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Fang, Youhan. "Efficient Markov Chain Monte Carlo Methods." Thesis, Purdue University, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10809188.
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<p> Generating random samples from a prescribed distribution is one of the most important and challenging problems in machine learning, Bayesian statistics, and the simulation of materials. Markov Chain Monte Carlo (MCMC) methods are usually the required tool for this task, if the desired distribution is known only up to a multiplicative constant. Samples produced by an MCMC method are real values in <i>N</i>-dimensional space, called the configuration space. The distribution of such samples converges to the target distribution in the limit. However, existing MCMC methods still face many challenges that are not well resolved. Difficulties for sampling by using MCMC methods include, but not exclusively, dealing with high dimensional and multimodal problems, high computation cost due to extremely large datasets in Bayesian machine learning models, and lack of reliable indicators for detecting convergence and measuring the accuracy of sampling. This dissertation focuses on new theory and methodology for efficient MCMC methods that aim to overcome the aforementioned difficulties.</p><p> One contribution of this dissertation is generalizations of hybrid Monte Carlo (HMC). An HMC method combines a discretized dynamical system in an extended space, called the state space, and an acceptance test based on the Metropolis criterion. The discretized dynamical system used in HMC is volume preserving—meaning that in the state space, the absolute Jacobian of a map from one point on the trajectory to another is 1. Volume preservation is, however, not necessary for the general purpose of sampling. A general theory allowing the use of non-volume preserving dynamics for proposing MCMC moves is proposed. Examples including isokinetic dynamics and variable mass Hamiltonian dynamics with an explicit integrator, are all designed with fewer restrictions based on the general theory. Experiments show improvement in efficiency for sampling high dimensional multimodal problems. A second contribution is stochastic gradient samplers with reduced bias. An in-depth analysis of the noise introduced by the stochastic gradient is provided. Two methods to reduce the bias in the distribution of samples are proposed. One is to correct the dynamics by using an estimated noise based on subsampled data, and the other is to introduce additional variables and corresponding dynamics to adaptively reduce the bias. Extensive experiments show that both methods outperform existing methods. A third contribution is quasi-reliable estimates of effective sample size. Proposed is a more reliable indicator—the longest integrated autocorrelation time over all functions in the state space—for detecting the convergence and measuring the accuracy of MCMC methods. The superiority of the new indicator is supported by experiments on both synthetic and real problems.</p><p> Minor contributions include a general framework of changing variables, and a numerical integrator for the Hamiltonian dynamics with fourth order accuracy. The idea of changing variables is to transform the potential energy function as a function of the original variable to a function of the new variable, such that undesired properties can be removed. Two examples are provided and preliminary experimental results are obtained for supporting this idea. The fourth order integrator is constructed by combining the idea of the simplified Takahashi-Imada method and a two-stage Hessian-based integrator. The proposed method, called two-stage simplified Takahashi-Imada method, shows outstanding performance over existing methods in high-dimensional sampling problems.</p><p>
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Murray, Iain Andrew. "Advances in Markov chain Monte Carlo methods." Thesis, University College London (University of London), 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.487199.
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Probability distributions over many variables occur frequently in Bayesian inference, statistical physics and simulation studies. Samples from distributions give insight into their typical behavior and can allow approximation of any quantity of interest, such as expectations or normalizing constants. Markov chain Monte Carlo (MCMC), introduced by Metropolis et al. (1953), allows r sampling from distributions with intractable normalization, and remains one of most important tools for approximate computation with probability distributions. I While not needed by MCMC, normalizers are key quantities: in Bayesian statistics marginal likelihoods are needed for model comparison; in statistical physics many physical quantities relate to the partition function. In this thesis we propose and investigate several new Monte Carlo algorithms, both for evaluating normalizing constants and for improved sampling of distributions. Many MCMC correctness proofs rely on using reversible transition operators; this can lead to chains exploring by slow random walks. After reviewing existing MCMC algorithms, we develop a new framework for constructing non-reversible transition operators from existing reversible ones. Next we explore and extend MCMC-based algorithms for computing normalizing constants. In particular we develop a newMCMC operator and Nested Sampling approach for the Potts model. Our results demonstrate that these approaches can be superior to finding normalizing constants by annealing methods and can obtain better posterior samples. Finally we consider 'doubly-intractable' distributions with extra unknown normalizer terms that do not cancel in standard MCMC algorithms. We propose using several deterministic approximations for the unknown terms, and investigate their interaction with sampling algorithms. We then develop novel exact-sampling-based MCMC methods, the Exchange Algorithm and Latent Histories. For the first time these algorithms do not require separate approximation before sampling begins. Moreover, the Exchange Algorithm outperforms the only alternative sampling algorithm for doubly intractable distributions.
