The Markov Chain Monte Carlo (MCMC) method has been widely used in practice since the 1950's in areas such as biology, statistics, and physics. However, it is only in the last few decades that powerful techniques for obtaining rigorous performance guarantees with respect to the running time have been developed. Today, with only a few notable exceptions, most known algorithms for approximately uniform sampling and approximate counting rely on the MCMC method. This thesis focuses on algorithms that use MCMC combined with an algorithm from optimization called simulated annealing, for sampling and counting problems. Annealing is a heuristic for finding the global optimum of a function over a large search space. It has recently emerged as a powerful technique used in conjunction with the MCMC method for sampling problems, for example in the estimation of the permanent and in algorithms for computing the volume of a convex body. We examine other applications of annealing to sampling problems as well as scenarios when it fails to converge in polynomial time. We consider the problem of randomly generating 0-1 contingency tables. This is a well-studied problem in statistics, as well as the theory of random graphs, since it is also equivalent to generating a random bipartite graph with a prescribed degree sequence. Previously, the only algorithm known for all degree sequences was by reduction to approximating the permanent of a 0-1 matrix. We give a direct and more efficient combinatorial algorithm which relies on simulated annealing. Simulated tempering is a variant of annealing used for sampling in which a temperature parameter is randomly raised or lowered during the simulation. The idea is that by extending the state space of the Markov chain to a polynomial number of progressively smoother distributions, parameterized by temperature, the chain could cross bottlenecks in the original space which cause slow mixing. We show that simulated tempering mixes torpidly for the 3-state ferromagnetic Potts model on the complete graph. Moreover, we disprove the conventional belief that tempering can slow fixed temperature algorithms by at most a polynomial in the number of temperatures and show that it can converge at a rate that is slower by at least an exponential factor.

Doctoral thesis is focused on the unconventional methods of the discrete probability estimation of categorical quantity from its observed values. The gradient of quasinorm and so-called line estimation were emlopyed for these estimations. Bootstrap method was used for the improvement of accuracy. Theoretical results for selected quasinorms were illustrated on specific examples.

Thesis is based on original self-made application programmed at VBA in MS Excel 2007. The reason to build this application was integration of simulation Monte Carlo and chosen optimization methods. The application allows do simulation of the knapsack problem and of the assignment problem with uncertainty. The parameters of these models are possible to set up as changing values in dependence of chosen probability distribution. Output of the simulation is a probability recommendation which objects should be used. Choose of objects depend on optimized models. Results of both models are represented by statistical indexes, tables of parameters and graph.

Master's thesis is focused on solution of the statistical problem to find a probability distribution of a discrete random variable on the basis of the observed data. These estimates are obtained by minimizing pseudo-quasinorm which is introduced here.The thesis further focuses on atributes of this pseudo-quasinorm. It also contains practical application of these methods.

Dissertation thesis is focused on solution of the statistical problem to find a probability distribution of a discrete random variable on the basis of the observed data. These estimates are obtained by minimizing quasi-norms with given constraints. The thesis further focuses on deriving confidence intervals for estimated probabilities. It also contains practical application of these methods.

In many agricultural and biological contexts, the response variable is a nonnegative integer value which we wish to explain or analyze in terms of a set of covariates. Unlike the Gaussian linear model, the response variable is discrete with a distribution that places probability mass at natural numbers only. The Poisson regression is the standard model for count data. However, assumptions of this model forces the equality between mean and variance, which may be implausible in many applications. Motivated by experimental data sets, this work intended to develop more realistic methods for the analysis of count data. We proposed a novel parametrization of the COM-Poisson distribution and explored the regression models based on it. We extended the model to allow the dispersion, as well as the mean, depending on covariates. A set of count statistical models, namely COM-Poisson, Gamma-count, discrete Weibull, generalized Poisson, double Poisson and Poisson-Tweedie, was reviewed and compared, considering the dispersion, zero-inflation, and heavy tail indexes, together with the results of data analyzes. The computational routines developed in this dissertation were organized in two R packages available on GitHub. / Em diversos estudos agrícolas e biológicos, a variável resposta é um número inteiro não negativo que desejamos explicar ou analisar em termos de um conjunto de covariáveis. Diferentemente do modelo linear Gaussiano, a variável resposta é discreta com distribuição de probabilidade definida apenas em valores do conjunto dos naturais. O modelo Poisson é o modelo padrão para dados em forma de contagens. No entanto, as suposições desse modelo forçam que a média seja igual a variância, o que pode ser implausível em muitas aplicações. Motivado por conjuntos de dados experimentais, este trabalho teve como objetivo desenvolver métodos mais realistas para a análise de contagens. Foi proposta uma nova reparametrização da distribuição COM-Poisson e explorados modelos de regressão baseados nessa distribuição. Uma extensão desse modelo para permitir que a dispersão, assim como a média, dependa de covariáveis, foi proposta. Um conjunto de modelos para contagens, nomeadamente COM-Poisson, Gamma-count, Weibull discreto, Poisson generalizado, duplo Poisson e Poisson-Tweedie, foi revisado e comparado, considerando os índices de dispersão, inflação de zero e cauda pesada, juntamente com os resultados de análises de dados. As rotinas computacionais desenvolvidas nesta dissertação foram organizadas em dois pacotes R disponíveis no GitHub.

