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Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group The University of the Basque Country ACML, Singapore, November 4th, 2012

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Page 1: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Probabilistic Modeling on Rankings

Jose A. Lozano Ekhine Irurozki

Intelligent Systems GroupThe University of the Basque Country

ACML, Singapore, November 4th, 2012

Page 2: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

Page 3: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Problems we are interested in

Identity-tracking

Page 4: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Problems we are interested in

Preferences

Page 5: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Problems we are interested in

Information retrieval

Page 6: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Probabilistic Modeling on Ranking

RemarksSocial scienceMachine learning: ECML, ICML, UAI, NIPS, JMLRNOT “learning to rank”A model can be explained from many different points ofviewProbabilistic graphical model bias

Page 7: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Permutations

What are permutations?A permutation σ can be seen as a bijection between theset 1,2, . . . ,n onto itself:

σ : 1,2, . . . ,n −→ 1,2, . . . ,ni 7−→ σ(i)

Interpretationσ(i) represents the rank associated to the i-th elementσ−1(i) represents the i-th ranked element

Sn is the set of permutations of n elements, then (Sn,o)forms the symmetric group:

σ1o σ2(i) = σ1(σ2(i))

Page 8: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Permutations

Notation and representation

The identity permutation (1 2 . . . n) will be denoted by eA permutation σ can be equivalently represented as apermutation matrix M = [mij ] where:

mij =

1 if σ(i) = j0 otherwise

Page 9: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Learning probability distributions over permutations

(1 2 3 4 5 6 7)(2 3 1 7 6 5 4)(6 7 1 3 4 2 5)(5 2 7 6 3 4 1)(5 1 7 2 4 3 6)(3 5 7 1 2 4 6)...

p : Sn −→ [0,1]

σ 7−→ p(σ)

Page 10: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Learning probability distributions over permutations

(1 2 3 4 5 6 7− − − − . . .)(2 3 1 7 6 5 4− − − − . . .)(6 7 1 3 4 2 5− − − − . . .)(5 2 7 6 3 4 1− − − − . . .)(5 1 7 2 4 3 6− − − − . . .)(3 5 7 1 2 4 6− − − − . . .)...

p : Sn −→ [0,1]

σ 7−→ p(σ)

Page 11: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Learning probability distributions over permutations

1 2 4, 5

2, 3, 1 7 6, 5, 4

6, 7, 1, 3 4 2

5, 2 7, 6 3, 4 1...

p : Sn −→ [0,1]

σ 7−→ p(σ)

Page 12: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Learning probability distributions over permutations

(1 2 3 4 − − . . .− − 5 6 7)(2 3 1 7 − − . . .− − 6 5 4)(6 7 1 3 − − . . .− − 4 2 5)(5 2 7 6 − − . . .− − 3 4 1)(5 1 7 2 − − . . .− − 4 3 6)(3 5 7 1 − − . . .− − 2 4 6)...

p : Sn −→ [0,1]

σ 7−→ p(σ)

Page 13: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Learning probability distributions over permutations

1 5

1 7

3 4

5 7...

p : Sn −→ [0,1]

σ 7−→ p(σ)

Page 14: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Representing probability distributions overpermutation

ProblemsHow many parameters are needed?

p(1 2 3 4 5) , p(2 1 3 4 5)

p(3 2 1 4 5) , . . .

. . . , p(5 4 3 2 1)

5!− 1 parameters. In general n!− 1

Page 15: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Representing probability distributions overpermutation

ProblemsHow many parameters are needed?

p(1 2 3 4 5) , p(2 1 3 4 5)

p(3 2 1 4 5) , . . .

. . . , p(5 4 3 2 1)

5!− 1 parameters. In general n!− 1

Page 16: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks

X1 X2

X3 X6

X4 X5

p(x) = p(x1) · p(x2) · p(x3|x1, x2) · p(x4|x3) · p(x5|x3, x6) · p(x6)

Page 17: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks

X1θ1 X2 θ2

X3θ3 X6 θ6

X4θ4 X5 θ5

p(x) = p(x1) · p(x2) · p(x3|x1, x2) · p(x4|x3) · p(x5|x3, x6) · p(x6)

Page 18: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks

θ4 = (θ41,θ42) X1θ1 X2 θ2

X3θ3 X6 θ6

X4θ4 X5 θ5

p(x) = p(x1) · p(x2) · p(x3|x1, x2) · p(x4|x3) · p(x5|x3, x6) · p(x6)

