text: p 20 50 100 150 200 BMA 0.01 0.1 0.3 0.7 1.2 MAP 14.5 64.1 241.7 538.8 955.4 Table 2: Memory usage in megabytes for versus number of nodes (p) with 100, 000 iterations for our sampling algorithms with both BMA and MAP estimations. CMC algorithm is computationally more expensive but converges faster than the RJMCMC algorithm. 5.2 Comparison with frequentist methods Here we present the summary of our comparison of the Bayesian approaches of our BDgraph package with two frequentist approaches, the graphical lasso (glasso) (Friedman et al., 2008) and the Meinshausen-Buhlmann graph estima- tion (mb) (Meinshausen and Buhlmann, 2006), both of which are available in the huge package. Table 3 presents the averaged F -score with standard errors in parentheses, 1 corresponding to our Bayesian approach based on the BDMCMC algorithm and the two frequentist methods, glasso and mb, with diﬀerent regularization pa- rameter selection approaches available in the huge package. For our Bayesian approach, the inferred graph is selected based on a BMA estimator with cut- point 0.5 as a default of function select in our BDgraph package. For the glasso and mb methods, the inferred graph is selected by determining the reg- ularization parameter λ which controls the sparsity of the inferred graph. The inferred graphs in the glasso and mb methods are quite sensitive to the values of λ, since diﬀerent λ’s lead to diﬀerent graph structures (Vujaˇci´c et al., 2015). For the glasso methods, we consider three diﬀerent regularization parameter selection approaches for choosing λ: rotation information criterion (ric) (Zhao et al., 2012), the stability approach to regularization selection (stars) (Liu et al., 2010), and the extended Bayesian information criterion (ebic) (Foygel and Dr- ton, 2010). These approaches are available in the huge package. For the mb method, we consider two regularization parameter selection approaches, namely ric and star. Based on Table 3, we see that our Bayesian method performs better than the glasso and mb approaches except for hub graphs. The results in the table show the performance of both the glasso and mb methods is quite sensitive to their regularization parameter selection approaches. Both the glasso and mb methods have good performance in some cases and substantially deteriorate in other cases. For example, the mb approach with stars criterion for scale-free graphs has good performance, but not for circle and random graphs. To compare the performance of our Bayesian method for both BDMCMC and RJMCMC algorithms with both glasso and mb, the average ROC curves for various p and n are presented in Figures 7 and 8. For each ﬁgure four curves are presented, corresponding to our BDMCMC and RJMCMC algorithms and glasso and mb methods from the huge package. We can see that the perfor- mance of BDMCMC and RJMCMC is comparable and that both algorithms in most cases outperform the glasso and mb methods. As can be seen from the ROC curves the BDMCMC performs better than the RJMCMC algorithm. From a theoretical point of view BDMCMC and RJMCMC should asymptoti- 20 ##LTLine## ##LTLine## ##LTLine##

text: p

label: table

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text: the

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Comprehensive R Archive Network (CRAN) at http://CRAN.R-project.org. We use the igraph package for graph visualization and the Matrix package for memory-optimization, using the sparse matrix output. To speed up computations, we eﬃciently implement the BDgraph package by linking the C++ code to R. For the C++ code, we use the highly optimized LA- PACK (Anderson et al., 1999) and BLAS (Lawson et al., 1979) linear algebra libraries on systems that provide them. The use of these libraries signiﬁcantly improves program speed. We design the BDgraph package to provide a Bayesian framework for undi- rected graph estimation of diﬀerent types of datasets such as continuous, discrete or mixed data. The package facilitates a pipeline for analysis by three functional modules; see Figure 1. These modules are as follows: Figure 1: Conﬁguration of the BDgraph package which includes three main parts: (M1) data generation, (M2) several search algorithms, (M3) various functions to eval- uate convergence of the search algorithms, estimation of the true graph, assessment and comparison of the results and graph visualization. Module 1. Data simulation: Function bdgraph.sim simulates multivari- ate Gaussian, discrete and mixed data with diﬀerent undirected graph struc- tures, including “random”, “cluster”, “hub”, “scale-free”, “circle” and “ﬁxed” graphs. Users can determine the sparsity of the graph structure and can gener- ate mixed data, including “count”, “ordinal”, “binary”, “Gaussian” and “non- Gaussian” variables. Module 2. Method: The function bdgraph provides two estimation meth- ods with two diﬀerent algorithms for sampling from the posterior distributions: (1) Graph estimation for the multivariate data that follow the Gaussian- ity assumption, based on the Gaussian graphical models (GGMs), and using the birth-death MCMC sampling algorithm described in Mohammadi and Wit (2015b). (2) Graph estimation for non-Gaussian, discrete, and mixed data, based on Gaussian copula graphical models (GCGMs), and using the birth-death MCMC sampling algorithm, described in Mohammadi et al. (2015). Note, in the bdgraph function, besides the BDMCMC algorithms, we imple- mented the Reversible Jump MCMC (RJMCMC) algorithms (Green, 1995), as an alternative. Module 3. Result: Includes four types of functions: (1) Graph selection: The functions select, plinks, and pgraph provide the selected graph, the posterior link inclusion probabilities and the posterior probability of each graph, respectively. 3 ##LTFigure##

text: Comprehensive

label: paragraph

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text: the

text: igraph

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text: Matrix

text: package

text: for

text: memory-optimization,

text: using

text: the

text: sparse

text: matrix

text: output.

text: To

text: speed

text: up

text: computations,

text: we

text: eﬃciently

text: implement

text: the

text: BDgraph

text: package

text: by

text: linking

text: the

text: C++

text: code

text: to

text: R.

text: For

text: the

text: C++

text: code,

text: we

text: use

text: the

text: highly

text: optimized

text: LA-

text: PACK

text: (Anderson

text: et

text: al.,

text: and

text: BLAS

text: (Lawson

text: et

text: al.,

text: linear

text: algebra

text: libraries

text: on

text: systems

text: that

text: provide

text: them.

text: The

text: use

text: of

text: these

text: libraries

text: signiﬁcantly

text: improves

text: program

text: speed.

text: We

text: design

text: the

text: BDgraph

text: package

text: to

text: provide

text: a

text: Bayesian

text: framework

text: for

text: undi-

text: rected

text: graph

text: estimation

text: of

text: diﬀerent

text: types

text: of

text: datasets

text: such

text: as

text: continuous,

text: discrete

text: or

text: mixed

text: data.

text: The

text: package

text: facilitates

text: a

text: pipeline

text: for

text: analysis

text: by

text: three

text: functional

text: modules;

text: see

text: Figure

text: These

text: modules

text: are

text: as

text: follows:

text: Figure

text: Conﬁguration

text: of

text: the

text: BDgraph

text: package

text: which

text: includes

text: three

text: main

text: parts:

text: data

text: generation,

text: several

text: search

text: algorithms,

text: various

text: functions

text: to

text: eval-

text: uate

text: convergence

text: of

text: the

text: search

text: algorithms,

text: estimation

text: of

text: the

text: true

text: graph,

text: assessment

text: and

text: comparison

text: of

text: the

text: results

text: and

text: graph

text: visualization.

text: Module

text: Data

text: simulation:

text: Function

text: bdgraph.sim

text: simulates

text: multivari-

text: ate

text: Gaussian,

text: discrete

text: and

text: mixed

text: data

text: with

text: diﬀerent

text: undirected

text: graph

text: struc-

text: tures,

text: including

text: “random”,

text: “cluster”,

text: “hub”,

text: “scale-free”,

text: “circle”

text: and

text: “ﬁxed”

text: graphs.

