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Inference in Bayesian Networks
Šimeček, Josef ; Rozman, Jaroslav (referee) ; Zbořil, František (advisor)
This master's thesis deals with demonstration of various approaches to probabilistic inference in Bayesian networks. Basics of probability theory, introduction to Bayesian networks, methods for Bayesian inference and applications of Bayesian networks are described in theoretical part. Inference techniques are explained and complemented by their algorithm. Techniques are also illustrated on example. Practical part contains implementation description, experiments with demonstration applications and conclusion of the results.
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Two Composition Operators for Belief Functions Revisited
Jiroušek, Radim ; Kratochvíl, Václav ; Shenoy, P. P.
In probability theory, compositional models are as powerful as Bayesian networks. However, the relation between belief-function graphical models and the corresponding compositional models is much more complicated due to several reasons. One of them is that there are two composition operators for belief functions. This paper deals with their main properties and presents sufficient conditions under which they yield the same results.
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Computing the Decomposable Entropy of Graphical Belief Function Models
Jiroušek, Radim ; Kratochvíl, Václav ; Shenoy, P. P.
In 2018, Jiroušek and Shenoy proposed a definition of entropy for Dempster-Shafer (D-S) belief functions called decomposable entropy. Here, we provide an algorithm for computing the decomposable entropy of directed graphical D-S belief function models. For undirected graphical belief function models, assuming that each belief function in the model is non-informative to the others, no algorithm is necessary. We compute the entropy of each belief function and add them together to get the decomposable entropy of the model. Finally, the decomposable entropy generalizes Shannon’s entropy not only for the probability of a single random variable but also for multinomial distributions expressed as directed acyclic graphical models called Bayesian networks.
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Subjective well-being and the individual material situation in Central Europe: A Bayesian network approach
Švorc, Jan ; Vomlel, Jiří
The objective of this paper is to explore the associations between the subjective well-being (SWB) and the subjective and objective measures of the individual material situation in the four post-communist countries of Central Europe (the Czech Republic, Hungary, Poland, and Slovakia). The material situation is measured by income, relative income compared to others, relative income compared to one’s own past, perceived economic strain, financial problems, material deprivation, and housing problems. Our analysis is based on empirical data from the third wave of European Quality of Life Study conducted in 2011. Bayesian networks as a graphical representation of the relations between SWB and the material situation have been constructed in five versions. The models have been assessed using the Bayesian Information Criterion (BIC) and SWB prediction accuracy, and compared\nwith Ordinal Logistic Regression (OLR). Expert knowledge, as well as three different algorithms (greedy, Gobnilp, and Tree-augmented Naive Bayes) were used for learning the network structures. Network parameters were learned using the EM algorithm. Parameters based on OLR were learned for a version of the expert model. The Gobnilp model, the Markov equivalent to the greedy model, is BIC optimal. The OLR predicts SWB slightly better than the other models. We conclude that the objective material conditions' influence on SWB is rather indirect, through the subjective situational assessment of various aspects related to the individual material conditions.
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Bayesian Networks for the Analysis of Subjective Well-Being
Švorc, Jan ; Vomlel, Jiří
We use Bayesian Networks to model the influence of diverse socio-economic factors on subjective well-being and their interrelations. The classical statistical analysis aims at finding significant explanatory variables, while Bayesian Networks can also help sociologists to explain and visualize the problem in its complexity. Using Bayesian Networks the sociologists may get a deeper insight into the interplay of all measured factors and their influence on the variable of a special interest. In the paper we present several Bayesian Network models -- each being optimal from a different perspective. We show how important it is to pay a special attention to a local structure of conditional probability tables. Finally, we present results of an experimental evaluation of the suggested approaches based on real data from a large international survey. We believe that the suggested approach is well applicable to other sociological problems and that Bayesian Networks represent a new valuable tool for sociological research.
