameter estimationMaximum likelihood estimationExpectation maximizationExpectation maximization examplesAlternative estimation methods
AC–IM–MdR Click Models for Web Search 55
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Click probabilities Applications Parameter estimation
Alternative estimation methods
Bayesian inference
Probit link
Matrix factorization
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Click probabilities Applications Parameter estimation
Course overview
Basic Click Models
Parameter Estimation Evaluation
Data and ToolsResultsApplications
Advanced Models
Recent Studies
Future Research
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Click probabilities Applications Parameter estimation
Next lecture
Basic Click Models
Parameter Estimation Evaluation
Data and ToolsResultsApplications
Advanced Models
Recent Studies
Future Research
AC–IM–MdR Click Models for Web Search 58
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Click probabilities Applications Parameter estimation
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 Click Models for Web Search 59