A data modeling procedure called Mixture model, is introduced beneficial to the characteristics of our data. Mixture models have been proved flexible and easy to use, a situation which can be confirmed from the majority of papers and books which have been published the last twenty years. The models are estimated using a Bayesian inference through an efficient Markov Chain Monte Carlo (MCMC) algorithm, known as Gibbs Sampling. The focus of the paper is on models for network-telecommunication lab data (not time dependent data) and on the valid predictions we can accomplish. We categorize our variables (based on their distribution) in three cases, a mixture of Normal distributions with known allocation, a mixture of Negative Binomial Distributions with known allocations and a mixture of Normal distributions with unknown allocation.

The multinomial logit model (MNL) possesses a latent variable representation in terms of random variables following a multivariate logistic distribution. Based on multivariate finite mixture approximations of the multivariate logistic distribution, various data-augmented Metropolis-Hastings algorithms are developed for a Bayesian inference of the MNL model.

This thesis develops a new Bayesian approach to structural break modeling. The focuses of the approach are the modeling of in-sample structural breaks and forecasting time series allowing out-of-sample breaks. Our model has some desirable features. First, the number of regimes is not fixed and is treated as a random variable in our model. Second, our model adopts a hierarchical prior for regime coefficients, which allows for the regime coefficients of one regime to contain information about regime coefficients of other regimes. However, the regime coefficients can be analytically integrated out of the posterior distribution and therefore we only need to deal with one level of the hierarchy. Third, the implementation of our model is simple and the computational cost is low. Our model is applied to two different time series: S&P 500 monthly returns and U.S. real GDP quarterly growth rates. We linked breaks detected by our model to certain historical events.

<div>Markov jump processes are continuous-time stochastic processes widely used in a variety of applied disciplines. Inference typically proceeds via Markov chain Monte Carlo (MCMC), the state-of-the-art being a uniformization-based auxiliary variable Gibbs sampler. This was designed for situations where the process parameters are known, and Bayesian inference over unknown parameters is typically carried out by incorporating it into a larger Gibbs sampler. This strategy of sampling parameters given path, and path given parameters can result in poor Markov chain mixing.</div><div><br></div><div>In this thesis, we focus on the problem of path and parameter inference for Markov jump processes.</div><div><br></div><div>In the first part of the thesis, a simple and efficient MCMC algorithm is proposed to address the problem of path and parameter inference for Markov jump processes. Our scheme brings Metropolis-Hastings approaches for discrete-time hidden Markov models to the continuous-time setting, resulting in a complete and clean recipe for parameter and path inference in Markov jump processes. In our experiments, we demonstrate superior performance over Gibbs sampling, a more naive Metropolis-Hastings algorithm we propose, as well as another popular approach, particle Markov chain Monte Carlo. We also show our sampler inherits geometric mixing from an ‘ideal’ sampler that is computationally much more expensive.</div><div><br></div><div>In the second part of the thesis, a novel collapsed variational inference algorithm is proposed. Our variational inference algorithm leverages ideas from discrete-time Markov chains, and exploits a connection between Markov jump processes and discrete-time Markov chains through uniformization. Our algorithm proceeds by marginalizing out the parameters of the Markov jump process, and then approximating the distribution over the trajectory with a factored distribution over segments of a piecewise-constant function. Unlike MCMC schemes that marginalize out transition times of a piecewise-constant process, our scheme optimizes the discretization of time, resulting in significant computational savings. We apply our ideas to synthetic data as well as a dataset of check-in recordings, where we demonstrate superior performance over state-of-the-art MCMC methods.</div><div><br></div>

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23003号 / 理博第4680号 / 新制||理||1671(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 福間 將文, 教授 畑 浩之, 教授 田中 貴浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Tato práce popisuje a aplikuje parametrické a neparametrické modely spolehlivosti na cenzorovaná data. Ukazuje implementaci spolehlivosti v metodologii Six Sigma. Metody jsou využity pro přežití/spolehlivost reálných technických dat.

Parameter inference for stochastic systems is considered as one of the fundamental classical problems in the domain of computational systems biology. The problem becomes challenging and often analytically intractable with the large number of uncertain parameters. In this scenario, Markov Chain Monte Carlo (MCMC) algorithms have been proved to be highly effective. For a stochastic system, the most accurate description of the kinetics is given by the Chemical Master Equation (CME). Unfortunately, analytical solution of CME is often intractable even for considerably small amount of chemically reacting species due to its super exponential state space complexity. As a solution, Stochastic Simulation Algorithm (SSA) using Monte Carlo approach was introduced to simulate the chemical process defined by the CME. SSA is an exact stochastic method to simulate CME but it also suffers from high time complexity due to simulation of every reaction. Therefore computation of likelihood function (based on exact CME) in MCMC becomes expensive which alternately makes the rejection step expensive. In this generic work, we introduce different approximations of CME as a pre-conditioning step to the full MCMC to make rejection cheaper. The goal is to avoid expensive computation of exact CME as far as possible. We show that, with effective pre-conditioning scheme, one can save a considerable amount of exact CME computations maintaining similar convergence characteristics. Additionally, we investigate three different sampling schemes (dense sampling, longer sampling and i.i.d sampling) under which convergence for MCMC using exact CME for parameter estimation can be analyzed. We find that under i.i.d sampling scheme, better convergence can be achieved than that of dense sampling of the same process or sampling the same process for longer time. We verify our theoretical findings for two different processes: linear birth-death and dimerization.Apart from providing a framework for parameter inference using CME, this work also provides us the reasons behind avoiding CME (in general) as a parameter estimation technique for so long years after its formulation

Joint models for the association of a longitudinal binary and a longitudinal continuous process are proposed for situations in which their association is of direct interest. The models are parameterized such that the dependence between the two processes is characterized by unconstrained regression coefficients. Bayesian variable selection techniques are used to parsimoniously model these coefficients. A Markov chain Monte Carlo (MCMC) sampling algorithm is developed for sampling from the posterior distribution, using data augmentation steps to handle missing data. Several technical issues are addressed to implement the MCMC algorithm efficiently. The models are motivated by, and are used for, the analysis of a smoking cessation clinical trial in which an important question of interest was the effect of the (exercise) treatment on the relationship between smoking cessation and weight gain.

In der Arbeit wird ein neuartiges Lernverfahren für Gauß-Mischverteilungen entwickelt. Es basiert auf der Technik der Markov-Chain Monte-Carlo Verfahren und ist in der Lage, in einem Zuge die Größe der Mischverteilung sowie deren Parameter zu bestimmen. Das Verfahren zeichnet sich sowohl durch eine gute Anpassung an die Trainingsdaten als auch durch eine gute Generalisierungsleistung aus. Ausgehend von einer Beschreibung der stochastischen Grundlagen und einer Analyse der Probleme, die beim Lernen von Gauß-Mischverteilungen auftreten, wird in der Abeit das neue Lernverfahren schrittweise entwickelt und seine Eigenschaften untersucht. Ein experimenteller Vergleich mit bekannten Lernverfahren für Gauß-Mischverteilungen weist die Eignung des neuen Verfahrens auch empirisch nach.

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