Page 19: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks

θ4 = (θ41,θ42)

θ411 = p(X4 = 0 | X3 = 0)θ412 = p(X4 = 1 | X3 = 0)

X1θ1 X2 θ2

X3θ3 X6 θ6

X4θ4 X5 θ5

p(x) = p(x1) · p(x2) · p(x3|x1, x2) · p(x4|x3) · p(x5|x3, x6) · p(x6)

Page 20: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks

θ4 = (θ41,θ42)

θ411 = p(X4 = 0 | X3 = 0)θ412 = p(X4 = 1 | X3 = 0)

X1θ1 X2 θ2

X3θ3 X6 θ6

X4θ4 X5 θ5General case

θijk = p(Xi = xki |Pai = paj

i)

p(x) = p(x1) · p(x2) · p(x3|x1, x2) · p(x4|x3) · p(x5|x3, x6) · p(x6)

Page 21: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1 X1 X2

X3 X4

Page 22: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1 X1 X2

X3 X4

Page 23: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 X2

X3 X4

Page 24: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 X2

X3 X4

Page 25: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 = 1⇒ X4 6= 1

X1 X2

X3 X4

Page 26: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 = 1⇒ X4 6= 1

X1 X2

X3 X4

Page 27: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 = 1⇒ X4 6= 1

X1 X2

X3 X4

Page 28: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Bayesian networks for permutations

X1 = 1⇒ X2 6= 1

X1 = 1⇒ X3 6= 1

X1 = 1⇒ X4 6= 1

X1 X2

X3 X4

p(x) = p(x1) · p(x2|x1) · p(x3|x1, x2) · p(x4|x1, x2, x3)

Page 29: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Inference over permutations

Recommender systems (Sun and Lebanon, 2012)What is the most preferred film given the current personalrankings?

arg m«axi 6=3,7,9,15,4

p(σ−1(1) = i |3 7, 9, 15 4)

What is the most probable order of the films given thecurrent rankings?

arg m«axσ

p(σ |3 7, 9, 15 4)

Page 30: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Introduction

Inference over permutations

Information retrievalLabel ranking (Cheng and Hüllermeier, 2009; Chen et al.,2010): finding the ranking that maximizes the probability

Page 31: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

Page 32: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1

B2

B3

B4

B5

Page 33: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→

B2

B3

B4

B5

Page 34: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2

B3

B4

B5

Page 35: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3

B4

B5

Page 36: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

Page 37: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,_,_)

Page 38: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,_,_)

Page 39: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,1,_)

Page 40: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,1,_)

Page 41: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,1,_)

Page 42: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,1,2)

Page 43: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(_,_,_,1,2)

Page 44: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6

B2 −→ 4.9

B3 −→ 9.3

B4 −→ 2.1

B5 −→ 3.4

(4,3,5,1,2)

Page 45: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

Ranking biscuits in relation with its sweetness

B1 −→ 5.6 ∼ X1

B2 −→ 4.9 ∼ X2

B3 −→ 9.3 ∼ X3

B4 −→ 2.1 ∼ X4

B5 −→ 3.4 ∼ X5

(4,3,5,1,2)

Page 46: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

BasicsEach item is associated with a true continue value:sweetness of a cookie, loudness of a sound, etc.A judge assesses the cookies or the sounds and classifythem.Errors are produced because of the lack of exactness ofthe sensorial apparatus of the judge.The output of the assessment is a classification of theitems.

Page 47: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

BasicsCodify a permutation as a real-valued vectorGiven a real vector (x1, x2, . . . , xn) of length n apermutation can be obtained by ranking the positions usingthe values xi , (i = 1, . . . ,n)Given

(2.35, 3.42, 9.35, 0.32, 11.54, 10.42, 5.23, 4.2, 7.8)

the permutation obtained is (2 3 7 1 9 8 5 4 6)

Page 48: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Order Statistic Models

DefinitionGiven X1,X2, . . . ,Xn random variables with a continuous jointdistribution F (x1, . . . , xn) we can define a random ranking σ insuch a way that σ(i) is the rank that Xi occupied betweenX1,X2, . . . ,Xn. In this way:

P(π) = P(Xσ−1(1) < Xσ−1(2) < . . . < Xσ−1(n))