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Graham, Matthew McKenzie. "Auxiliary variable Markov chain Monte Carlo methods." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/28962.
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Markov chain Monte Carlo (MCMC) methods are a widely applicable class of algorithms for estimating integrals in statistical inference problems. A common approach in MCMC methods is to introduce additional auxiliary variables into the Markov chain state and perform transitions in the joint space of target and auxiliary variables. In this thesis we consider novel methods for using auxiliary variables within MCMC methods to allow approximate inference in otherwise intractable models and to improve sampling performance in models exhibiting challenging properties such as multimodality. We first consider the pseudo-marginal framework. This extends the Metropolis–Hastings algorithm to cases where we only have access to an unbiased estimator of the density of target distribution. The resulting chains can sometimes show ‘sticking’ behaviour where long series of proposed updates are rejected. Further the algorithms can be difficult to tune and it is not immediately clear how to generalise the approach to alternative transition operators. We show that if the auxiliary variables used in the density estimator are included in the chain state it is possible to use new transition operators such as those based on slice-sampling algorithms within a pseudo-marginal setting. This auxiliary pseudo-marginal approach leads to easier to tune methods and is often able to improve sampling efficiency over existing approaches. As a second contribution we consider inference in probabilistic models defined via a generative process with the probability density of the outputs of this process only implicitly defined. The approximate Bayesian computation (ABC) framework allows inference in such models when conditioning on the values of observed model variables by making the approximation that generated observed variables are ‘close’ rather than exactly equal to observed data. Although making the inference problem more tractable, the approximation error introduced in ABC methods can be difficult to quantify and standard algorithms tend to perform poorly when conditioning on high dimensional observations. This often requires further approximation by reducing the observations to lower dimensional summary statistics. We show how including all of the random variables used in generating model outputs as auxiliary variables in a Markov chain state can allow the use of more efficient and robust MCMC methods such as slice sampling and Hamiltonian Monte Carlo (HMC) within an ABC framework. In some cases this can allow inference when conditioning on the full set of observed values when standard ABC methods require reduction to lower dimensional summaries for tractability. Further we introduce a novel constrained HMC method for performing inference in a restricted class of differentiable generative models which allows conditioning the generated observed variables to be arbitrarily close to observed data while maintaining computational tractability. As a final topicwe consider the use of an auxiliary temperature variable in MCMC methods to improve exploration of multimodal target densities and allow estimation of normalising constants. Existing approaches such as simulated tempering and annealed importance sampling use temperature variables which take on only a discrete set of values. The performance of these methods can be sensitive to the number and spacing of the temperature values used, and the discrete nature of the temperature variable prevents the use of gradient-based methods such as HMC to update the temperature alongside the target variables. We introduce new MCMC methods which instead use a continuous temperature variable. This both removes the need to tune the choice of discrete temperature values and allows the temperature variable to be updated jointly with the target variables within a HMC method.
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Xu, Jason Qian. "Markov Chain Monte Carlo and Non-Reversible Methods." Thesis, The University of Arizona, 2012. http://hdl.handle.net/10150/244823.
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The bulk of Markov chain Monte Carlo applications make use of reversible chains, relying on the Metropolis-Hastings algorithm or similar methods. While reversible chains have the advantage of being relatively easy to analyze, it has been shown that non-reversible chains may outperform them in various scenarios. Neal proposes an algorithm that transforms a general reversible chain into a non-reversible chain with a construction that does not increase the asymptotic variance. These modified chains work to avoid diffusive backtracking behavior which causes Markov chains to be trapped in one position for too long. In this paper, we provide an introduction to MCMC, and discuss the Metropolis algorithm and Neal’s algorithm. We introduce a decaying memory algorithm inspired by Neal’s idea, and then analyze and compare the performance of these chains on several examples.
