**804**

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**Two Algorithms for Risk-averse Reformulation of Multi-stage Stochastic Programming Problems**

Šmíd, Martin; Kozmík, Václav

2018 - English
Many real-life applications lead to risk-averse multi-stage stochastic problems, therefore effective solution of these problems is of great importance. Many tools can be used to their solution (GAMS, Coin-OR, 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 one-time, not easy to modify for different applications. We overcome these problems by providing a front-end 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 multi-stage with Multi-period Mean-CVaR or Nested Mean-CVaR criteria. In the present paper, we describe the algorithms, transforming these problems into their deterministic equivalents.
Keywords:
*Multi-stage stochastic programming; deterministic equivalent; multi-period CVaR; nested CVaR; optimization algorithm*
Fulltext is available at external website.
Two Algorithms for Risk-averse Reformulation of Multi-stage Stochastic Programming Problems

Many real-life applications lead to risk-averse multi-stage stochastic problems, therefore effective solution of these problems is of great importance. Many tools can be used to their solution (GAMS, ...

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**Problem of competing risks with covariates: Application to an unemployment study**

Volf, Petr

2018 - English
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.
Keywords:
*statistical analysis; competing risks; unemployment study*
Fulltext is available at external website.
Problem of competing risks with covariates: Application to an unemployment study

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 ...

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**Dynamic Bayesian Networks for the Classification of Sleep Stages**

Vomlel, Jiří; Kratochvíl, Václav

2018 - English
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.
Keywords:
*Dynamic Bayesian Network; Sleep Analysis*
Fulltext is available at external website.
Dynamic Bayesian Networks for the Classification of Sleep Stages

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 ...

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**Comparison of Shenoy’s Expectation Operator with Probabilistic Transforms and Perez’ Barycenter**

Jiroušek, R.; Kratochvíl, Václav

2018 - English
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.
Keywords:
*expected utility; Dempster-Shafer theory; Shenoy's operator*
Fulltext is available at external website.
Comparison of Shenoy’s Expectation Operator with Probabilistic Transforms and Perez’ Barycenter

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 ...

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**Employing Bayesian Networks for Subjective Well-being Prediction**

Švorc, Jan; Vomlel, Jiří

2018 - English
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.
Keywords:
*Subjective well-being; Bayesian networks*
Fulltext is available at external website.
Employing Bayesian Networks for Subjective Well-being Prediction

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 ...

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**Representations of Bayesian Networks by Low-Rank Models**

Tichavský, Petr; Vomlel, Jiří

2018 - English
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 so-called Kruskal form as a sum of rank-one 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 well-known Bayesian networks.\n\n
Keywords:
*canonical polyadic tensor decomposition; conditional probability tables; marginal probability tables*
Fulltext is available at external website.
Representations of Bayesian Networks by Low-Rank Models

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 ...

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**Risk-sensitive and Mean Variance Optimality in Continuous-time Markov Decision Chains**

Sladký, Karel

2018 - English
In this note we consider continuous-time 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 (so-called risk-sensitive models). If the risk sensitivity coefficient equals zero (risk-neutral 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 risk-sensitive case, i.e. if the risk-sensitivity coefficient is non-zero, for a given value of the risk-sensitivity 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.
Keywords:
*continuous-time Markov decision chains; exponential utility functions; certainty equivalent; mean-variance optimality; connections between risk-sensitive and risk-neutral optimality*
Fulltext is available at external website.
Risk-sensitive and Mean Variance Optimality in Continuous-time Markov Decision Chains

In this note we consider continuous-time Markov decision processes with finite state and actions spaces where the stream of rewards generated by the Markov processes is evaluated by an exponential ...

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**Multi-Objective Optimization Problems with Random Elements - Survey of Approaches**

Kaňková, Vlasta

2018 - English
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 (one-stage), dynamic with finite or infinite horizon, single-objective or multi-objective. We focus on one-stage multi-objective 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 well-known mean-risk 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.
Keywords:
*multi-objective optimization problems; random element; mean-risk model; deterministic approach; stochastic multi-objective problems; constraints set; relaxation*
Fulltext is available at external website.
Multi-Objective Optimization Problems with Random Elements - Survey of Approaches

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 ...

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**On attempts to characterize facet-defining inequalities of the cone of exact games**

Studený, Milan; Kroupa, Tomáš; Kratochvíl, Václav

2018 - English
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.
Keywords:
*exact game; extremity; irreducible; balanced*
Fulltext is available at external website.
On attempts to characterize facet-defining inequalities of the cone of exact games

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 ...

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**Gradient Descent Parameter Learning of Bayesian Networks under Monotonicity Restrictions**

Plajner, Martin; Vomlel, Jiří

2018 - English
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.
Keywords:
*Bayesian networks; Learning model parameters; monotonicity condition*
Fulltext is available at external website.
Gradient Descent Parameter Learning of Bayesian Networks under Monotonicity Restrictions

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 ...

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