Os estudos iniciais de análise combinatória e probabilidade tem uma forte relação com os jogos de azar, lembramos um jogo com dados praticado por Antoine Gombaud (Chavalier de Méré). Conta que Chavalier após uma bem sucedida estratégia (lançar um dado quatro vezes e obter um 6), conseguindo ganhos significativos, modificou o jogo para dois dados e venceria caso ocorresse um duplo 6 em 24 lançamentos, e neste acumula prejuízo. Detalhe que marca seu contanto com Blaise Pascal. Isto estimula o estudo de probabilidade em espaços discretos. Os conceitos probabilísticos discretos (conjunto enumerável finito) utilizados por Pascal na resolução do problema de Méré não são suficientes para responder a problemas de natureza contínua. Por exemplo, o problema das agulhas do francês Georges Louis Leclerc (conde de Buffon) e outras situações que envolvem o cálculo de probabilidade em segmentos de retas, áreas de figuras planas ou volumes de sólidos, assim como em um jogo aplicado durante uma feira de matemática para estudantes do ensino básico (6o ao 9o ano) do ensino fundamental II. Utilizando o jogo “GIROU GANHOU” é possível explorar o conceito de probabilidade geométrica, comparar o resultado da aplicação com os cálculos realizados e abordar a esperança matemática quando o jogo for realizado uma quantidade significativa de vezes. A esperança é uma expectativa de ganho “médio”, uma convergência, em torno de um resultado “esperado”. Neste faremos uma caracterização de probabilidade geométrica e esperança matemática, por fim aplicaremos tais conceitos na resolução de problemas de natureza continua (geométrica). / The initial studies of combinatorial analysis and probability have a strong relationship with gambling, we recall a game with data practiced by Antoine Gombaud (Chavalier de Méré). He says that after a successful strategy (throwing a die four times and get a 6), achieving significant gains, he modified the game to two dice and would win if there were a double 6 in 24 throws, and in this accumulates loss. Detail marking his astounding with Blaise Pascal. This stimulates the study of probability in discrete spaces. The discrete probabilistic concepts (finite enumerable set) used by Pascal in solving the Méré problem are not sufficient to respond to problems of a continuous nature. For example, the problem of French needles Georges Louis Leclerc (count of Buffon) and other situations involving the calculation of probability in segments of straight lines, areas of flat figures or volumes of solids, as well as in a game applied during a fair of mathematics for primary school students (6th to 9th grade) of elementary education II. Using the “TURNEDWON” game it is possible to explore the concept of geometric probability, compare the result of the application with the calculations made and approach the mathematical hope when the game is performed a significant amount of times. Hope is an expectation of “ middle ” gain, a convergence, around an “ expected ” result. In this we will make a characterization of geometric probability and mathematical hope, finally we will apply these concepts in the resolution of problems of a continuous (geometric) nature.

The diploma thesis describes the bootstrap method and its applications in the estimate accuracy statement, in the confidence intervals generation and in the testing of statistical hypotheses. Further the method of the discrete probability estimation of the categorical quantity is presented, making use the gradient of the quasi-norm hereof distribution. On concrete examples the bootstrap method is applied in the confidence intervals forming of the categorical quantity probability function. The diploma thesis was supported by the project of MŠMT of the Czech Republic no. 1M06047 "Centre for Quality and Reliability of Production", by the grant of Grant Agency of the Czech Republic (Czech Science Foundation) reg. no. 103/08/1658 "Advanced optimum design of composed concrete structures" and by the research plan of MŠMT of the Czech Republic no. MSM0021630519 "Progressive reliable and durable structures".

The diploma thesis describes the bootstrap method and its applications in the confidence intervals generation, in the testing of statistical hypotheses and in the regression analysis. We present the confidence interval for individual value. Further the method of discrete probability estimation of the categorical quantity is presented, making use the gradient and the line estimate.