Unconstrained Xi ’s (and therefore F ) produce all kind ofdistributions over Sn

Page 49: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Models

n-1 parameter modelAssumption: All the Xi are independent (it produces aproper subset of the distribution over permutations)All the distributions are similar Fi(x) = F (x − µi)

The parameters of the model are (µ2 − µ1, . . . , µn − µ1)

Most common models: Fi Gaussian, Fi Gumbel (Luce’sModel)

Page 50: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Models

Learning I

MLE of the parameters: complex numerical integralproblem:

p(σ) = P(Xσ−1(1) < Xσ−1(2) < . . . < Xσ−1(n)) (1)

=

∫Ω

F (x1, x2, . . . , xn)dx1 · · · dxn (2)

where Ω = (x1, x2, . . . , xn)|xσ−1(1) < xσ−1(2) < . . . <xσ−1(n) xi ∈ R i = 1, . . . ,n

Page 51: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Models

Learning Approaches

Böckenholt (1993): Numerical integration (n ≤ 4)Yao & Böckenholt (1999): Bayesian Gibbs sampling(n ≤ 10)Limited information (Maydeu-Olivares, 1999; 2001; 2003):pair, triplets and tetrads comparisons frequencies (n ≤ 10)

Page 52: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Thurstone Models

Thurstone Models

Sampling

It depends on the complexity of F (x1, . . . , xn)

TrueSkillTM (Herbrich et al. 2007)

Page 53: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

Page 54: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1

O2

O3

O4

O5

Page 55: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1

O2

O3

O4

O5

Page 56: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1

O2

O3

O4

O5

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1

O2 −→ w2

O3 −→ w3

O4 −→ w4

O5 −→ w5

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2

O3 −→ w3

O4 −→ w4

O5 −→ w5

Page 59: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,_,_)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,_,_)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,_)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,_)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,_)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,_)

Page 66: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,2)

Page 67: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,2)

Page 68: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(_,_,_,1,2)

Page 69: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(4,3,5,1,2)

Page 70: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Ranking objects with Plackett-Luce

O1 −→ w1w1

w1+w2+w3+w4+w5

O2 −→ w2w2

w1+w2+w3+w4+w5

O3 −→ w3w3

w1+w2+w3+w4+w5

O4 −→ w4w4

w1+w2+w3+w4+w5

O5 −→ w5w5

w1+w2+w3+w4+w5

(4,3,5,1,2)

p((4, 3, 5, 1, 2)) =w4

w1 + w2 + w3 + w4 + w5·

w5

w1 + w2 + w3 + w5·

w2

w1 + w2 + w3·

w1

w1 + w3

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

Definition procedure

Parameters: W = (w1,w2, . . . ,wn) with∑n

i=1 wi = 1 andwi > 0wi is the probability of chosen object iProcedure:

1 An object is chosen using W2 Update W and chose another object

Model

P(σ) =n−1∏i=1

wσ−1(i)∑nj=i wσ−1(j)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce Model

PropertiesThe choice probability ratio between two items isindependent of any other items in the setEasy to extend to partial rankings:

P(σ) =n′−1∏i=1

wσ−1(i)∑n′j=i wσ−1(j)

Plackett-Luce model can be seen as a Thurstone modelwith Fi Gumbel

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Plackett-Luce model

Learning

Given a dataset σ1, σ2, . . . , σN find WTwo main approaches have been developed

MM algorithm (Hunter, 2004)Bayesian approach by means of message passing (Guiverand Snelson, 2009)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Minorization-Maximization (MM) algorithmIt is an iterative algorithm to calculate ML parametersEM algorithm can be considered as a special case of MMThe key idea: Find a function Q(w|w‘) such that(minorization):

Q(w|w‘) ≤ L(w)

with equality if w = w‘

In this case it happens that:

if Q(w|w′) ≥ Q(w′|w′) then L(w) ≥ L(w′)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Minorization-Maximization (MM) algorithm

The algorithm is as follows:1 Give an initial guess for the parameters w0

2 Set l = 03 Construct function Q(w|wl)

4 Find wl+1 = argmaxwQ(w|wl)

5 If ||wl+1 −wl || < θ stop. Otherwise set l = l + 1 and go tostep 3

Page 76: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Minorization-Maximization (MM) algorithm

