رتبه بندی ارزیابی موسسات بر اساس یک شبکه بیزی با داشتن یک متغیر پنهان

This paper proposes a new probabilistic graphical model which contains an unobservable latent variable that affects all other observable variables, and the proposed model is applied to ranking evaluation of institutions using a set of performance indicators. Linear Gaussian models are used to express the causal relationship among variables. The proposed iterative method uses a combined causal discovery algorithm of score-based and constraint-based methods to find the network structure, while Gibbs sampling and regression analysis are conducted to estimate the parameters. The latent variable representing ranking scores of institutions is estimated, and the rankings are determined by comparing the estimated scores. The interval estimate of the ranking of an institution is finally obtained from a repetitive procedure. The proposed procedure was applied to a real data set as well as artificial data sets.

Ranking higher education institutions began in the United States in the early 20th century, and it has become increasingly more influential with time. Most ranking systems select the performance indicators (research, education, etc.), assign a weight to each indicator and calculate the weighted score for each institution; this is the ‘weight-and-sum’ approach. However, this method sometimes becomes controversial because assigning weights to performance indicators is subjective. Thus demand for a new quantitative method of ranking is on the rise although there are many issues other than raking methods as pointed out in Vught and Westerheijden [34]. Some studies have used approaches other than the weight-and-sum approach to rank institutions. These methods include an empirical Bayes approach to ranking schools based on student achievement [18], Bayesian analysis for ranking institutions [15], a latent-variable technique for university ranking based on several indicators [16] and the analytic hierarchy process for determining weights of indicators [23]. In addition, a method based on a supervised naïve Bayes structure uses the mixture of truncated exponentials, which applies the rank information of an expert [12]. A Bayesian latent variable model has been proposed for estimating the top ranked SNPs detected from genetic association studies [13], which may also be applied to the institution ranking problem. When using the ranking evaluation systems other than the weight-and-sum approach, a structure analysis is necessary which considers the relationship among performance indicators. However, few studies have been conducted on the structure analysis for ranking evaluation. The ordinary modeling techniques for the structure analysis are structural equation models (SEMs) based either on linear structure relationships or partial least squares and Bayesian networks (BNs) [20]. Especially, BNs can be used as probabilistic inference engines, building models of domains that have intrinsic uncertainty. BNs are graphical models based on the notion of conditional independence that subsumes a wide range of statistical models including regression models, factor analysis models, and structural equation models. In a BN, a directed acyclic graph (DAG) represents a set of conditional independence constraints among a given number of variables and their related conditional probability distributions. The procedures for developing BNs involve learning the structure (the relationships between variables) first, and then parameterizing the associated conditional distributions. Graphical models, in particular those based on DAGs, have natural causal interpretations and thus form a language in which causal concepts can be discussed and analyzed in precise terms [21]. The conditional independence assumptions in a BN yield models more compact than those based on full joint probability distributions, thus reducing computational complexity when the number of variables is large [30]. Lately, BNs have been used in various applications such as risk management [3], resource allocation decisions [10] and [11], IT implementation [20], species distribution [1], higher education [12], and health risk assessment [22]. In this paper, we propose a new Bayesian network model which has an unobservable latent variable that affects all other observable variables. To solve the ranking problem, the latent variable represents the ranking variable to be finally estimated, and the observable variables indicate performance indicators of each institution. Our first task is to identify the causal relation among these variables, and the second task is to determine the institution rankings in terms of intervals. Causal discovery algorithms (CDAs) for a BN can be classified into three different categories: the score-based approach, the constraint-based approach, and the combined approach [35]. Score-based methods select a model with the highest posterior probability when the prior of each model is given. Constraint-based methods use statistical methods to detect associations and independencies among variables. Each method has a trade-off between time complexity and accuracy. Score-based methods generally provide accuracy but their time complexity is very high. Constraint-based methods give much lower time complexity, but they may not be accurate if conditional independence (CI) tests fail. The combined methods use both concepts to build the graph by compromising the benefit in terms of time complexity and accuracy. These methods cannot be directly applied to our new graphical model which has a latent variable, so a new approach needs to be developed for our purpose. The proposed approach consists of two phases and each phase is repeated until the network structure converges. First, latent rankings are obtained using Gibbs sampling and gradient descent from a given network structure. Second, the updated network structure is found by the revised version of Multiple Search (MS) algorithm [7] using the latent rankings. The final rankings in terms of interval estimates are obtained based on the convergent structure. The rest of the paper is organized as follows. The BN under consideration is modeled in Section 2. The proposed method of learning the BN is described in Section 3. The proposed method is applied to real data in Section 4. Section 5 reports on a further experiment with artificial data to check the accuracy and consistency of the proposed new Bayesian network model. Conclusions are presented in Section 6.

In this paper, we considered a new type of Bayesian network containing an unobservable latent variable which affects all observable variables. We applied the proposed method to the task of ranking institutions and determining the rankings in terms of intervals based only on the institutions’ scores in the performance indicators. The linear Gaussian models are utilized to express the causal relationship among variables in the network. The proposed iterative method finds the network structure of variables by using a combined algorithm of the revised MS and the BIC scoring function, estimating the parameters by using Gibbs sampling, gradient descent and regression analysis. The latent variable that represents the ranking score of an institution is estimated, and the final rankings are determined by comparing those scores. The interval estimate and the median estimate of the ranking of an instruction are obtained by repeating the estimation process. The analyses of a real case and some artificial data sets demonstrate the performance of the proposed method. To sum up, the proposed method has some advantages as a ranking system. First, the proposed method compresses the information of data (performance indicators in our case) into a single latent variable. Second, the proposed method does not require weights for indicators in advance, and therefore, it avoids the controversial subjective assignment of such weights. Third, using the interval estimate rather than the point estimate makes it possible to compare institutions statistically. However, the proposed method still has some limitations in computation time and in convergence. We also need to handle some semantic constraints in the network structure when utilizing an expert knowledge. On the other hand, the proposed method can be applied to a more general model where a latent variable has both in-degree and out-degree to all observable variables. If we consider a latent variable as a clustering node or a class node, then we can apply the proposed method to an unsupervised clustering problem or a classification problem. These issues should be considered in the future work.