text: Users

text: can

text: determine

text: the

text: sparsity

text: of

text: the

text: graph

text: structure

text: and

text: can

text: gener-

text: ate

text: mixed

text: data,

text: including

text: “count”,

text: “ordinal”,

text: “binary”,

text: “Gaussian”

text: and

text: “non-

text: Gaussian”

text: variables.

text: Module

text: Method:

text: The

text: function

text: bdgraph

text: provides

text: two

text: estimation

text: meth-

text: ods

text: with

text: two

text: diﬀerent

text: algorithms

text: for

text: sampling

text: from

text: the

text: posterior

text: distributions:

text: Graph

text: estimation

text: for

text: the

text: multivariate

text: data

text: that

text: follow

text: the

text: Gaussian-

text: ity

text: assumption,

text: based

text: on

text: the

text: Gaussian

text: graphical

text: models

text: (GGMs),

text: and

text: using

text: the

text: birth-death

text: MCMC

text: sampling

text: algorithm

text: described

text: in

text: Mohammadi

text: and

text: Wit

text: Graph

text: estimation

text: for

text: non-Gaussian,

text: discrete,

text: and

text: mixed

text: data,

text: based

text: on

text: Gaussian

text: copula

text: graphical

text: models

text: (GCGMs),

text: and

text: using

text: the

text: birth-death

text: MCMC

text: sampling

text: algorithm,

text: described

text: in

text: Mohammadi

text: et

text: al.

text: Note,

text: in

text: the

text: bdgraph

text: function,

text: besides

text: the

text: BDMCMC

text: algorithms,

text: we

text: imple-

text: mented

text: the

text: Reversible

text: Jump

text: MCMC

text: (RJMCMC)

text: algorithms

text: (Green,

text: as

text: an

text: alternative.

text: Module

text: Result:

text: Includes

text: four

text: types

text: of

text: functions:

text: Graph

text: selection:

text: The

text: functions

text: select,

text: plinks,

text: and

text: pgraph

text: provide

text: the

text: selected

text: graph,

text: the

text: posterior

text: link

text: inclusion

text: probabilities

text: and

text: the

text: posterior

text: probability

text: of

text: each

text: graph,

text: respectively.

text: ##LTFigure##


text: 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 0.1 8.0 6.0 4.0 2.0 0.0 p=150, n=300, random p=150, n=300, scale−free p=150, n=300, cluster BDMCMC BDMCMC BDMCMC RJMCMC RJMCMC RJMCMC glasso glasso glasso mb mb mb 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p=150, n=300, hub p=150, n=300, AR2 p=150, n=300, circle BDMCMC BDMCMC BDMCMC RJMCMC RJMCMC RJMCMC glasso glasso glasso mb mb mb 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p=200, n=200, random p=200, n=200, scale−free p=200, n=200, cluster BDMCMC BDMCMC BDMCMC RJMCMC RJMCMC RJMCMC glasso glasso glasso mb mb mb 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p=200, n=200, hub p=200, n=200, AR2 p=200, n=200, circle BDMCMC BDMCMC BDMCMC RJMCMC RJMCMC RJMCMC glasso glasso glasso mb mb mb 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 8: ROC curves present the performances of the two sampling algorithms from the BDgraph package and two frequentist methods from the huge package for p = { } 150, 200 and varying graph structures and with with 50 replications. 24 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (2) Convergence check: The functions plotcoda and traceplot provide sev- eral visualization plots to monitor the convergence of the sampling algorithms. (3) Comparison and goodness-of-ﬁt: The functions compare and plotroc provide several comparison measures and an ROC plot for model comparison. (4) Visualization: The plotting functions plot.bdgraph and plot.sim pro- vide visualizations of the simulated data and estimated graphs. 3 Methodological background In Section we brieﬂy explain the Gaussian graphical model for multivariate data. Then we illustrate the birth-death MCMC algorithm for sampling from the joint posterior distribution over Gaussian graphical models; for more details see Mohammadi and Wit (2015b). In Section we brieﬂy describe the Gaussian copula graphical model (Dobra and Lenkoski, 2011), which can deal with non- Gaussian, discrete or mixed data. Then we explain the birth-death MCMC algorithm which is designed for the Gaussian copula graphical models; for more details see Mohammadi et al. (2015). 3.1 Bayesian Gaussian graphical models In graphical models, each random variable is associated with a node and conditional dependence relationships among random variables are presented as { } a graph G = (V, E) in which V = 1, 2, ..., p speciﬁes a set of nodes and a set ⊂ × of existing links E V V (Lauritzen, 1996). Our focus here is on undirected ∈ ⇔ ∈ graphs, in which (i, j) E (j, i) E. The absence of a link between two nodes speciﬁes the pairwise conditional independence of those two variables given the remaining variables, while a link between two variables determines their conditional dependence. In Gaussian graphical models (GGMs), we assume that the observed data N − follow multivariate Gaussian distribution (µ, K 1). Here we assume µ = 0. p Let Z = (Z(1), ..., Z(n))T be the observed data of n independent samples, then the likelihood function is (cid:26) (cid:27) | ∝ | | −1 Pr(Z K, G) K n/2 exp tr(KU) , (1) 2 (cid:48) where U = Z Z. In GGMs, conditional independence is implied by the form of the precision matrix. Based on the pairwise Markov property, variables i and j are condi- tionally independent given the remaining variables, if and only if K = 0. This ij property implies that the links in graph G = (V, E) correspond with the nonzero { | (cid:54) } elements of the precision matrix K; this means that E = (i, j) K = 0 . Given P ij graph G, the precision matrix K is constrained to the cone of symmetric G ∈ positive deﬁnite matrices with elements K equal to zero for all (i, j) / E. ij We consider the G-Wishart distribution W (b, D) to be a prior distribution G for the precision matrix K with density (cid:26) (cid:27) | 1 | | − −1 ∈ P Pr(K G) = K (b 2)/2 exp tr(DK) 1(K ), (2) I (b, D) 2 G G 4 ##LTLine## ##LTLine## ##LTLine##

text: Convergence

text: check:

text: The

text: functions

text: plotcoda

text: and

text: traceplot

text: provide

text: sev-

text: eral

text: visualization

text: plots

text: to

text: monitor

text: the

text: convergence

text: of

text: the

text: sampling

text: algorithms.

text: Comparison

text: and

text: goodness-of-ﬁt:

text: The

text: functions

text: compare

text: and

text: plotroc

text: provide

text: several

text: comparison

text: measures

text: and

text: an

text: ROC

text: plot

text: for

text: model

text: comparison.

text: Visualization:

text: The

text: plotting

text: functions

text: plot.bdgraph

text: and

text: plot.sim

text: pro-

text: vide

text: visualizations

text: of

text: the

text: simulated

text: data

text: and

text: estimated

text: graphs.

label: section

text: Methodological

text: background

text: In

text: Section

text: we

text: brieﬂy

text: explain

text: the

text: Gaussian

text: graphical

text: model

text: for

text: multivariate

text: data.

text: Then

text: we

text: illustrate

text: the

text: birth-death

text: MCMC

text: algorithm

text: for

text: sampling

text: from

text: the

text: joint

text: posterior

text: distribution

text: over

text: Gaussian

text: graphical

text: models;

text: for

text: more

text: details

text: see

text: Mohammadi

text: and

text: Wit

text: In

text: Section

text: we

text: brieﬂy

text: describe

text: the

text: Gaussian

text: copula

text: graphical

text: model

text: (Dobra

text: and

text: Lenkoski,

text: which

text: can

text: deal

text: with

text: non-

text: Gaussian,

text: discrete

text: or

text: mixed

text: data.