The algorithm is as follows:1 Give an initial guess for the parameters w0

2 Set l = 03 Construct function Q(w|wl)

4 Find wl+1 = argmaxwQ(w|wl)

5 If ||wl+1 −wl || < θ stop. Otherwise set l = l + 1 and go tostep 3

How to construct Q(w|wl)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Minorization-Maximization (MM) algorithm

The algorithm is as follows:1 Give an initial guess for the parameters w0

2 Set l = 03 Construct function Q(w|wl)

4 Find wl+1 = argmaxwQ(w|wl)

5 If ||wl+1 −wl || < θ stop. Otherwise set l = l + 1 and go tostep 3

How to construct Q(w|wl)

How to calculate argmaxwQ(w|wl)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

MM for P-L

Maximum Likelihood function given a sample σ1, . . . , σN:

ln L(w|σ1, . . . , σN) =N∑

k=1

n−1∑i=1

ln w(σk )−1(i) − lnn∑

j=i

w(σk )−1(j)

Constructing Q

∀x , y > 0 we have − ln x ≥ 1− ln y − xy

Taking y in the previous formula as wl and applying it to L weobtain Q

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

MM for P-L. Function Q

Q(w|wl) =N∑

k=1

n−1∑i=1

(ln w(σk )−1(i) −

∑nj=i w(σk )−1(j)∑nj=i w l

(σk )−1(j)

)

Page 80: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

MM for P-L. Maximize QWe maximize Q by deriving and equaling to 0:

w l+1r =

hr∑Nk=1

∑n−1i=1 δkir [

∑nj=i w l

(σk )−1(j)]−1

where hr is the number of rankings in which the r−th item isranked higher than last and

δkir =

1 if an item higher than i − 1 is ranked r

in the k -th permutation

0 otherwise

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Comments on the MM approachTo guarantee convergence several properties have to becomplied with. For instance:

...in every possible partition of the items into twononempty subsets, some item in the second setranks higher than some item in the first set atleast once..

It can be applied to partial rankings without any changesIt can be combined with Newton-Raphson methodHunter (2004) reports experiments with up to n = 80

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Bayesian Approach (Guiver and Snelson, 2009)Try to avoid the overfitting that can suffer the MLparameters in some situationsGive a method that can be applied globally without theconstraints of the MM methodBayesian approach: assume a prior distribution over theparameters and carry out inference to obtain thea-posteriori distribution given the dataset

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Learning

Bayesian Approach (Guiver and Snelson, 2009)The authors assume a Gamma distribution as a prior:

wi ∼ Gamma(v |α0, β0)

Assume a full factorization for the posteriorp(w) ≈

∏ni=1 p(wi)

Use a message-passing algorithm (powerExpectation-Propagation) to carry out inference

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Probabilistic Modeling on Rankings

Plackett-Luce Model

P-L Inference

InferenceSampling is trivial in this modelMarginal calculation can be exponential: P(σ−1(n) = i)

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Probabilistic Modeling on Rankings

Plackett-Luce Model

Machine Learning Applications

ML ApplicationsCheng et al. (2010): Label rankingChen et al. (2007): Document ranking

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Ranking induced by pair-comparisons

DefinitionBabington-Smith proposes a model based on paircomparisons:

pi,j = probability of preferring item i to item j

if only that comparison were to be made

Assuming no ties, for n objects there are(

n2

)possible

comparisons

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Ranking induced by pair-comparisons

DefinitionGiven an ordering it is easy to get the pair comparisons:

(312)⇒ 3 > 1, 3 > 2, 1 > 2

The opposite is not true in general:

3 > 1, 1 > 2, 2 > 3

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Ranking induced by pair-comparisons

DefinitionA ranking is obtained carrying out all kind of comparisonsuntil a consistent set of comparisons is obtained:

P(σ) ∝∏

(i,j)|σ−1(i)<σ−1(j)

pi,j

For instance given the ordering (1 3 4 2):

p(1 3 4 2) ∝ p1,3 · p1,4 · p1,2 · p3,4 · p3,2 · p4,2

The number of parameters is quadratic n(n − 1)/2There is no closed form for the normalization constantSimplifications....

