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Cross-entropy based combination of discrete probability distributions for distributed decision making
Sečkárová, Vladimíra ; Kárný, Miroslav (advisor) ; Jurečková, Jana (referee) ; Janžura, Martin (referee)
Dissertation abstract Title: Cross-entropy based combination of discrete probability distributions for distributed de- cision making Author: Vladimíra Sečkárová Author's email: seckarov@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague Supervisor: Ing. Miroslav Kárný, DrSc., The Institute of Information Theory and Automation of the Czech Academy of Sciences Supervisor's email: school@utia.cas.cz Abstract: In this work we propose a systematic way to combine discrete probability distributions based on decision making theory and theory of information, namely the cross-entropy (also known as the Kullback-Leibler (KL) divergence). The optimal combination is a probability mass function minimizing the conditional expected KL-divergence. The ex- pectation is taken with respect to a probability density function also minimizing the KL divergence under problem-reflecting constraints. Although the combination is derived for the case when sources provided probabilistic type of information on the common support, it can applied to other types of given information by proposed transformation and/or extension. The discussion regarding proposed combining and sequential processing of available data, duplicate data, influence...
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Cross-entropy based combination of discrete probability distributions for distributed decision making
Sečkárová, Vladimíra ; Kárný, Miroslav (advisor)
Dissertation abstract Title: Cross-entropy based combination of discrete probability distributions for distributed de- cision making Author: Vladimíra Sečkárová Author's email: seckarov@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague Supervisor: Ing. Miroslav Kárný, DrSc., The Institute of Information Theory and Automation of the Czech Academy of Sciences Supervisor's email: school@utia.cas.cz Abstract: In this work we propose a systematic way to combine discrete probability distributions based on decision making theory and theory of information, namely the cross-entropy (also known as the Kullback-Leibler (KL) divergence). The optimal combination is a probability mass function minimizing the conditional expected KL-divergence. The ex- pectation is taken with respect to a probability density function also minimizing the KL divergence under problem-reflecting constraints. Although the combination is derived for the case when sources provided probabilistic type of information on the common support, it can applied to other types of given information by proposed transformation and/or extension. The discussion regarding proposed combining and sequential processing of available data, duplicate data, influence...
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Recursive estimation of models relating discrete-valued variables to continuous-valued ones applied to trading with futures
Svoboda, Miroslav ; Kárný, Miroslav (advisor) ; Hurt, Jan (referee)
This bachelor thesis deals with recursive estimation of a dependence of the models with discrete variables on variables that are either discretely or continuously distributed. To this purpose Bayes formula, described in the first chapter, is used, to which an additional assumption of conditional independence is added so that it can be used dynamically. The second chapter describes an approximation algorithm, which is used for recursive approximation of the density of random variable that has been estimated by the Bayesian equation. The third chapter deals with the application of the whole model on a special form of logistic regression. Results are shown on the examples using simulated data. At last, the model along with approximation algorithm is applied on a trading with futures. Powered by TCPDF (www.tcpdf.org)
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Bayesian modeling of market price using autoregression model
Šindelář, Jan ; Kárný, Miroslav (advisor) ; Pawlas, Zbyněk (referee) ; Šmíd, Martin (referee)
1 Bayesian modeling of market price using autoregression model 1Šindelář Jan Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Miroslav Kárný, DrSc. Abstract: In the thesis we present a novel solution of Bayesian filtering in autoregression model with Laplace distributed innovations. Estimation of regression models with lep- tokurtically distributed innovations has been studied before in a Bayesian framework [2], [1]. Compared to previously conducted studies, the method described in this article leads to an exact solution for density specifying the posterior distribution of parameters. Such a solution was previously known only for a very limited class of innovation distributions. In the text an algorithm leading to an effective solution of the problem is also proposed. The algorithm is slower than the one for the classical setup, but due to increasing com- putational power and stronger support of parallel computing, it can be executed in a reasonable time for models, where the number of parameters isn't very high. Keywords: Bayesian, Autoregression, Optimal Trading, Time Series References [1] P. Congdon. Bayesian statistical modelling. Wiley, 2006. [2] A. Zellner. Bayesian and Non-Bayesian analysis of the regression model with multivari- ate Student-t error term. Journal...
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Výběr volitelných parametrů částečného zapomínání
Votava, Adam ; Kárný, Miroslav (advisor) ; Šmíd, Martin (referee)
Presented work deals with the choice of optional parameters determining partial forgetting. The main objective is to design an algorithm for the development of the optional parameters in time in the optimal way, which would be better than usage of constant parameters. For this purpose, the Bayesian dynamic decision making, general principles of tracking the slowly varying parameters via forgetting and partial forgetting method are presented. To make computations feasible the exponential family of probability distribution functions is used. Applied algorithm is described mathematically using Bayesian learning. The stress is laid on the forgetting factor's choice, that is regarded as a Bayesian hypothesis testing. Moreover, the set of hypotheses on the forgetting factor varies in time. To hypotheses, forgetting is also applied. The presented methods are then applied to the normal regression model. However, the generality of the theoretical part allows the application to other models, e.g. Markov chain model, too. The algorithm is then programmed within the Python environment and tested on the real traffic data and on the simulated data as well.
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Sdílení pravděpodobnostní informace bayesovských agentů
Kalenkovich, Evgeny ; Kárný, Miroslav (advisor) ; Lachout, Petr (referee)
A need for combining probability distribution arises in many decision-theoretical problems. In this work we follow articles [14] and [15] in pursuing the supra Bayesian approach [9]. A method for combining nite discrete distributions is introduced, as well as a way to deal with incomplete information and bounded continuous distributions. In the discrete case our approach is along the lines of, but di erent at a few key points from the thesis [20]. The result is a shifted arithmetic mean of pmfs, which is discrepant from the usual arithmetic pooling (see [9] for details).
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Metody aproximace plně pravděpodobnostního návrhu rozhodování za neúplné znalosti
Pištěk, Miroslav ; Andrýsek, Josef (referee) ; Kárný, Miroslav (advisor)
In this thesis, we introduce an efficient algorithm for an optimal decision strategy approximation. It approximates the optimal equations of dynamic programming without omitting the principal uncertainty stemming from an uncomplete knowledge of a controlled system. Thus, the algorithm retains the ability to constantly verify the actual knowledge, which is the essence of dual control. An integral part of solution proposed is a reduction of memory demands using HDMR approximation. We have developed a general method for numerical solution of linear integral equations based on this approximation, and applied it to solve a linearized variant of optimal equations. To achieve such a variant, it was necessary to apply a different control design called fully probabilistic design which allows easier finding of a linearized approximation. The result of this method is a pair of linear algebraic systems for the upper and lower bound on the central function describing the optimal strategy. One illustrative example has been completely resolved.
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Supra-bayesovská kombinace pravděpodobnostních distribucí
Sečkárová, Vladimíra ; Komárek, Arnošt (referee) ; Kárný, Miroslav (advisor)
In this work we study problems of sharing of probabilistic information by using Supra-Bayesian approach. In 1st Chapter the methods and formulas used in the work are mentioned. 2nd Chapter contains the introduction to discussed topic. In 3rd Chapter the main method of sharing the probabilistic information, which is based on common domain, is derived. In 4th Chapter the types of given knowledge pieces are specified, which are then transformed into probabilistic terms and extended on the whole domain. In 5th Chapter the results from the previous chapters are assessed.
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