text: Then

text: we

text: explain

text: the

text: birth-death

text: MCMC

text: algorithm

text: which

text: is

text: designed

text: for

text: the

text: Gaussian

text: copula

text: graphical

text: models;

text: for

text: more

text: details

text: see

text: Mohammadi

text: et

text: al.

text: Bayesian

text: Gaussian

text: graphical

text: models

text: In

text: graphical

text: models,

text: each

text: random

text: variable

text: is

text: associated

text: with

text: a

text: node

text: and

text: conditional

text: dependence

text: relationships

text: among

text: random

text: variables

text: are

text: presented

text: as

text: a

text: graph

text: G

text: (V,

text: E)

text: in

text: which

text: V

text: p

text: speciﬁes

text: a

text: set

text: of

text: nodes

text: and

text: a

text: set

text: of

text: existing

text: links

text: E

text: V

text: V

text: (Lauritzen,

text: Our

text: focus

text: here

text: is

text: on

text: undirected

text: graphs,

text: in

text: which

text: (i,

text: j)

text: E

text: (j,

text: i)

text: E.

text: The

text: absence

text: of

text: a

text: link

text: between

text: two

text: nodes

text: speciﬁes

text: the

text: pairwise

text: conditional

text: independence

text: of

text: those

text: two

text: variables

text: given

text: the

text: remaining

text: variables,

text: while

text: a

text: link

text: between

text: two

text: variables

text: determines

text: their

text: conditional

text: dependence.

text: In

text: Gaussian

text: graphical

text: models

text: (GGMs),

text: we

text: assume

text: that

text: the

text: observed

text: data

text: N

text: follow

text: multivariate

text: Gaussian

text: distribution

text: (µ,

text: K

text: Here

text: we

text: assume

text: µ

text: p

text: Let

text: Z

text: Z(n))T

text: be

text: the

text: observed

text: data

text: of

text: n

text: independent

text: samples,

text: then

text: the

text: likelihood

text: function

text: is

text: −1

text: Pr(Z

text: K,

text: G)

text: K

text: exp

text: tr(KU)

text: where

text: U

text: Z

text: Z.

text: In

text: GGMs,

text: conditional

text: independence

text: is

text: implied

text: by

text: the

text: form

text: of

text: the

text: precision

text: matrix.

text: Based

text: on

text: the

text: pairwise

text: Markov

text: property,

text: variables

text: i

text: and

text: j

text: are

text: condi-

text: tionally

text: independent

text: given

text: the

text: remaining

text: variables,

text: if

text: and

text: only

text: if

text: K

text: This

text: ij

text: property

text: implies

text: that

text: the

text: links

text: in

text: graph

text: G

text: (V,

text: E)

text: correspond

text: with

text: the

text: nonzero

text: elements

text: of

text: the

text: precision

text: matrix

text: K;

text: this

text: means

text: that

text: E

text: (i,

text: j)

text: K

text: Given

text: P

text: ij

text: graph

text: G,

text: the

text: precision

text: matrix

text: K

text: is

text: constrained

text: to

text: the

text: cone

text: of

text: symmetric

text: G

text: positive

text: deﬁnite

text: matrices

text: with

text: elements

text: K

text: equal

text: to

text: zero

text: for

text: all

text: (i,

text: j)

text: E.

text: ij

text: We

text: consider

text: the

text: G-Wishart

text: distribution

text: W

text: (b,

text: D)

text: to

text: be

text: a

text: prior

text: distribution

text: G

text: for

text: the

text: precision

text: matrix

text: K

text: with

text: density

text: −1

text: P

text: Pr(K

text: G)

text: K

text: (b

text: exp

text: tr(DK)

text: I

text: (b,

text: D)

text: G

text: G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON BI-FREE DE FINETTI THEOREMS 5 I. Charlesworth, B. Nelson and P. Skoufranis introduced another pictorial representation for bi-noncrossing partitions, drawing the points on vertical lines instead of horizontal ones. Our point in keeping the horizontal drawing is to make the connection with the combinatorics of quantum groups easier, as will appear later on. Mimicking free probability, one would like to deﬁne "bi-noncrossing cumulants" κχ by induction using a deﬁning formula of the type 1χ ( ) = ∑ ( ) ∈ ( ) (1) E T . . . T κχ T , . . . , T , 1 n σ 1 n ′ σ BNC χ ′ E ′ where κχ should be computable using only elements of the form κχ for restrictions χ of χ. To do this, σ 1 the authors of [2] ﬁrst deﬁne "bi-moment functionEs" and then use Mχöbius inversion. As we will see, these π moment functions look like the usual ones except that they take care of the left-right structure. To avoid confusion, they w=ill(be⊗de)noted by =a c(url⊗y le)tter while we will denote by E , for a noncrossing partition p, p the usual nested moment function as deﬁned e.g. in [10, Def 2.1.1]. For the sake of simplicity, we will from now on write Lb ε b ∈1{ and} Rb ε 1∈ b . ( ) ( ) ∈ ( ) ⩽ ⩽ Deﬁnition 2.8. Let χ (cid:96), r n, let π BEN(C χ( ) and cho(o)s)e ∈an element T χ i Cχ i for every 1 i n. We deﬁne an element ji ji χ 1 χ n T , . . . , T B ( ( ) ( )π) j1( ○ ( )jn ○ ( )) in three steps : ( ) ( ) (1) Permute the tuple Tjχ1 1 ,−. .(. ,)Tjχn n to Tjχsχsχ1 1 , . . . , Tjχsχsχn n− (. ) (2) Apply the expectation E to this tuple. Using the properties of usual multiplicative functions, s 1 π − ′ χ this can be written as a nesting of the expectation E, ′since s 1 π is noncrossing. χ (3) Blockwise, permute back the elements according to s 1, where χ is the restriction of χ to the block. χ Then replace each nEested expectation b by Lb if the leftmost element of the block is a left element and by R if the ﬁrst element of the block is a right element. b Note that the functions are completely determined by E. Let us illustrate this seemingly complicated π deﬁnition on the same partition as in Example 2.7. E ( ) Example 2.9. We consider χ and π as in Example 2.7. Let us compute ( ) T , T , T , T , T , T , T , T , T . π 1 2 3 4 5 6 7 8 9 (1) Permute th(e elements to T1, T3, T4, T7, T)9=, T8,(T6,(T5(, T2(. ( ) ( ) )) ) (2) Compute E T , T , T , T , T , T , T , T , T E T E T E T T T E T T T T . p 1 3 4 7 9 (8 (6 ( 5 (2 ( ) 1 ( 3 ) 4))7 )9 8 6 5 2 (3) Permute back the elements blockwise to E T E T E T T T E T T T T 1 3 4 7 5 6 8 9 2 E ( ) = ( ) and then replace by the corresponding action of B to( g(et( )) ( ) ( ) T , T , T , T , T , T , T , T , T E T L T . π 1 2 3 4 5 6 7 8 9 1 E T3 LE T4T7 T5RE T6T8 T9 2 Because T is a right element, it commutes with every L , so that we recover [2, Ex 5.1.2]. 2 b Remark 2.10. The deﬁnition of the operator-valued bi-moment function in [2] is diﬀerent from Deﬁnition 2.8 and much more involved. However, [2, Thm 5.1.4] asserts that these bi-free moment functions are completely determined by E and the properties of bi-multiplicative functions. This means that they coincide with Deﬁnition 2.8. We took here advantage of this deep result to give the "easy" deﬁnition of the bi-moment functions.