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Ranking induced by pair-comparisons

Mallows-Bradley-Terry modelsThe model comes defined by a vector of weights(v1, . . . , vn) each one associated with an item:

pi,j =vi

vi + vjwith

n∑i=1

vi = 1 and vi > 0

MM algorithms have been given to learn the MLEparameters (Hunter, 2004)

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Probabilistic Modeling on Rankings

Ranking Induced by Pair-comparisons

Ranking induced by pair-comparisons

Extensions to Mallows-Bradley-Terry modelsAgresti (1990) considers a model where the probability of ibeating j depends on which individual is listed first:

pij =

θwi/(θvi + vj) if i is home

vi/(vi + θvj) if j is home

Rao and Kupper (1967) allows ties:

P(i beats j) = vi/(vi + θvj)

P(j beats i) = vj/(θvi + vj)

P(j ties i) = (θ2 − 1)vivj/((θvi + vj)(vi + θvj))

Page 92: Probabilistic Modeling on Rankings - gipuzkoa€¦ · Probabilistic Modeling on Rankings Probabilistic Modeling on Rankings Jose A. Lozano Ekhine Irurozki Intelligent Systems Group

Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distance-based ranking models (Mallows models)

Definition

p(σ|θ, σ0) =1

Z (θ)e−θd(σ,σ0)

It is an exponential modeld is a distance between permutations such that:

d(σ, π) ≥ 0 ∀σ, π with equality iff σ = πRight-invariant property: d(σ, π) = d(σφ, πφ) ∀σ, π, φ

Two parameters:Central permutation σ0Spread parameter θ ≥ 0

Z (θ) is the partition function

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model

Equivalent to Gaussian distribution for permutations

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model

Equivalent to Gaussian distribution for permutations

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 π(2) = 3 σ(1) = 1 σ(2) = 3

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 < π(2) = 3 σ(1) = 1 σ(2) = 3

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 < π(2) = 3 σ(1) = 1 < σ(2) = 3

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 < π(2) = 3 σ(1) = 1 < σ(2) = 3

(1,3)→ π(1) = 2 π(3) = 1 σ(1) = 1 σ(3) = 4

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 < π(2) = 3 σ(1) = 1 < σ(2) = 3

(1,3)→ π(1) = 2 > π(3) = 1 σ(1) = 1 σ(3) = 4

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations I

Kendall-tau metric. Measures the number ofdisagreements between two permutations:

T (π, σ) =∑i<j

I(π(i)− π(j))(σ(i)− σ(j)) < 0

Example

π = (2 3 1 5 4) σ = (1 3 4 5 2)

(1,2)→ π(1) = 2 < π(2) = 3 σ(1) = 1 < σ(2) = 3

(1,3)→ π(1) = 2 > π(3) = 1 σ(1) = 1 < σ(3) = 4

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations II

Kendall-tau is equivalente to: minimum number of adjacentswaps to go from π−1 to σ−1.0 ≤ T (π, σ) ≤ n(n − 1)/2

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations III

Spearman rho metric:

R(π, σ) =

(n∑

i=1

(π(i)− σ(i))2

)1/2

Spearman footrule:

F (π, σ) =n∑

i=1

|π(i)− σ(i)|

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations IV

Hamming distance:

H(π, σ) =n∑

i=1

Iπ(i) 6= σ(i)

Cayley metric:

C(π, σ) = minimum number of transpositionsto go from π to σ

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Distances between permutations V

Ulam metric

U(π, σ) = n −maximum number of items rankedthe same relative order by π and σ

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model

Learning

Given a dataset σ1, σ2, . . . , σN find θ and σ0

Maximum likelihood estimationσ0 = arg minσ

∑Ni=1 d(σi , σ) (consensus ranking, NP-hard

for T )θ can be found by a standard numerical method for convexoptimization, given σ0

InferenceFor all distance d , Gibbs samplingThere are alternative methods depending on d

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model with Kendall distance

DefinitionIn this particular case the partition function can becalculated in a closed form:

p(π|θ, σ0) =(1− e−θ)n−1∏n−1

i=1 (1− e−(n−i+1)θ)e−θT (σ,σ0)

Learning. Kemeny ranking problem:σ0 = arg minσ

∑Ni=1 T (σi , σ)

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model with Kendall distance

Learning

Ali and Meila (2012) compare 104 algorithms in theKemeny ranking problem. Borda one of the best:

1 Calculate for all index i :

vi = 1/NN∑

j=1

σj (i)

2 Assign σ0(1) with the index of the smallest v1, . . . , vn,σ0(2) with second smallest and so on

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model

ProblemsUnimodalityEstimationPermutations at the same distance from σ0 have the sameprobability