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Zhang, Yichuan. "Scalable geometric Markov chain Monte Carlo." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20978.
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Markov chain Monte Carlo (MCMC) is one of the most popular statistical inference methods in machine learning. Recent work shows that a significant improvement of the statistical efficiency of MCMC on complex distributions can be achieved by exploiting geometric properties of the target distribution. This is known as geometric MCMC. However, many such methods, like Riemannian manifold Hamiltonian Monte Carlo (RMHMC), are computationally challenging to scale up to high dimensional distributions. The primary goal of this thesis is to develop novel geometric MCMC methods applicable to large-scale problems. To overcome the computational bottleneck of computing second order derivatives in geometric MCMC, I propose an adaptive MCMC algorithm using an efficient approximation based on Limited memory BFGS. I also propose a simplified variant of RMHMC that is able to work effectively on larger scale than the previous methods. Finally, I address an important limitation of geometric MCMC, namely that is only available for continuous distributions. I investigate a relaxation of discrete variables to continuous variables that allows us to apply the geometric methods. This is a new direction of MCMC research which is of potential interest to many applications. The effectiveness of the proposed methods is demonstrated on a wide range of popular models, including generalised linear models, conditional random fields (CRFs), hierarchical models and Boltzmann machines.
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Pereira, Fernanda Chaves. "Bayesian Markov chain Monte Carlo methods in general insurance." Thesis, City University London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342720.
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Cheal, Ryan. "Markov Chain Monte Carlo methods for simulation in pedigrees." Thesis, University of Bath, 1996. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362254.
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Durmus, Alain. "High dimensional Markov chain Monte Carlo methods : theory, methods and applications." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLT001/document.
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L'objet de cette thèse est l'analyse fine de méthodes de Monte Carlopar chaînes de Markov (MCMC) et la proposition de méthodologies nouvelles pour échantillonner une mesure de probabilité en grande dimension. Nos travaux s'articulent autour de trois grands sujets.Le premier thème que nous abordons est la convergence de chaînes de Markov en distance de Wasserstein. Nous établissons des bornes explicites de convergence géométrique et sous-géométrique. Nous appliquons ensuite ces résultats à l'étude d'algorithmes MCMC. Nous nous intéressons à une variante de l'algorithme de Metropolis-Langevin ajusté (MALA) pour lequel nous donnons des bornes explicites de convergence. Le deuxième algorithme MCMC que nous analysons est l'algorithme de Crank-Nicolson pré-conditionné, pour lequel nous montrerons une convergence sous-géométrique.Le second objet de cette thèse est l'étude de l'algorithme de Langevin unajusté (ULA). Nous nous intéressons tout d'abord à des bornes explicites en variation totale suivant différentes hypothèses sur le potentiel associé à la distribution cible. Notre étude traite le cas où le pas de discrétisation est maintenu constant mais aussi du cas d'une suite de pas tendant vers 0. Nous prêtons dans cette étude une attention toute particulière à la dépendance de l'algorithme en la dimension de l'espace d'état. Dans le cas où la densité est fortement convexe, nous établissons des bornes de convergence en distance de Wasserstein. Ces bornes nous permettent ensuite de déduire des bornes de convergence en variation totale qui sont plus précises que celles reportées précédemment sous des conditions plus faibles sur le potentiel. Le dernier sujet de cette thèse est l'étude des algorithmes de type Metropolis-Hastings par échelonnage optimal. Tout d'abord, nous étendons le résultat pionnier sur l'échelonnage optimal de l'algorithme de Metropolis à marche aléatoire aux densités cibles dérivables en moyenne Lp pour p ≥ 2. Ensuite, nous proposons de nouveaux algorithmes de type Metropolis-Hastings qui présentent un échelonnage optimal plus avantageux que celui de l'algorithme MALA. Enfin, nous analysons la stabilité et la convergence en variation totale de ces nouveaux algorithmes<br>The subject of this thesis is the analysis of Markov Chain Monte Carlo (MCMC) methods and the development of new methodologies to sample from a high dimensional distribution. Our work is divided into three main topics. The first problem addressed in this manuscript is the convergence of Markov chains in Wasserstein distance. Geometric and sub-geometric convergence with explicit constants, are derived under appropriate conditions. These results are then applied to thestudy of MCMC algorithms. The first analyzed algorithm is an alternative scheme to the Metropolis Adjusted Langevin algorithm for which explicit geometric convergence bounds are established. The second method is the pre-Conditioned Crank-Nicolson algorithm. It is shown that under mild assumption, the Markov chain associated with thisalgorithm is sub-geometrically ergodic in an appropriated Wasserstein distance. The second topic of this thesis is the study of the Unadjusted Langevin algorithm (ULA). We are first interested in explicit convergence bounds in total variation under different kinds of assumption on the potential associated with the target distribution. In