text: ON

text: BI-FREE

text: DE

text: FINETTI

text: THEOREMS

text: I.

text: Charlesworth,

text: B.

text: Nelson

text: and

text: P.

text: Skoufranis

text: introduced

text: another

text: pictorial

text: representation

text: for

text: bi-noncrossing

text: partitions,

text: drawing

text: the

text: points

text: on

text: vertical

text: lines

text: instead

text: of

text: horizontal

text: ones.

text: Our

text: point

text: in

text: keeping

text: the

text: horizontal

text: drawing

text: is

text: to

text: make

text: the

text: connection

text: with

text: the

text: combinatorics

text: of

text: quantum

text: groups

text: easier,

text: as

text: will

text: appear

text: later

text: on.

text: Mimicking

text: free

text: probability,

text: one

text: would

text: like

text: to

text: deﬁne

text: "bi-noncrossing

text: cumulants"

text: κχ

text: by

text: induction

text: using

text: a

text: deﬁning

text: formula

text: of

text: the

text: type

text: 1χ

label: equation

text: E

text: T

text: T

text: κχ

text: T

text: T

text: n

text: σ

text: n

text: σ

text: BNC

text: χ

text: E

text: where

text: κχ

text: should

text: be

text: computable

text: using

text: only

text: elements

text: of

text: the

text: form

text: κχ

text: for

text: restrictions

text: χ

text: of

text: χ.

text: To

text: do

text: this,

text: σ

text: the

text: authors

text: of

text: ﬁrst

text: deﬁne

text: "bi-moment

text: functionEs"

text: and

text: then

text: use

text: Mχöbius

text: inversion.

text: As

text: we

text: will

text: see,

text: these

text: π

text: moment

text: functions

text: look

text: like

text: the

text: usual

text: ones

text: except

text: that

text: they

text: take

text: care

text: of

text: the

text: left-right

text: structure.

text: To

text: avoid

text: confusion,

text: they

text: w=ill(be⊗de)noted

text: by

text: =a

text: c(url⊗y

text: le)tter

text: while

text: we

text: will

text: denote

text: by

text: E

text: for

text: a

text: noncrossing

text: partition

text: p,

text: p

text: the

text: usual

text: nested

text: moment

text: function

text: as

text: deﬁned

text: e.g.

text: in

text: Def

text: For

text: the

text: sake

text: of

text: simplicity,

text: we

text: will

text: from

text: now

text: on

text: write

text: Lb

text: ε

text: b

text: and}

text: Rb

text: ε

text: 1∈

text: b

text: Deﬁnition

text: Let

text: χ

text: r

text: n,

text: let

text: π

text: BEN(C

text: χ(

text: and

text: cho(o)s)e

text: ∈an

text: element

text: T

text: χ

text: i

text: Cχ

text: i

text: for

text: every

text: i

text: n.

text: We

text: deﬁne

text: an

text: element

text: ji

text: ji

text: χ

text: χ

text: n

text: T

text: T

text: B

text: )π)

text: )jn

text: in

text: three

text: steps

text: Permute

text: the

text: tuple

text: ,)Tjχn

text: n

text: to

text: Tjχsχsχn

text: n−

text: Apply

text: the

text: expectation

text: E

text: to

text: this

text: tuple.

text: Using

text: the

text: properties

text: of

text: usual

text: multiplicative

text: functions,

text: s

text: π

text: χ

text: this

text: can

text: be

text: written

text: as

text: a

text: nesting

text: of

text: the

text: expectation

text: E,

text: ′since

text: s

text: π

text: is

text: noncrossing.

text: χ

text: Blockwise,

text: permute

text: back

text: the

text: elements

text: according

text: to

text: s

text: where

text: χ

text: is

text: the

text: restriction

text: of

text: χ

text: to

text: the

text: block.

text: χ

text: Then

text: replace

text: each

text: nEested

text: expectation

text: b

text: by

text: Lb

text: if

text: the

text: leftmost

text: element

text: of

text: the

text: block

text: is

text: a

text: left

text: element

text: and

text: by

text: R

text: if

text: the

text: ﬁrst

text: element

text: of

text: the

text: block

text: is

text: a

text: right

text: element.

text: b

text: Note

text: that

text: the

text: functions

text: are

text: completely

text: determined

text: by

text: E.

text: Let

text: us

text: illustrate

text: this

text: seemingly

text: complicated

text: π

text: deﬁnition

text: on

text: the

text: same

text: partition

text: as

text: in

text: Example

text: E

text: Example

text: We

text: consider

text: χ

text: and

text: π

text: as

text: in

text: Example

text: Let

text: us

text: compute

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: π

text: Permute

text: th(e

text: elements

text: to

text: Compute

text: E

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: E

text: T

text: E

text: T

text: E

text: T

text: T

text: T

text: E

text: T

text: T

text: T

text: T

text: p

text: Permute

text: back

text: the

text: elements

text: blockwise

text: to

text: E

text: T

text: E

text: T

text: E

text: T

text: T

text: T

text: E

text: T

text: T

text: T

text: T

text: E

text: and

text: then

text: replace

text: by

text: the

text: corresponding

text: action

text: of

text: B

text: to(

text: g(et(

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: T

text: E

text: T

text: L

text: T

text: π

text: E

text: LE

text: Because

text: T

text: is

text: a

text: right

text: element,

text: it

text: commutes

text: with

text: every

text: L

text: so

text: that

text: we

text: recover

text: Ex

text: b

text: Remark

text: The

text: deﬁnition

text: of

text: the

text: operator-valued

text: bi-moment

text: function

text: in

text: is

text: diﬀerent

text: from

text: Deﬁnition

text: and

text: much

text: more

text: involved.

text: However,

text: Thm

text: asserts

text: that

text: these

text: bi-free

text: moment

text: functions

text: are

text: completely

text: determined

text: by

text: E

text: and

text: the

text: properties

text: of

text: bi-multiplicative

text: functions.

text: This

text: means

text: that

text: they

text: coincide

text: with

text: Deﬁnition

text: We

text: took

text: here

text: advantage

text: of

text: this

text: deep

text: result

text: to

text: give

text: the

text: "easy"

text: deﬁnition

text: of

text: the

text: bi-moment

text: functions.


