20181002 14:38 
MultiObjective Optimization Problems with Random Elements  Survey of Approaches
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and a decision parameter. Mostly, it is possible to influence the above mentioned situation only by an optimization model depending on a probability measure. This optimization problem can be static (onestage), dynamic with finite or infinite horizon, singleobjective or multiobjective. We focus on onestage multiobjective problems corresponding to applications those are suitable to evaluate simultaneously by a few objectives. The aim of the contribution is to give a survey of different approaches (as they are known from the literature) of the above mentioned applications. To this end we start with wellknown meanrisk model and continue with other known approaches. Moreover, we try to complete every model by a suitable application. Except an analysis of a choice of the objective functions type we try to discuss suitable constraints set with respect to the problem base, possible investigation and relaxation. At the end we mention properties of the problem in the case when the theoretical "underlying" probability measure is replaced by its "deterministic" or "stochastic" estimate.
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20181002 14:38 
Risksensitive and Mean Variance Optimality in Continuoustime Markov Decision Chains
Sladký, Karel
In this note we consider continuoustime Markov decision processes with finite state and actions spaces where the stream of rewards generated by the Markov processes is evaluated by an exponential utility function with a given risk sensitivitycoefficient (socalled risksensitive models). If the risk sensitivity coefficient equals zero (riskneutral case) we arrive at a standard Markov decision process. Then we can easily obtain necessary and sufficient mean reward optimality conditions and the variability can be evaluated by the mean variance of total expected rewards. For the risksensitive case, i.e. if the risksensitivity coefficient is nonzero, for a given value of the risksensitivity coefficient we establish necessary and sufficient optimality conditions for maximal (or minimal) growth rate of expectation of the exponential utility function, along with mean value of the corresponding certainty equivalent. Recall that in this case along with the total reward also its higher moments are taken into account.
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20181002 14:38 
Problem of competing risks with covariates: Application to an unemployment study
Volf, Petr
The study deals with the methods of statistical analysis in the situation of competing risks in the presence of regression. First, the problem of identification of marginal and joint distributions of competing random variables is recalled. The main objective is then to demonstrate that the parameters and, in particular, the correlation of competing variables, may depend on covariates. The approach is applied to solution of a real example with unemployment data. The model uses the Gauss copula and Cox’s regression model.
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20181002 14:38 
Representations of Bayesian Networks by LowRank Models
Tichavský, Petr ; Vomlel, Jiří
Conditional probability tables (CPTs) of discrete valued random variables may achieve high dimensions and Bayesian networks deﬁned as the product of these CPTs may become intractable by conventional methods of BN inference because of their dimensionality. In many cases, however, these probability tables constitute tensors of relatively low rank. Such tensors can be written in the socalled Kruskal form as a sum of rankone components. Such representation would be equivalent to adding one artiﬁcial parent to all random variables and deleting all edges between the variables. The most difﬁcult task is to ﬁnd such a representation given a set of marginals or CPTs of the random variables under consideration. In the former case, it is a problem of joint canonical polyadic (CP) decomposition of a set of tensors. The latter ﬁtting problem can be solved in a similar manner. We apply a recently proposed alternating direction method of multipliers (ADMM), which assures that the model has a probabilistic interpretation, i.e., that all elements of all factor matrices are nonnegative. We perform experiments with several wellknown Bayesian networks.\n\n
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20181002 14:38 
Two Algorithms for Riskaverse Reformulation of Multistage Stochastic Programming Problems
Šmíd, Martin ; Kozmík, Václav
Many reallife applications lead to riskaverse multistage stochastic problems, therefore effective solution of these problems is of great importance. Many tools can be used to their solution (GAMS, CoinOR, APML or, for smaller problems, Excel), it is, however, mostly up to researcher to reformulate the problem into its deterministic equivalent. Moreover, such solutions are usually onetime, not easy to modify for different applications. We overcome these problems by providing a frontend software package, written in C++, which enables to enter problem definitions in a way close to their mathematical definition. Creating of a deterministic equivalent (and its solution) is up to the computer. In particular, our code is able to solve linear multistage with Multiperiod MeanCVaR or Nested MeanCVaR criteria. In the present paper, we describe the algorithms, transforming these problems into their deterministic equivalents.
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20180730 11:02 
On attempts to characterize facetdefining 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) nonredundant description of these cones by means of the socalled facetdefining 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 facetdefining inequalities for these cones. We introduce the concept of an irreducible minbalanced set system and conjecture that the facetdefining inequalities for the cone of totally balanced games correspond to these set systems. The conjecture concerning exact games is that the facetdefining inequalities for this cone are those which correspond to irreducible minbalanced 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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20180730 11:02 
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 DempsterShafer theory of evidence. Up to now, there was no direct way to get the expected value of a utility function in DS 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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20180730 11:02 
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 nonconflicting parts, as well as with W. Liu’s degree of conflict.
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20180730 11:02 
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 30year 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 thirtyyear 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, DempsterShafer Theory, Belief Functions, Probabilistic Logic, Game Theory, etc.
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20180619 19:26 
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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