particular, we pay attention to the dependence of the algorithm on the dimension of the state space. The case of fixed step sizes as well as the case of nonincreasing sequences of step sizes are dealt with. When the target density is strongly log-concave, explicit bounds in Wasserstein distance are established. These results are then used to derived new bounds in the total variation distance which improve the one previously derived under weaker conditions on the target density.The last part tackles new optimal scaling results for Metropolis-Hastings type algorithms. First, we extend the pioneer result on the optimal scaling of the random walk Metropolis algorithm to target densities which are differentiable in Lp mean for p ≥ 2. Then, we derive new Metropolis-Hastings type algorithms which have a better optimal scaling compared the MALA algorithm. Finally, the stability and the convergence in total variation of these new algorithms are studied
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Durmus, Alain. "High dimensional Markov chain Monte Carlo methods : theory, methods and applications." Electronic Thesis or Diss., Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLT001.
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L'objet de cette thèse est l'analyse fine de méthodes de Monte Carlopar chaînes de Markov (MCMC) et la proposition de méthodologies nouvelles pour échantillonner une mesure de probabilité en grande dimension. Nos travaux s'articulent autour de trois grands sujets.Le premier thème que nous abordons est la convergence de chaînes de Markov en distance de Wasserstein. Nous établissons des bornes explicites de convergence géométrique et sous-géométrique. Nous appliquons ensuite ces résultats à l'étude d'algorithmes MCMC. Nous nous intéressons à une variante de l'algorithme de Metropolis-Langevin ajusté (MALA) pour lequel nous donnons des bornes explicites de convergence. Le deuxième algorithme MCMC que nous analysons est l'algorithme de Crank-Nicolson pré-conditionné, pour lequel nous montrerons une convergence sous-géométrique.Le second objet de cette thèse est l'étude de l'algorithme de Langevin unajusté (ULA). Nous nous intéressons tout d'abord à des bornes explicites en variation totale suivant différentes hypothèses sur le potentiel associé à la distribution cible. Notre étude traite le cas où le pas de discrétisation est maintenu constant mais aussi du cas d'une suite de pas tendant vers 0. Nous prêtons dans cette étude une attention toute particulière à la dépendance de l'algorithme en la dimension de l'espace d'état. Dans le cas où la densité est fortement convexe, nous établissons des bornes de convergence en distance de Wasserstein. Ces bornes nous permettent ensuite de déduire des bornes de convergence en variation totale qui sont plus précises que celles reportées précédemment sous des conditions plus faibles sur le potentiel. Le dernier sujet de cette thèse est l'étude des algorithmes de type Metropolis-Hastings par échelonnage optimal. Tout d'abord, nous étendons le résultat pionnier sur l'échelonnage optimal de l'algorithme de Metropolis à marche aléatoire aux densités cibles dérivables en moyenne Lp pour p ≥ 2. Ensuite, nous proposons de nouveaux algorithmes de type Metropolis-Hastings qui présentent un échelonnage optimal plus avantageux que celui de l'algorithme MALA. Enfin, nous analysons la stabilité et la convergence en variation totale de ces nouveaux algorithmes<br>The subject of this thesis is the analysis of Markov Chain Monte Carlo (MCMC) methods and the development of new methodologies to sample from a high dimensional distribution. Our work is divided into three main topics. The first problem addressed in this manuscript is the convergence of Markov chains in Wasserstein distance. Geometric and sub-geometric convergence with explicit constants, are derived under appropriate conditions. These results are then applied to thestudy of MCMC algorithms. The first analyzed algorithm is an alternative scheme to the Metropolis Adjusted Langevin algorithm for which explicit geometric convergence bounds are established. The second method is the pre-Conditioned Crank-Nicolson algorithm. It is shown that under mild assumption, the Markov chain associated with thisalgorithm is sub-geometrically ergodic in an appropriated Wasserstein distance. The second topic of this thesis is the study of the Unadjusted Langevin algorithm (ULA). We are first interested in explicit convergence bounds in total variation under different kinds of assumption on the potential associated with the target distribution. In particular, we pay attention to the dependence of the algorithm on the dimension of the state space. The case of fixed step sizes as well as the case of nonincreasing sequences of step sizes are dealt with. When the target density is strongly log-concave, explicit bounds in Wasserstein distance are established. These results are then used to derived new bounds in the total variation distance which improve the one previously derived under weaker conditions on the target density.The last part tackles new optimal scaling results for Metropolis-Hastings type algorithms. First, we extend the pioneer result on the optimal scaling of the random walk Metropolis algorithm to target densities which are differentiable in Lp mean for p ≥ 2. Then, we derive new Metropolis-Hastings type algorithms which have a better optimal scaling compared the MALA algorithm. Finally, the stability and the convergence in total variation of these new algorithms are studied
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Wu, Miaodan. "Markov chain Monte Carlo methods applied to Bayesian data analysis." Thesis, University of Cambridge, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.625087.