text: ON BI-FREE DE FINETTI THEOREMS 7 ⩽ ⩽ ∶ → ( ) Let us consider the classical permuta(tion) group S represented on CN as permutation matrices. For N 1 i, j N, let v S C be the function se(nd+in)g a permutation matrix σ to its i, j -th coeﬃcient. Then, ij N the functions v∗ generate the algeb∶ra C( +S) —→of al(l (co)ntinuous) functions on S . M(oreo)ve=r, the family v ij N N ij obviously satisﬁes the deﬁning relations of C S . Hence, by the+deﬁnition of a universal C*-algebra, there N is a surjective -homomorphism Φ C S C S completely determined by Φ u v . M+oreover, N N ij ij one can prove that Φ respects the quantum group structure of S and the group structure of S and that N N S is the biggest classical group admitting such a quotient map. This is one of the reasons why S can be N N seen as a quantum version of the permutation group. 3. Quantum invariance and bi-freeness for families of pairs In this section, we will study the consequences of quantum b+i-invariance in two diﬀerent settings, ﬁrst assuming some bi-noncommutative structure on the pairs and then taking a more general approach. This ﬁrst requires the introduction o+f a speciﬁc family of actions of S on families of pairs. N + It is not clear at ﬁrst how S can be used to deﬁne the symmetries of bi-freeness. The path we take here N is to consider, instead of a usual quantum group action, a linear action of S . A similar idea was used by N W. Liu in [7] to characterize boolean independence, even though he was using quantum semigroups instead of quantum groups. + ∶ → ⊗ ( + ) ( ⊗ ) ○ = ( ⊗ ) ○ Deﬁnition 3.1. A linear action of S on a vector space V is a linear map β V V C S such that N ( ⊗ ) ○ = N ( ) β id β id ∆ β. + If ϕ is a linear functional on V , then it is said to be invariant under β if ϕ id β ϕ.1 . C S N Let us insist that β is not assumed to be multiplicative, hence is not a quan(tum g)roup action in the usual sense. We will have to deal in the= s(equel wit)h moments o∈f{mon}omials, i.e. of products of elements indexed by tuples. To make things easier we introduce some shorthand notations. If x(cid:96), xr is a family of pairs of = ( ) ( ) j j j random variables then, for any I i , . . . , i and any χ (cid:96), r n, we set 1 n ( ) = ( ) = ( ) ∈ { } xχ xχ 1 . . . xχ n . I i1 in If u is a matrix whose coeﬃcient=s are operators then, fo=r any I i , . . . , i , any J j , . . . , j and ij ij ( ) ( )1 n 1 n any χ ((cid:96), r) =n,(we set ) ( ) ( ) ( ) u u . . . u and uχ u , IJ i1j1 injn IJ sχ I sχ J where s I i , . . . , i and similarly f∑or =s J∑. Eventually, when summing on tuples, we will set χ sχ 1 sχ n χ = N N . I i1,...,in 1 Remark 3.2. With the notations above, the formula for the coproduct of a monomial of coeﬃcients of u is ( ) = ∑ ⊗ given by N ∆ u u u . + IJ IK KJ K Our quantum symm(Aetrie)s for bi-freeness will be given by the following (linear a)ction of S : N ⩽ ⩽ A M ⊂ A Deﬁnition 3.3. Let , ϕ be a noncommutat+ive prMobability space, let x(cid:96), xr be a ﬁnite family of j j 1 j N pairs of random variables in and let be the algebra that they generate. The bi-free quantum permutation action, is the linear action β of S on given by N ( N ) = ∑ ⊗ N (3) β xχ xχ uχ . N J I IJ I

text: ON

text: BI-FREE

text: DE

text: FINETTI

text: THEOREMS

text: Let

text: us

text: consider

text: the

text: classical

text: permuta(tion)

text: group

text: S

text: represented

text: on

text: CN

text: as

text: permutation

text: matrices.

text: For

text: N

text: i,

text: j

text: N,

text: let

text: v

text: S

text: C

text: be

text: the

text: function

text: se(nd+in)g

text: a

text: permutation

text: matrix

text: σ

text: to

text: its

text: i,

text: j

text: -th

text: coeﬃcient.

text: Then,

text: ij

text: N

text: the

text: functions

text: v∗

text: generate

text: the

text: algeb∶ra

text: C(

text: +S)

text: —→of

text: al(l

text: (co)ntinuous)

text: functions

text: on

text: S

text: M(oreo)ve=r,

text: the

text: family

text: v

text: ij

text: N

text: N

text: ij

text: obviously

text: satisﬁes

text: the

text: deﬁning

text: relations

text: of

text: C

text: S

text: Hence,

text: by

text: the+deﬁnition

text: of

text: a

text: universal

text: C*-algebra,

text: there

text: N

text: is

text: a

text: surjective

text: -homomorphism

text: Φ

text: C

text: S

text: C

text: S

text: completely

text: determined

text: by

text: Φ

text: u

text: v

text: M+oreover,

text: N

text: N

text: ij

text: ij

text: one

text: can

text: prove

text: that

text: Φ

text: respects

text: the

text: quantum

text: group

text: structure

text: of

text: S

text: and

text: the

text: group

text: structure

text: of

text: S

text: and

text: that

text: N

text: N

text: S

text: is

text: the

text: biggest

text: classical

text: group

text: admitting

text: such

text: a

text: quotient

text: map.

text: This

text: is

text: one

text: of

text: the

text: reasons

text: why

text: S

text: can

text: be

text: N

text: N

text: seen

text: as

text: a

text: quantum

text: version

text: of

text: the

text: permutation

text: group.

text: Quantum

text: invariance

text: and

text: bi-freeness

text: for

text: families

text: of

text: pairs

text: In

text: this

text: section,

text: we

text: will

text: study

text: the

text: consequences

text: of

text: quantum

text: b+i-invariance

text: in

text: two

text: diﬀerent

text: settings,

text: ﬁrst

text: assuming

text: some

text: bi-noncommutative

text: structure

text: on

text: the

text: pairs

text: and

text: then

text: taking

text: a

text: more

text: general

text: approach.

text: This

text: ﬁrst

text: requires

text: the

text: introduction

text: o+f

text: a

text: speciﬁc

text: family

text: of

text: actions

text: of

text: S

text: on

text: families

text: of

text: pairs.

text: N

text: It

text: is

text: not

text: clear

text: at

text: ﬁrst

text: how

text: S

text: can

text: be

text: used

text: to

text: deﬁne

text: the

text: symmetries

text: of

text: bi-freeness.

text: The

text: path

text: we

text: take

text: here

text: N

text: is

text: to

text: consider,

text: instead

text: of

text: a

text: usual

text: quantum

text: group

text: action,

text: a

text: linear

text: action

text: of

text: S

text: A

text: similar

text: idea

text: was

text: used

text: by

text: N

text: W.

text: Liu

text: in

text: to

text: characterize

text: boolean

text: independence,

text: even

text: though

text: he

text: was

text: using

text: quantum

text: semigroups

text: instead

text: of

text: quantum

text: groups.

text: Deﬁnition

text: A

text: linear

text: action

text: of

text: S

text: on

text: a

text: vector

text: space

text: V

text: is

text: a

text: linear

text: map

text: β

text: V

text: V

text: C

text: S

text: such

text: that

text: N

text: N

text: β

text: id

text: β

text: id

text: β.

text: If

text: ϕ

text: is

text: a

text: linear

text: functional

text: on

text: V

text: then

text: it

text: is

text: said

text: to

text: be

text: invariant

text: under

text: β

text: if

text: ϕ

text: id

text: β

text: C

text: S

text: N

text: Let

text: us

text: insist

text: that

text: β

text: is

text: not

text: assumed

text: to

text: be

text: multiplicative,

text: hence

text: is

text: not

text: a

text: quan(tum

text: g)roup

text: action

text: in

text: the

text: usual

text: sense.

text: We

text: will

text: have

text: to

text: deal

text: in

text: the=

text: s(equel

text: wit)h

text: moments

text: o∈f{mon}omials,

text: i.e.

text: of

text: products

text: of

text: elements

text: indexed

text: by

text: tuples.

text: To

text: make

text: things

text: easier

text: we

text: introduce

text: some

text: shorthand

text: notations.

text: If

text: xr

text: is

text: a

text: family

text: of

text: pairs

text: of

text: j

text: j

text: j

text: random

text: variables

text: then,

text: for

text: any

text: I

text: i

text: i

text: and

text: any

text: χ

text: r

text: n,

text: we

text: set

text: n

text: xχ

text: xχ

text: xχ

text: n

text: I

text: in

text: If

text: u

text: is

text: a

text: matrix

text: whose

text: coeﬃcient=s

text: are

text: operators

text: then,

text: fo=r

text: any

text: I

text: i

text: i

text: any

text: J

text: j

text: j

text: and

text: ij

text: ij

text: n

text: n

text: any

text: χ

text: r)

text: =n,(we

text: set

text: u

text: u

text: u

text: and

text: uχ

text: u

text: IJ

text: injn

text: IJ

text: sχ

text: I

text: sχ

text: J

text: where

text: s

text: I

text: i

text: i

text: and

text: similarly

text: f∑or

text: =s

text: J∑.