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Mallows model

ExtensionsMixture of Mallows models (Murphy and Martin, 2003; Leeand Yu, 2012; Lu and Boutilier, 2011)Two-side infinite extension (Gnedin and Grigori Olshanski,2012)Bao and Meila (2010) extends the Mallows model to infiniteitems. They learn the model from top-t lists of items andextend it to multi-modal dataGeneralized Mallows models

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Generalized Mallows models

DefinitionFligner and Verducci (1986): if a distance d can be writtenas:

d(π, σ) =n∑

i=1

Si(σ, π)

and the Si are independent under the uniform distribution,then the Mallows model is factorizable. Even more it canbe generalized to a n-parameters model:

P(σ) ∝ exp(−n∑

j=1

θjSj(σ, σ0))

where a different spread parameter θi is associated witheach position in the permutation

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Generalized Mallows models

Kendall distanceKendall distance can be decomposed as:

T (π, σ) =n−1∑i=1

Vi(π, σ)

where Vi(π, σ) is defined as follows:

n∑j=i+1

I(π(i)− π(j))(σ(i)− σ(j)) < 0

There exists a bijection between the set of vectors(v1, . . . , vn−1) and the set of permutations σ

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Generalized Mallows models

Kendall distanceIf we assign a different spread parameter θi to eachposition i then

T (σ, σ0) =n−1∑i=1

θiVi(σ, σ0)

and the model can be written as:

p(σ) =n−1∏i=1

1ψi(θi)

e−θi Vi (σ,σ0) =n−1∏i=1

1− e−θi

1− e−(n−i+1)θie−θi Vi (σ,σ0)

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Generalized Mallows models with Kendall distance

Sampling

The distribution of Vi (i = 1, . . . ,n − 1) under the model isknown:

p(Vi(σ, σ0) = r) =e−rθi

ψ(θi)

Learning

Mandhani and Meila (2009) gives a A* search algorithmwith an admissible heuristic.

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Generalized Mallows model

Cayley distanceDefine the following variables Xi variables:

cycleσ(i) = σk (i)|k = 0,1, . . .where σ0(i) = i and σk (i) = σk−1(σ(i))

Xi(σ) =

0 if i = m«axcycleσ(i)1 otherwise

Cayley distance can be written as:

C(π, σ) = C(πσ−1, σσ−1) = C(πσ−1,e) =n−1∑i=1

Xi(πσ−1)

A similar development that with Kendall can be given

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Probabilistic Modeling on Rankings

Distance-Based Ranking Models

Multi-stage ranking models

DefinitionAt each step a decision is takenA modal raking σ0 is assumed to existThe decision only depends on the number of items leftIt has n(n − 1)/2 parameters:

p(m, r) = probability of taken the m-th best decision at stage r

Let Vr = m if the (m + 1)-th best decision is taken at stager . Then

P(π) =n−1∏r=1

p(Vr , r) factorable model

Mallows and Plackett-Luce can be seen as multi-stagemodels

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Probabilistic Modeling on Rankings

Other Models

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Other Models

Models based on marginals

First-order maginals

First-order marginals Q = [qij ]:

qij = P(σ(i) = j)

Easy to learn and to representWhich distribution is represented by these marginals?

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Probabilistic Modeling on Rankings

Other Models

Models based on marginals

First-order maginals

First-order marginals Q = [qij ]:

qij = P(σ(i) = j)

Easy to learn and to representWhich distribution is represented by these marginals?Infinite distributions

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Probabilistic Modeling on Rankings

Other Models

Models based on marginals

First-order marginals

Criterion to choose between them: The one with themaximum entropy

P(σ) = exp(n∑

i=1

Y(i,σ(i)) − 1)

where Y ∈ Rnxn

Obtaining Y is #P-hard (Agrawal et al., 2008)

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Probabilistic Modeling on Rankings

Other Models

Models based on marginals

High-order marginals

Huang et al. (2009) considers models based on high-ordermarginalsThey use the Fourier transform over the symmetric groupto carry out inference tasksIrurozki et al. (2011) give a first approach to learning thesekind of models

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Probabilistic Modeling on Rankings

Other Models

Non-parametric model I

Based on Mallows kernelsLebanon and Mao (2008) give a probabilistic approachbased on Mallows kernels with Kendall distance

p(σ) =1

NZ (θ)