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Paul, Rajib. "Theoretical And Algorithmic Developments In Markov Chain Monte Carlo." The Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=osu1218184168.
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Khedri, Shiler. "Markov chain Monte Carlo methods for exact tests in contingency tables." Thesis, Durham University, 2012. http://etheses.dur.ac.uk/5579/.
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This thesis is mainly concerned with conditional inference for contingency tables, where the MCMC method is used to take a sample of the conditional distribution. One of the most common models to be investigated in contingency tables is the independence model. Classic test statistics for testing the independence hypothesis, Pearson and likelihood chi-square statistics rely on large sample distributions. The large sample distribution does not provide a good approximation when the sample size is small. The Fisher exact test is an alternative method which enables us to compute the exact p-value for testing the independence hypothesis. For contingency tables of large dimension, the Fisher exact test is not practical as it requires counting all tables in the sample space. We will review some enumeration methods which do not require us to count all tables in the sample space. However, these methods would also fail to compute the exact p-value for contingency tables of large dimensions. \cite{DiacStur98} introduced a method based on the Grobner basis. It is quite complicated to compute the Grobner basis for contingency tables as it is different for each individual table, not only for different sizes of table. We also review the method introduced by \citet{AokiTake03} using the minimal Markov basis for some particular tables. \cite{BuneBesa00} provided an algorithm using the most fundamental move to make the irreducible Markov chain over the sample space, defining an extra space. The algorithm is only introduced for $2\times J \times K$ tables using the Rasch model. We introduce direct proof for irreducibility of the Markov chain achieved by the Bunea and Besag algorithm. This is then used to prove that \cite{BuneBesa00} approach can be applied for some tables of higher dimensions, such as $3\times 3\times K$ and $3\times 4 \times 4$. The efficiency of the Bunea and Besag approach is extensively investigated for many different settings such as for tables of low/moderate/large dimensions, tables with special zero pattern, etc. The efficiency of algorithms is measured based on the effective sample size of the MCMC sample. We use two different metrics to penalise the effective sample size: running time of the algorithm and total number of bits used. These measures are also used to compute the efficiency of an adjustment of the Bunea and Besag algorithm which show that it outperforms the the original algorithm for some settings.
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Ibrahim, Adriana Irawati Nur. "New methods for mode jumping in Markov chain Monte Carlo algorithms." Thesis, University of Bath, 2009. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.500720.
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Standard Markov chain Monte Carlo (MCMC) sampling methods can experience problem sampling from multi-modal distributions. A variety of sampling methods have been introduced to overcome this problem. The mode jumping method of Tjelmeland & Hegstad (2001) tries to find a mode and propose a value from that mode in each mode jumping attempt. This approach is inefficient in that the work needed to find each mode and model the distribution in a neighbourhood of the mode is carried out repeatedly during the sampling process. We shall propose a new mode jumping approach which retains features of the Tjelmeland & Hegstad (2001) method but differs in that it finds the modes in an initial search, then uses this information to jump between modes effectively in the sampling run. Although this approach does not allow a second chance to find modes in the sampling run, we can show that the overall probability of missing a mode in our approach is still low. We apply our methods to sample from distributions which have continuous variables, discrete variables, a mixture of discrete and continuous variables and variable dimension. We show that our methods work well in each case and in general, are better than the MCMC sampling methods commonly used in these cases and also, are better than the Tjelmeland & Hegstad (2001) method in particular.