text: Eventually,

text: when

text: summing

text: on

text: tuples,

text: we

text: will

text: set

text: χ

text: sχ

text: sχ

text: n

text: χ

text: N

text: N

text: I

text: Remark

text: With

text: the

text: notations

text: above,

text: the

text: formula

text: for

text: the

text: coproduct

text: of

text: a

text: monomial

text: of

text: coeﬃcients

text: of

text: u

text: is

text: given

text: by

text: N

text: u

text: u

text: u

text: IJ

text: IK

text: KJ

text: K

text: Our

text: quantum

text: symm(Aetrie)s

text: for

text: bi-freeness

text: will

text: be

text: given

text: by

text: the

text: following

text: (linear

text: a)ction

text: of

text: S

text: N

text: A

text: M

text: A

text: Deﬁnition

text: Let

text: ϕ

text: be

text: a

text: noncommutat+ive

text: prMobability

text: space,

text: let

text: xr

text: be

text: a

text: ﬁnite

text: family

text: of

text: j

text: j

text: j

text: N

text: pairs

text: of

text: random

text: variables

text: in

text: and

text: let

text: be

text: the

text: algebra

text: that

text: they

text: generate.

text: The

text: bi-free

text: quantum

text: permutation

text: action,

text: is

text: the

text: linear

text: action

text: β

text: of

text: S

text: on

text: given

text: by

text: N

text: N

text: N

text: β

text: xχ

text: xχ

text: uχ

text: N

text: J

text: I

text: IJ

text: I


text: Learning to Rank Academic Experts in the DBLP Dataset Catarina Moreira Pa´vel Calado Bruno Martins catarina.p.moreira@ist.utl.pt pavel.calado@ist.utl.pt bruno.g.martins@ist.utl.pt Instituto Superior T´ecnico, INESC-ID Av. Professor Cavaco Silva, 2744-016 Porto Salvo, Portugal The original publication is available at: Expert Systems, Wiley Online Library http://onlinelibrary.wiley.com/doi/10.1111/exsy.12062/abstract Abstract Expert ﬁnding is an information retrieval task that is concerned with the search for the most knowledgeable people with respect to a speciﬁc topic, and the search is based on documents that describe people’s activities. The task involves taking a user query as input and returning a list of people who are sorted by their level of expertise with respect to the user query. Despite recent interest in the area, the current state-of-the-art techniques lack in principled approaches for optimally combining diﬀerent sources of evidence. This article proposes two frameworks for combining multiple estimators of expertise. These estimators are derived from textual contents, from graph-structure of the citation patterns for the community of experts, and from proﬁle information about the experts. More speciﬁcally, this article explores the use of supervised learning to rank methods, as well as rank aggregation approaches, for combing all of the estimators of expertise. Several supervised learning algorithms, which are representative of the pointwise, pairwise and listwise approaches, were tested, and various state-of-the-art data fusion techniques were also explored for the rank aggregation framework. Experiments that were performed on a dataset of academic publications from the Computer Science domain attest the adequacy of the proposed approaches. 1 Introduction The search for people who are knowledgeable about speciﬁc topic areas and who are within the scope of speciﬁc user communities, with basis on documents that describe people’s activities, is an information retrieval problem that has been receiving increasing attention (Serdyukov 2009). 1

text: Learning

label: title

text: to

text: Rank

text: Academic

text: Experts

text: in

text: the

text: DBLP

text: Dataset

text: Catarina

label: author

text: Moreira

text: Pa´vel

text: Calado

text: Bruno

text: Martins

text: catarina.p.moreira@ist.utl.pt

text: pavel.calado@ist.utl.pt

text: bruno.g.martins@ist.utl.pt

text: Instituto

text: Superior

text: T´ecnico,

text: INESC-ID

text: Av.

text: Professor

text: Cavaco

text: Silva,

text: Porto

text: Salvo,

text: Portugal

text: The

text: original

text: publication

text: is

text: available

text: at:

text: Expert

text: Systems,

text: Wiley

text: Online

text: Library

text: Abstract

text: Expert

label: abstract

text: ﬁnding

text: is

text: an

text: information

text: retrieval

text: task

text: that

text: is

text: concerned

text: with

text: the

text: search

text: for

text: the

text: most

text: knowledgeable

text: people

text: with

text: respect

text: to

text: a

text: speciﬁc

text: topic,

text: and

text: the

text: search

text: is

text: based

text: on

text: documents

text: that

text: describe

text: people’s

text: activities.

text: The

text: task

text: involves

text: taking

text: a

text: user

text: query

text: as

text: input

text: and

text: returning

text: a

text: list

text: of

text: people

text: who

text: are

text: sorted

text: by

text: their

text: level

text: of

text: expertise

text: with

text: respect

text: to

text: the

text: user

text: query.

text: Despite

text: recent

text: interest

text: in

text: the

text: area,

text: the

text: current

text: state-of-the-art

text: techniques

text: lack

text: in

text: principled

text: approaches

text: for

text: optimally

text: combining

text: diﬀerent

text: sources

text: of

text: evidence.

text: This

text: article

text: proposes

text: two

text: frameworks

text: for

text: combining

text: multiple

text: estimators

text: of

text: expertise.

text: These

text: estimators

text: are

text: derived

text: from

text: textual

text: contents,

text: from

text: graph-structure

text: of

text: the

text: citation

text: patterns

text: for

text: the

text: community

text: of

text: experts,

text: and

text: from

text: proﬁle

text: information

text: about

text: the

text: experts.

text: More

text: speciﬁcally,

text: this

text: article

text: explores

text: the

text: use

text: of

text: supervised

text: learning

text: to

text: rank

text: methods,

text: as

text: well

text: as

text: rank

text: aggregation

text: approaches,

text: for

text: combing

text: all

text: of

text: the

text: estimators

text: of

text: expertise.

text: Several

text: supervised

text: learning

text: algorithms,

text: which

text: are

text: representative

text: of

text: the

text: pointwise,

text: pairwise

text: and

text: listwise

text: approaches,

text: were

text: tested,

text: and

text: various

text: state-of-the-art

text: data

text: fusion

text: techniques

text: were

text: also

text: explored

text: for

text: the

text: rank

text: aggregation

text: framework.

text: Experiments

text: that

text: were

text: performed

text: on

text: a

text: dataset

text: of

text: academic

text: publications

text: from

text: the

text: Computer

text: Science

text: domain

text: attest

text: the

text: adequacy

text: of

text: the

text: proposed

text: approaches.