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Question Selection Methods for Adaptive Testing with Bayesian Networks
Plajner, Martin ; Magauina, A. ; Vomlel, Jiří
The performance of Computerized Adaptive Testing systems, which are used for testing of human knowledge, relies heavily on methods selecting correct questions for tested students. In this article we propose three different methods selecting questions with Bayesian networks as students’ models. We present the motivation to use these methods and their mathematical description. Two empirical datasets, paper tests of specific topics in mathematics and Czech language for foreigners, were collected for the purpose of methods’ testing. All three methods were tested using simulated testing procedure and results are compared for individual methods. The comparison is done also with the sequential selection of questions to provide a relation to the classical way of testing. The proposed methods are behaving much better than the sequential selection which verifies the need to use a better selection method. Individually, our methods behave differently, i.e., select different questions but the success rate of model’s predictions is very similar for all of them. This motivates further research in this topic to find an ordering between methods and to find the best method which would provide the best possible selections in computerized adaptive tests.
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Proceedings of the 11th Workshop on Uncertainty Processing
Kratochvíl, Václav ; Vejnarová, Jiřina
The Workshop on Uncertainty Processing, better known under its abbreviation WUPES, celebrates its 30-year anniversary this year. In 1988, when the first Workshop took place, Czechoslovakia was still a communist country and a part of the Soviet bloc. Since then, many things have changed. For example, Czechoslovakia no longer exists as a country (because in 1993 it was peacefully split into two independent countries - Czechia and Slovakia). From this perspective, it is hard to believe that we have several participants who have attended most workshops in the the thirty-year history of WUPES. As of now, the Program Committee has accepted, based on the extended abstracts, 21 papers to be presented at the Workshop, and 19 out of them are to be published in the present Conference Proceedings. These papers cover diverse topics, such as information processing, decision making, and data analysis, but what is common to most of them is that they are related to uncertainty calculus - Bayesian Networks, Dempster-Shafer Theory, Belief Functions, Probabilistic Logic, Game Theory, etc.
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Gradient Descent Parameter Learning of Bayesian Networks under Monotonicity Restrictions
Plajner, Martin ; Vomlel, Jiří
Learning parameters of a probabilistic model is a necessary step in most machine learning modeling tasks. When the model is complex and data volume is small the learning process may fail to provide good results. In this paper we present a method to improve learning results for small data sets by using additional information about the modelled system. This additional information is represented by monotonicity conditions which are restrictions on parameters of the model. Monotonicity simplifies the learning process and also these conditions are often required by the user of the system to hold. \n\nIn this paper we present a generalization of the previously used algorithm for parameter learning of Bayesian Networks under monotonicity conditions. This generalization allows both parents and children in the network to have multiple states. The algorithm is described in detail as well as monotonicity conditions are.\n\nThe presented algorithm is tested on two different data sets. Models are trained on differently sized data subsamples with the proposed method and the general EM algorithm. Learned models are then compared by their ability to fit data. We present empirical results showing the benefit of monotonicity conditions. The difference is especially significant when working with small data samples. The proposed method outperforms the EM algorithm for small sets and provides comparable results for larger sets.
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Employing Bayesian Networks for Subjective Well-being Prediction
Švorc, Jan ; Vomlel, Jiří
This contribution aims at using Bayesian networks for modelling the relations between the individual subjective well-being (SWB) and the individual material situation. The material situation is approximated by subjective measures (perceived economic strain, subjective evaluation of the income relative to most people in the country and to own past) and objective measures (household’s income, material deprivation, financial problems and housing defects). The suggested Bayesian network represents the relations among SWB and the variables approximating the material situation. The structure is established based on the expertise gained from literature, whereas the parameters are learnt based on empirical data from 3rd edition of European Quality of Life Study for the Czech Republic, Hungary, Poland and Slovakia conducted in 2011. Prediction accuracy of SWB is tested and compared with two benchmark models whose structures are learnt using Gobnilp software and a greedy algorithm built in Hugin software. SWB prediction accuracy of the expert model is 66,83%, which is significantly different from no information rate of 55,16%. It is slightly lower than the two machine learnt benchmark models.
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A machine learning method for incomplete and imbalanced medical data
Salman, I. ; Vomlel, Jiří
Our research reported in this paper is twofold. In the first part of the paper we use\nstandard statistical methods to analyze medical records of patients suffering myocardial\ninfarction from the third world Syria and a developed country - the Czech Republic.\nOne of our goals is to find whether there are statistically significant differences between\nthe two countries. In the second part of the paper we present an idea how to deal with\nincomplete and imbalanced data for tree-augmented naive Bayesian (TAN). All results\npresented in this paper are based on a real data about 603 patients from a hospital in\nthe Czech Republic and about 184 patients from two hospitals in Syria.
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