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Barata, Teresa Cordeiro Ferreira Nunes. "Two examples of curve estimation using Markov Chain Monte Carlo methods." Thesis, University of Cambridge, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.612139.
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Nemirovsky, Danil. "Monte Carlo methods and Markov chain based approaches for PageRank computation." Nice, 2010. http://www.theses.fr/2010NICE4018.
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Vaičiulytė, Ingrida. "Study and application of Markov chain Monte Carlo method." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2014~D_20141209_112440-55390.
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Markov chain Monte Carlo adaptive methods by creating computationally effective algorithms for decision-making of data analysis with the given accuracy are analyzed in this dissertation. The tasks for estimation of parameters of the multivariate distributions which are constructed in hierarchical way (skew t distribution, Poisson-Gaussian model, stable symmetric vector law) are described and solved in this research. To create the adaptive MCMC procedure, the sequential generating method is applied for Monte Carlo samples, introducing rules for statistical termination and for sample size regulation of Markov chains. Statistical tasks, solved by this method, reveal characteristics of relevant computational problems including MCMC method.
Effectiveness of the MCMC algorithms is analyzed by statistical modeling method, constructed in the dissertation. Tests made with sportsmen data and financial data of enterprises, belonging to health-care industry, confirmed that numerical properties of the method correspond to the theoretical model. The methods and algorithms created also are applied to construct the model for sociological data analysis. Tests of algorithms have shown that adaptive MCMC algorithm allows to obtain estimators of examined distribution parameters in lower number of chains, and reducing the volume of calculations approximately two times. The algorithms created in this dissertation can be used to test the systems of stochastic type and to solve other statistical... [to full text]<br>Disertacijoje nagrinėjami Markovo grandinės Monte-Karlo (MCMC) adaptavimo metodai, skirti efektyviems skaitiniams duomenų analizės sprendimų priėmimo su iš anksto nustatytu patikimumu algoritmams sudaryti. Suformuluoti ir išspręsti hierarchiniu būdu sudarytų daugiamačių skirstinių (asimetrinio t skirstinio, Puasono-Gauso modelio, stabiliojo simetrinio vektoriaus dėsnio) parametrų vertinimo uždaviniai. Adaptuotai MCMC procedūrai sukurti yra pritaikytas nuoseklaus Monte-Karlo imčių generavimo metodas, įvedant statistinį stabdymo kriterijų ir imties tūrio reguliavimą. Statistiniai uždaviniai išspręsti šiuo metodu leidžia atskleisti aktualias MCMC metodų skaitmeninimo problemų ypatybes. MCMC algoritmų efektyvumas tiriamas pasinaudojant disertacijoje sudarytu statistinio modeliavimo metodu. Atlikti eksperimentai su sportininkų duomenimis ir sveikatos industrijai priklausančių įmonių finansiniais duomenimis patvirtino, kad metodo skaitinės savybės atitinka teorinį modelį. Taip pat sukurti metodai ir algoritmai pritaikyti sociologinių duomenų analizės modeliui sudaryti. Atlikti tyrimai parodė, kad adaptuotas MCMC algoritmas leidžia gauti nagrinėjamų skirstinių parametrų įvertinius per mažesnį grandžių skaičių ir maždaug du kartus sumažinti skaičiavimų apimtį. Disertacijoje sukonstruoti algoritmai gali būti pritaikyti stochastinio pobūdžio sistemų tyrimui ir kitiems statistikos uždaviniams spręsti MCMC metodu.
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Gausland, Eivind Blomholm. "Parameter Estimation in Extreme Value Models with Markov Chain Monte Carlo Methods." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2010. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-10032.