text: Introduction

text: The

text: search

text: for

text: people

text: who

text: are

text: knowledgeable

text: about

text: speciﬁc

text: topic

text: areas

text: and

text: who

text: are

text: within

text: the

text: scope

text: of

text: speciﬁc

text: user

text: communities,

text: with

text: basis

text: on

text: documents

text: that

text: describe

text: people’s

text: activities,

text: is

text: an

text: information

text: retrieval

text: problem

text: that

text: has

text: been

text: receiving

text: increasing

text: attention

text: (Serdyukov


text: the query topics. Then, according to the data fusion method that is used, all of the individual scores of each feature will be combined into a single value. For example, if one uses the CombSUM data fusion method, then the scores of each feature will be summed. The authors are then ranked by the resulting summation score. 5 Features for Estimating Author Expertise The considered set of features for estimating the degree of expertise of a person toward a given query can be divided into three groups, namely, the textual features, proﬁle features and citation graph features. The textual features are similar to those used in standard text retrieval systems and also in previous learning to rank experiments (e.g., TF-IDF and BM25 scores). The proﬁle information features correspond to importance estimates for the authors, which are derived from their proﬁle information (e.g., the number of papers published). Finally, the graph features correspond to importance and relevance estimates that are computed from citation counts and citation graphs. 5.1 Textual Similarity Features Similar to previous expert ﬁnding studies that are based on document-centric approaches, we also use textual similarities between the query and the contents of the documents to build estimates of expertise. In the domain of academic digital libraries, the associations between documents and experts can easily be obtained from the authorship information that is associated with the publications. For each topic-expert pair, we used the Okapi BM25 document-scoring function to compute the textual similarity features. Okapi BM25 is a state-of-the-art IR ranking mechanism that is composed of several simpler scoring functions with diﬀerent parameters and components (e.g., term frequency and inverse document frequency). It can be computed through the formula shown in Equation 5, where T erms(q) represents the set of terms from query q, Docs(a) is the set of documents that have a as an author, Freq(i, d ) is the number of occurrences j | | A of term i in document d , d is the number of terms in document d , and is the average length j j j of the documents in the collection. The values given to the parameters k and b were 1.2 and 1 0.75, respectively. Most of the previous IR experiments use these default values for the k and b 1 parameters. 18

text: the

text: query

text: topics.

text: Then,

text: according

text: to

text: the

text: data

text: fusion

text: method

text: that

text: is

text: used,

text: all

text: of

text: the

text: individual

text: scores

text: of

text: each

text: feature

text: will

text: be

text: combined

text: into

text: a

text: single

text: value.

text: For

text: example,

text: if

text: one

text: uses

text: the

text: CombSUM

text: data

text: fusion

text: method,

text: then

text: the

text: scores

text: of

text: each

text: feature

text: will

text: be

text: summed.

text: The

text: authors

text: are

text: then

text: ranked

text: by

text: the

text: resulting

text: summation

text: score.

text: Features

text: for

text: Estimating

text: Author

text: Expertise

text: The

text: considered

text: set

text: of

text: features

text: for

text: estimating

text: the

text: degree

text: of

text: expertise

text: of

text: a

text: person

text: toward

text: a

text: given

text: query

text: can

text: be

text: divided

text: into

text: three

text: groups,

text: namely,

text: the

text: textual

text: features,

text: proﬁle

text: features

text: and

text: citation

text: graph

text: features.

text: The

text: textual

text: features

text: are

text: similar

text: to

text: those

text: used

text: in

text: standard

text: text

text: retrieval

text: systems

text: and

text: also

text: in

text: previous

text: learning

text: to

text: rank

text: experiments

text: (e.g.,

text: TF-IDF

text: and

text: scores).

text: The

text: proﬁle

text: information

text: features

text: correspond

text: to

text: importance

text: estimates

text: for

text: the

text: authors,

text: which

text: are

text: derived

text: from

text: their

text: proﬁle

text: information

text: (e.g.,

text: the

text: number

text: of

text: papers

text: published).

text: Finally,

text: the

text: graph

text: features

text: correspond

text: to

text: importance

text: and

text: relevance

text: estimates

text: that

text: are

text: computed

text: from

text: citation

text: counts

text: and

text: citation

text: graphs.

text: Textual

text: Similarity

text: Features

text: Similar

text: to

text: previous

text: expert

text: ﬁnding

text: studies

text: that

text: are

text: based

text: on

text: document-centric

text: approaches,

text: we

text: also

text: use

text: textual

text: similarities

text: between

text: the

text: query

text: and

text: the

text: contents

text: of

text: the

text: documents

text: to

text: build

text: estimates

text: of

text: expertise.

text: In

text: the

text: domain

text: of

text: academic

text: digital

text: libraries,

text: the

text: associations

text: between

text: documents

text: and

text: experts

text: can

text: easily

text: be

text: obtained

text: from

text: the

text: authorship

text: information

text: that

text: is

text: associated

text: with

text: the

text: publications.

text: For

text: each

text: topic-expert

text: pair,

text: we

text: used

text: the

text: Okapi

text: document-scoring

text: function

text: to

text: compute

text: the

text: textual

text: similarity

text: features.

text: Okapi

text: is

text: a

text: state-of-the-art

text: IR

text: ranking

text: mechanism

text: that

text: is

text: composed

text: of

text: several

text: simpler

text: scoring

text: functions

text: with

text: diﬀerent

text: parameters

text: and

text: components

text: (e.g.,

text: term

text: frequency

text: and

text: inverse

text: document

text: frequency).

text: It

text: can

text: be

text: computed

text: through

text: the

text: formula

text: shown

text: in

text: Equation

text: where

text: T

text: erms(q)

text: represents

text: the

text: set

text: of

text: terms

text: from

text: query

text: q,

text: Docs(a)

text: is

text: the

text: set

text: of

text: documents

text: that

text: have

text: a

text: as

text: an

text: author,

text: Freq(i,

text: d

text: is

text: the

text: number

text: of

text: occurrences

text: j

text: A

text: of

text: term

text: i

text: in

text: document

text: d

text: d

text: is

text: the

text: number

text: of

text: terms

text: in

text: document

text: d

text: and

text: is

text: the

text: average

text: length

text: j

text: j

text: j

text: of

text: the

text: documents

text: in

text: the

text: collection.

text: The

text: values

text: given

text: to

text: the

text: parameters

text: k

text: and

text: b

text: were

text: and

text: respectively.

text: Most

text: of

text: the

text: previous

text: IR

text: experiments

text: use

text: these

text: default

text: values

text: for

text: the

text: k

text: and

text: b

text: parameters.


text: than all of the other approaches that were tested. On the other hand, the listwise SVMmap algorithm, with an RBF kernel, performed almost as good as the Additive Groves method. This ﬁnding makes sense because the goal of SVMmap is to optimise the Mean Average Precision scores. In fact, SVMmap outperformed Additive Groves when ranking the top 5 experts (P@5), which shows that this listwise method can also be successfully used in the context of expert ﬁnding. Table 4 also shows competitive results for the pairwise approach of SVMrank with a linear kernel and for RankBoost. This ﬁnding leads to the conclusion that pairwise learning to rank algorithms that are based on the formalisms of support vector machines and on the boosting framework can also be applied successfully in the task of expert ﬁnding. With respect to the worst results that were obtained, Table 4 shows that AdaRank and RankNet were not as successful as the other algorithms. For RankNet, an easy explanation can be found in the use of gradient descent in the backpropagation algorithm. Gradient descent is an optimisation algorithm that attempts to ﬁnd the minimum of a function by taking steps that are proportional to the negative gradient of the function. However, this method has the problem of not being able to ﬁnd a global minimum because it can get stuck in a local minimum and, therefore, a good optimisation might never be achieved. In addition, the cost function that is used in RankNet is general and does not correspond to any speciﬁc information-retrieval metric. For the case of AdaRank, its greedy feature selection method did not perform very well in this experiment. Although AdaRank and RankBoost are based on the same paradigm, i.e., boosting, they use it in very diﬀerent ways. For example, RankBoost contains an extra parameter, which is the number of threshold candidates. AdaRank, on the other hand, makes a greedy linear com- bination of weak rankers. RankBoost not only makes a linear combination of weak rankers, but also considers whether an expert who is returned by a query is above a speciﬁc threshold (this approach decreases the weight of the weak rankers of correctly ranked experts and consequently builds more accurate ranking functions). Table 4 presents the results of the baseline models that were proposed by Balog et al. (2006), namely the candidate-based Model 1 and the document-based Model 2. To make the comparison fair, we used the code that was made publicly available by Balog7. Experiments revealed that Model 1 and Model 2 have similar performances on such an academic dataset, but they achieved a lower performance when compared to all of the algorithms that were tested with the supervised learning approach. In Model 1, when an author publishes a paper that contains a set of words that 7http://code.google.com/p/ears/ 28 ##LTLine##

text: than

text: all

text: of

text: the

text: other

text: approaches

text: that

text: were

text: tested.

