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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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On attempts to characterize facet-defining inequalities of the cone of exact games
Studený, Milan ; Kroupa, Tomáš ; Kratochvíl, Václav
The sets of balanced, totally balanced, exact and supermodular games play an important role in cooperative game theory. These sets of games are known to be polyhedral cones. The (unique) non-redundant description of these cones by means of the so-called facet-defining inequalities is known in cases of balanced games and supermodular games, respectively. The facet description of the cones of exact games and totally balanced games are not known and we present conjectures about what are the facet-defining inequalities for these cones. We introduce the concept of an irreducible min-balanced set system and conjecture that the facet-defining inequalities for the cone of totally balanced games correspond to these set systems. The conjecture concerning exact games is that the facet-defining inequalities for this cone are those which correspond to irreducible min-balanced systems on strict subsets of the set of players and their conjugate inequalities. A consequence of the validity of the conjectures would be a novel result saying that a game m is exact if and only if m and its reflection are totally balanced.
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Comparison of Shenoy’s Expectation Operator with Probabilistic Transforms and Perez’ Barycenter
Jiroušek, R. ; Kratochvíl, Václav
Shenoy’s paper published in this Proceedings of WUPES 2018 introduces an operator that gives instructions how to compute an expected value in the Dempster-Shafer theory of evidence. Up to now, there was no direct way to get the expected value of a utility function in D-S theory. If eeded, one had to find a probability mass function corresponding to the considered belief function, and then - using this probability mass function - to compute the classical probabilistic expectation. In this paper, we take four different approaches to defining probabilistic representatives of a belief function and compare which one yields to the best approximations of Shenoy’s expected values of various utility functions. The achieved results support our conjecture that there does not exist a probabilistic representative of a belief function that would yield the same expectations as the Shenoy’s new operator.
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About Two Consonant Conflicts of Belief Functions
Daniel, M. ; Kratochvíl, Václav
General belief functions usually bear some internal conflict which comes mainly from disjoint focal elements. Analogously, there is often some conflict between two (or more) belief functions. After the recent observation of hidden conflicts (seminar CJS’17 [17]), appearing at belief functions with disjoint focal elements, importance of interest in conflict of belief functions has increased. This theoretical contribution introduces a new approach to conflicts (of belief functions). Conflicts are considered independently of any combination rule and of any distance measure. Consonant conflicts are based on consonant approximations of belief functions in general; two special cases of the consonant approach based on consonant inverse pignistic and consonant inverse plausibility transforms are discussed. Basic properties of the newly defined conflicts are presented, analyzed and briefly compared with our original approaches to conflict (combinational conflict, plausibility conflict and comparative conflict), with the recent conflict based on non-conflicting parts, as well as with W. Liu’s degree of conflict.
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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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Dynamic Bayesian Networks for the Classification of Sleep Stages
Vomlel, Jiří ; Kratochvíl, Václav
Human sleep is traditionally classified into five (or six) stages. The manual classification is time consuming since it requires knowledge of an extensive set of rules from manuals and experienced experts. Therefore automatic classification methods appear useful for this task. In this paper we extend the approach based on Hidden Markov Models by relating certain features not only to the current time slice but also to the previous one. Dynamic Bayesian Networks that results from this generalization are thus capable of modeling features related to state transitions. Experiments on real data revealed that in this way we are able to increase the prediction accuracy.
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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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Hidden Auto-Conflict in the Theory of Belief Functions
Daniel, M. ; Kratochvíl, Václav
Hidden conflicts of belief functions in some cases where the sum of all multiples of conflicting belief masses being equal to zero were observed. Relationships of hidden conflicts and auto-conflicts of belief functions are pointed out. We are focused on hidden auto-conflicts here - on hidden conflicts appearing when three or more numerically same belief functions are combined. Hidden auto-conflict is a kind of internal conflict. Degrees of hidden auto-conflicts and full non-conflictness are defined and analysed. Finally, computational issues of hidden auto-conflicts and non-conflictness are presented.
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Basic facts concerning extreme supermodular functions
Studený, Milan
Elementary facts and observations on the cone of supermodular set functions are recalled. The manuscript deals with such operations with set functions which preserve supermodularity\nand the emphasis is put on those such operations which even preserve extremality (of a supermodular function). These involve a few self-transformations of the cone of supermodular set functions. Moreover, projections to the (less-dimensional) linear space of set functions for a subset of the variable set are discussed. Finally, several extensions to the (more-dimensional) linear space of set functions for a superset of the variable set are shown to be both preserving supermodularity and extremality.
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