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<p>In this thesis I have studied how to estimate parameters in an extreme value model with Markov Chain Monte Carlo (MCMC) given a data set. This is done with synthetic Gaussian time series generated by spectral densities, called spectrums, with a "box" shape. Three different spectrums have been used. In the acceptance probability in the MCMC algorithm, the likelihood have been built up by dividing the time series into blocks consisting of a constant number of points. In each block, only the maximum value, i.e. the extreme value, have been used. Each extreme value will then be interpreted as independent. Since the time series analysed are generated the way they are, there exists theoretical values for the parameters in the extreme value model. When the MCMC algorithm is tested to fit a model to the generated data, the true parameter values are already known. For the first and widest spectrum, the method is unable to find estimates matching the true values for the parameters in the extreme value model. For the two other spectrums, I obtained good estimates for some block lengths, others block lengths gave poor estimates compared to the true values. Finally, it looked like an increasing block length gave more accurate estimates as the spectrum became more narrow banded. A final simulation on a time series generated by a narrow banded spectrum, disproved this hypothesis.</p>
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Niederberger, Theresa. "Markov chain Monte Carlo methods for parameter identification in systems biology models." Diss., Ludwig-Maximilians-Universität München, 2012. http://nbn-resolving.de/urn:nbn:de:bvb:19-157798.
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Xu, Xiaojin. "Methods in Hypothesis Testing, Markov Chain Monte Carlo and Neuroimaging Data Analysis." Thesis, Harvard University, 2013. http://dissertations.umi.com/gsas.harvard:10927.
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This thesis presents three distinct topics: a modified K-S test for autocorrelated data, improving MCMC convergence rate with residual augmentations, and resting state fMRI data analysis. In Chapter 1, we present a modified K-S test to adjust for sample autocorrelation. We first demonstrate that the original K-S test does not have the nominal type one error rate when applied to autocorrelated samples. Then the notion of mixing conditions and Billingsley's theorem are reviewed. Based on these results, we suggest an effective sample size formula to adjust sample autocorrelation. Extensive simulation studies are presented to demonstrate that this modified K-S test has the nominal type one error as well as reasonable power for various autocorrelated samples. An application to an fMRI data set is presented in the end. In Chapter 2 of this thesis, we present the work on MCMC sampling. Inspired by a toy example of random effect model, we find there are two ways to boost the efficiency of MCMC algorithms: direct and indirect residual augmentations. We first report theoretical investigations under a class of normal/independece models, where we find an intriguing phase transition type of phenomenon. Then we present an application of the direct residual augmentations to the probit regression, where we also include a numerical comparison with other existing algorithms.<br>Statistics
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Demiris, Nikolaos. "Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods." Thesis, University of Nottingham, 2004. http://eprints.nottingham.ac.uk/10078/.
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This thesis is concerned with statistical methodology for the analysis of stochastic SIR (Susceptible->Infective->Removed) epidemic models. We adopt the Bayesian paradigm and we develop suitably tailored Markov chain Monte Carlo (MCMC) algorithms. The focus is on methods that are easy to generalise in order to accomodate epidemic models with complex population structures. Additionally, the models are general enough to be applicable to a wide range of infectious diseases. We introduce the stochastic epidemic models of interest and the MCMC methods we shall use and we review existing methods of statistical inference for epidemic models. We develop algorithms that utilise multiple precision arithmetic to overcome the well-known numerical problems in the calculation of the final size distribution for the generalised stochastic epidemic. Consequently, we use these exact results to evaluate the precision of asymptotic theorems previously derived in the literature. We also use the exact final size probabilities to obtain the posterior distribution of the threshold parameter $R_0$. We proceed to develop methods of statistical inference for an epidemic model with two levels of mixing. This model assumes that the population is partitioned into subpopulations and permits infection on both local (within-group) and global (population-wide) scales. We adopt two different data augmentation algorithms. The first method introduces an appropriate latent variable, the \emph{final severity}, for which we have asymptotic information in the event of an outbreak among a population with a large number of groups. Hence, approximate inference can be performed conditional on a ``major'' outbreak, a common assumption for stochastic processes with threshold behaviour such as epidemics and branching processes. In the last part of this thesis we use a \emph{random graph} representation of the epidemic process and we impute more detailed information about the infection spread. The augmented state-space contains aspects of the infection spread that have been impossible to obtain before. Additionally, the method is exact in the sense that it works for any (finite) population and group sizes and it does not assume that the epidemic is above threshold. Potential uses of the extra information include the design and testing of appropriate prophylactic measures like different vaccination strategies. An attractive feature is that the two algorithms complement each other in the sense that when the number of groups is large the approximate method (which is faster) is almost as accurate as the exact one and can be used instead. Finally, it is straightforward to extend our methods to more complex population structures like overlapping groups, small-world and scale-free networks
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Spade, David Allen. "Investigating Convergence of Markov Chain Monte Carlo Methods for Bayesian Phylogenetic Inference." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1372173121.