N∑i=1

e−θd(σ,σi )

They were able to learn from partially ranked dataEfficient learningLacks: Marginalization, conditioning

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Probabilistic Modeling on Rankings

Other Models

Non-parametric model II

Modifying Mallows kernels

Sun and Lebanon (2012) uses a triangular kernel based onKendall distance in to solve problems in recommendationsystemsThey can deal with all kind of partial rankings

Based on the Fourier transformBarbosa y Kondor (2010) gives another kernel approachbased on the use of the Fourier transform

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Probabilistic Modeling on Rankings

Independence

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Independence

Luce modelA ranking σ is built by selecting first selecting the preferreditem, then among the remaining items, the second preferredand so on. Given n items, each with weight wi

P(σ) =n−1∏k=1

Pik ,...,in(σ(k)) =n−1∏k=1

wσ−1(k)∑nj=k wσ−1(j)

L-decomposability induces a conditional independence

P(σ−1(k) = ik |σ−1(1) = i1, . . . , σ−1k−1(k − 1) = ik−1) =

Pik ,...,in(ik )Thurstone and the Mallows model based on Kendall, Hammingor Spearman’s footrule are also L-decomposable

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Probabilistic Modeling on Rankings

Independence

DefinitionIndependence based on relative rankings.The subsets A ⊂ 1, . . . ,n and B = Ac are independent underdistribution p(σ) if

p(σ) = pA(σA) · pB(σB)

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Probabilistic Modeling on Rankings

Independence

First order marginal independence

Necessary (not sufficient) condition for full independenceGiven this collection of permutations, the frequency matrix is

1 2 5 3 42 1 4 3 51 2 3 4 51 2 3 5 41 2 5 4 32 1 5 3 4

First order independence detectionFinding the independent sets is clustering problem

Approximation qualityWhat if every time item 2 is in position 1, item 3 is in position 4?

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Probabilistic Modeling on Rankings

Independence

Riffle independence (Huang et al. 2010)

Riffle shuffle

DefinitionDistribution p is said to be riffle independent ifp(σ) = m(τ(A,B)) · pA(σA) · pB(σB)

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Probabilistic Modeling on Rankings

Independence

Riffle independence (Huang et al. 2010)

Riffle shuffle

DefinitionDistribution p is said to be riffle independent ifp(σ) = m(τ(A,B)) · pA(σA) · pB(σB)

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Probabilistic Modeling on Rankings

Independence

SimplificationsDifferent settings of interleaving distribution lead to simplermodels

To take homeNatural criterion in the ranking domainFirst generate the rankings on the independent subsetsand the shuffle those rankings to obtain the complete one

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Probabilistic Modeling on Rankings

Independence

SummaryL-decomposability: the probability of selecting an item ateach stage does not depend on the order of the alreadyselected itemsFirst order independence: a subset of items A always mapinto a subset of the positions X (|A| = |X |)Riffle independence: given 2 riffle independent subsets Aand B and items a1,a2 ∈ A and b ∈ B knowing that a1 isranked before a2 does not give any info about where b isranked

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Probabilistic Modeling on Rankings

Datasets and Software

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Datasets and Software

Application of ranking has changed in history, frompsychology to machine learningSo has the size and amount of data, from small datasets tohuge amount of data

Kinds of dataPartial ranking: not every items has received a rank, top 5rankingRank with ties: a set of items is preferred to another set butthere is not order within the items in the setImplicit data: the acts of users are known (buying film A orvisiting B web page) but not their opinion about it, there isno rating or ranking no negative feedback

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Probabilistic Modeling on Rankings

Datasets and Software

German sample (Croon, 1988)Small database of the permutations of 4 elements cited in(Bückenholt, 2002)

Idea (Fligner et al. 1986)

98 college students where asked to rank five words, namelythought, play, theory, dream, and attention regarding itsassociation with the word ideaPaper that cite it are (Gupta et al. 2002)

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Probabilistic Modeling on Rankings

Datasets and Software

Cars (Bückenholt, 2002)279 Spanish college students were asked to rank four compactcars according to their purchase preferences. The authorsprovide the ranking patterns’ observed frequencies in thissample

APA (Diaconis, 1988)The American Psychological Association dataset includes15449 ballots of the election of the president in 1980, 5738 ofwhich are complete rankings, in which the candidates areranked from most to least favoriteIt is very frequently used (Huang et. al 2010)