text: On

text: the

text: other

text: hand,

text: the

text: listwise

text: SVMmap

text: algorithm,

text: with

text: an

text: RBF

text: kernel,

text: performed

text: almost

text: as

text: good

text: as

text: the

text: Additive

text: Groves

text: method.

text: This

text: ﬁnding

text: makes

text: sense

text: because

text: the

text: goal

text: of

text: SVMmap

text: is

text: to

text: optimise

text: the

text: Mean

text: Average

text: Precision

text: scores.

text: In

text: fact,

text: SVMmap

text: outperformed

text: Additive

text: Groves

text: when

text: ranking

text: the

text: top

text: experts

text: which

text: shows

text: that

text: this

text: listwise

text: method

text: can

text: also

text: be

text: successfully

text: used

text: in

text: the

text: context

text: of

text: expert

text: ﬁnding.

text: Table

text: also

text: shows

text: competitive

text: results

text: for

text: the

text: pairwise

text: approach

text: of

text: SVMrank

text: with

text: a

text: linear

text: kernel

text: and

text: for

text: RankBoost.

text: This

text: ﬁnding

text: leads

text: to

text: the

text: conclusion

text: that

text: pairwise

text: learning

text: to

text: rank

text: algorithms

text: that

text: are

text: based

text: on

text: the

text: formalisms

text: of

text: support

text: vector

text: machines

text: and

text: on

text: the

text: boosting

text: framework

text: can

text: also

text: be

text: applied

text: successfully

text: in

text: the

text: task

text: of

text: expert

text: ﬁnding.

text: With

text: respect

text: to

text: the

text: worst

text: results

text: that

text: were

text: obtained,

text: Table

text: shows

text: that

text: AdaRank

text: and

text: RankNet

text: were

text: not

text: as

text: successful

text: as

text: the

text: other

text: algorithms.

text: For

text: RankNet,

text: an

text: easy

text: explanation

text: can

text: be

text: found

text: in

text: the

text: use

text: of

text: gradient

text: descent

text: in

text: the

text: backpropagation

text: algorithm.

text: Gradient

text: descent

text: is

text: an

text: optimisation

text: algorithm

text: that

text: attempts

text: to

text: ﬁnd

text: the

text: minimum

text: of

text: a

text: function

text: by

text: taking

text: steps

text: that

text: are

text: proportional

text: to

text: the

text: negative

text: gradient

text: of

text: the

text: function.

text: However,

text: this

text: method

text: has

text: the

text: problem

text: of

text: not

text: being

text: able

text: to

text: ﬁnd

text: a

text: global

text: minimum

text: because

text: it

text: can

text: get

text: stuck

text: in

text: a

text: local

text: minimum

text: and,

text: therefore,

text: a

text: good

text: optimisation

text: might

text: never

text: be

text: achieved.

text: In

text: addition,

text: the

text: cost

text: function

text: that

text: is

text: used

text: in

text: RankNet

text: is

text: general

text: and

text: does

text: not

text: correspond

text: to

text: any

text: speciﬁc

text: information-retrieval

text: metric.

text: For

text: the

text: case

text: of

text: AdaRank,

text: its

text: greedy

text: feature

text: selection

text: method

text: did

text: not

text: perform

text: very

text: well

text: in

text: this

text: experiment.

text: Although

text: AdaRank

text: and

text: RankBoost

text: are

text: based

text: on

text: the

text: same

text: paradigm,

text: i.e.,

text: boosting,

text: they

text: use

text: it

text: in

text: very

text: diﬀerent

text: ways.

text: For

text: example,

text: RankBoost

text: contains

text: an

text: extra

text: parameter,

text: which

text: is

text: the

text: number

text: of

text: threshold

text: candidates.

text: AdaRank,

text: on

text: the

text: other

text: hand,

text: makes

text: a

text: greedy

text: linear

text: com-

text: bination

text: of

text: weak

text: rankers.

text: RankBoost

text: not

text: only

text: makes

text: a

text: linear

text: combination

text: of

text: weak

text: rankers,

text: but

text: also

text: considers

text: whether

text: an

text: expert

text: who

text: is

text: returned

text: by

text: a

text: query

text: is

text: above

text: a

text: speciﬁc

text: threshold

text: (this

text: approach

text: decreases

text: the

text: weight

text: of

text: the

text: weak

text: rankers

text: of

text: correctly

text: ranked

text: experts

text: and

text: consequently

text: builds

text: more

text: accurate

text: ranking

text: functions).

text: Table

text: presents

text: the

text: results

text: of

text: the

text: baseline

text: models

text: that

text: were

text: proposed

text: by

text: Balog

text: et

text: al.

text: namely

text: the

text: candidate-based

text: Model

text: and

text: the

text: document-based

text: Model

text: To

text: make

text: the

text: comparison

text: fair,

text: we

text: used

text: the

text: code

text: that

text: was

text: made

text: publicly

text: available

text: by

text: Experiments

text: revealed

text: that

text: Model

text: and

text: Model

text: have

text: similar

text: performances

text: on

text: such

text: an

text: academic

text: dataset,

text: but

text: they

text: achieved

text: a

text: lower

text: performance

text: when

text: compared

text: to

text: all

text: of

text: the

text: algorithms

text: that

text: were

text: tested

text: with

text: the

text: supervised

text: learning

text: approach.

text: In

text: Model

text: when

text: an

text: author

text: publishes

text: a

text: paper

text: that

text: contains

text: a

text: set

text: of

text: words

text: that

text: ##LTLine##


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text: Y.,

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text: expert

text: search

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text: REtrieval

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text: Chapelle,

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text: papers)’.

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text: ence’.

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text: quantify

text: an

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text: in

text: ‘Proceedings

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text: the

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text: Academy

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text: Ji,

text: M.,

text: Han,

text: J.

text: Danilevsky,

text: M.

text: Ranking-based

text: classiﬁcation

text: of

text: heterogeneous

text: information

text: networks,

text: in

text: ‘Proceedings

text: of

text: the

text: ACM

text: International

text: Conference

text: on

text: Knowledge

text: Discovery

text: and

text: Data

text: Mining’.

text: Joachims,

text: T.

text: Training

text: linear

text: SVMs

text: in

text: linear

text: time,

text: in

text: ‘Proceedings

text: of

text: the

text: ACM

text: Conference

text: on

text: Knowledge

text: Discovery

text: and

text: Data

text: Mining’.

text: Liu,

text: T.-Y.

text: ‘Learning

text: to

text: rank

text: for

text: information

text: retrieval’,

text: Foundations

text: of

text: Trends

text: Information

text: Retrieval

text: Liu,

text: X.,

text: Bollen,

text: J.,

text: Nelson,

text: M.

text: L.

text: de

text: Sompel,

text: H.

text: V.

text: Co-authorship

text: networks

text: in

text: the

text: digital

text: library

text: research

text: community,

text: in

text: ‘Information

text: Processing

text: and

text: Management’

text: Macdonald,

text: C.

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