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Probabilistic Modeling on Rankings

Datasets and Software

Paired comparisons (Hunter, 2003)NFL dataset, results of the 1997 NFL seasonNASCAR dataset, results of the 2002 car season, 36 races and83 divers took part

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Probabilistic Modeling on Rankings

Datasets and Software

Sushi (Kamishima et al. 2010)This information collected via web comprises user informationas well as their preferences in sushi, including

1025 partial rankings on 100 different kinds of sushi1025 full rankings on 10 different kinds of sushi (items areranked by every ranker)

Cited by (Huang et. al 2010)

Jester (Goldberg et al. 2001)

4.1 Million continuous ratings (-10.00 to +10.00) of 100 jokesfrom 73421 usersAppears in (Meila et al. 2010)

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Probabilistic Modeling on Rankings

Datasets and Software

Both are offline

Netflix (Bennett et al. 07)Movie recomender system. In 2009 1000000 dollar prize wasgiven to Bellkor’s programatic chaos for besting Netflix’srecomenderoffline for privacy reasons

WikiLensWikiLens was a generalized collaborative recommender systemthat allowed its community to define item types (movie, bookalbum, restaurant, web) and categories for each type, and thenrate and get recommendations for items

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Probabilistic Modeling on Rankings

Datasets and Software

Book crossing (Ziegler, 2005)

Collected from the Book-Crossing community.Contains 278.858 users (with demographic information)providing 1.149.780 ratings (explicit / implicit) about 271.379books

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Probabilistic Modeling on Rankings

Datasets and Software

MovieLensRankings with-ties

MovieLens 100k: 100000 ratings (1-5) from 1000 users on1700 movies.MovieLens 1M: 1 million ratings from 6000 users on 4000movies.MovieLens 10M: 10 million ratings and 100000 tagapplications applied to 10000 movies by 72000 users. Inthe dataset, the movies are linked to movie reviewsystems. Each movie does have its IMDb and RTidentifiers, English and Spanish titles, picture URLs,genres, directors, actors (ordered by ”popularity”), RTaudience’ and experts’ ratings and scores, countries, andfilming locations.

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Probabilistic Modeling on Rankings

Datasets and Software

HetRec 2011 conferenceHighly sparse databases

movielens 2k: extension of MovieLens10M dataset, whichcontains personal ratings and tags about movies.delicious 2k: implicit data from Delicious socialbookmarking system. Each of the 1867 users hasbookmarks, tag assignments, i.e. tuples [user, tag,bookmark], and contact relations within the dataset socialnetwork. Each bookmark has a title and URL.lastfm 2k: implicit dataset in which each user has a list ofmost listened music artists, tag assignments, i.e. tuples[user, tag, artist], and friend relations within the datasetsocial network. Each artist has a Last.fm URL and apicture URL.

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Probabilistic Modeling on Rankings

Datasets and Software

Non parametric modelsThe SnOB software provides general functions orpermutations such as Fast Fourier transformThe props toolbox (based on SnOB) provides someefficient inference algorithms for distributions overpermutations and examples as described in the paper(Huang et al. 2009)

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Probabilistic Modeling on Rankings

Datasets and Software

Mallows modelpyMallows is a set of Python routines for fitting and simulatingfrom a generalized Mallows model. Learning algorithmsimplemented for both full and partial rankings.

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Probabilistic Modeling on Rankings

Datasets and Software

Paired comparisonsprefmod fits and simulates data as pairs comparisons,Bradley-Terry models and others, and their extension. Thepackage includes datasets for testingBradleyTerry2, eba, psychotree are other R packages forpaired comparisonsMMBT is a matlab packages for the estimation of theBradley-Terry model and and its extension to multicomparison case

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Probabilistic Modeling on Rankings

Conclusions

Outline of the talk

1 Introduction

2 Thurstone Models

3 Plackett-Luce Model

4 Ranking Induced by Pair-comparisons

5 Distance-Based Ranking Models

6 Other Models

7 Independence

8 Datasets and Software

9 Conclusions

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Probabilistic Modeling on Rankings

Conclusions

Conclusions

Many process are based on orderings and rankingNew technologies allow to store thousand partial rankingsof thousand of itemsNew probabilistic approaches need to be developed todeal with themTwo ways:

To adapt well-known probabilistic modelsTo create new specific models for the new kind of data