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Li, Yuanzhi. "Bayesian Models for Repeated Measures Data Using Markov Chain Monte Carlo Methods." DigitalCommons@USU, 2016. https://digitalcommons.usu.edu/etd/6997.
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Bayesian models for repeated measures data are fitted to three different data an analysis projects. Markov Chain Monte Carlo (MCMC) methodology is applied to each case with Gibbs sampling and / or an adaptive Metropolis-Hastings (MH ) algorithm used to simulate the posterior distribution of parameters. We implement a Bayesian model with different variance-covariance structures to an audit fee data set. Block structures and linear models for variances are used to examine the linear trend and different behaviors before and after regulatory change during year 2004-2005. We proposed a Bayesian hierarchical model with latent teacher effects, to determine whether teacher professional development (PD) utilizing cyber-enabled resources lead to meaningful student learning outcomes measured by 8th grade student end-of-year scores (CRT scores) for students with teachers who underwent PD. Bayesian variable selection methods are applied to select teacher learning instrument variables to predict teacher effects. We fit a Bayesian two-part model with the first-part a multivariate probit model and the second-p art a log-normal regression to a repeated measures health care data set to analyze the relationship between Body Mass Index (BMI) and health care expenditures and the correlation between the probability of expenditures and dollar amount spent given expenditures. Models were fitted to a training set and predictions were made on both the training set and the test set.
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Browne, William J. "Applying MCMC methods to multi-level models." Thesis, University of Bath, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.268210.
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Tu, Zhuowen. "Image Parsing by Data-Driven Markov Chain Monte Carlo." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1038347031.
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Veitch, John D. "Applications of Markov Chain Monte Carlo methods to continuous gravitational wave data analysis." Thesis, Connect to e-thesis to view abstract. Move to record for print version, 2007. http://theses.gla.ac.uk/35/.
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Thesis (Ph.D.) - University of Glasgow, 2007.<br>Ph.D. thesis submitted to Information and Mathematical Sciences Faculty, Department of Mathematics, University of Glasgow, 2007. Includes bibliographical references. Print version also available.
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Walker, Neil Rawlinson. "A Bayesian approach to the job search model and its application to unemployment durations using MCMC methods." Thesis, University of Newcastle Upon Tyne, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299053.
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Higdon, David. "Spatial applications of Markov chain Monte Carlo for Bayesian inference /." Thesis, Connect to this title online; UW restricted, 1994. http://hdl.handle.net/1773/8942.
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Olvera, Astivia Oscar Lorenzo. "On the estimation of the polychoric correlation coefficient via Markov Chain Monte Carlo methods." Thesis, University of British Columbia, 2013. http://hdl.handle.net/2429/44349.
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Bayesian statistics is an alternative approach to traditional frequentist statistics that is rapidly gaining
adherents across different scientific fields. Although initially only accessible to statisticians or
mathematically-sophisticated data analysts, advances in modern computational power are helping to
make this new paradigm approachable to the everyday researcher and this dissemination is helping
open doors to problems that have remained unsolvable or whose solution was extremely complicated
through the use of classical statistics. In spite of this, many researchers in the behavioural or
educational sciences are either unaware of this new approach or just vaguely familiar with some of its
basic tenets. The primary purpose of this thesis is to take a well-known problem in psychometrics, the
estimation of the polychoric correlation coefficient, and solve it using Bayesian statistics through the
method developed by Albert (1992). Through the use of computer simulations this method is compared
to traditional maximum likelihood estimation across various sample sizes, skewness levels and numbers
of discretisation points for the latent variable, highlighting the cases where the Bayesian approach is
superior, inferior or equally effective to the maximum likelihood approach. Another issue that is
investigated is a sensitivity analysis of sorts of the prior probability distributions where a skewed
(bivariate log-normal) and symmetric (bivariate normal) priors are used to calculate the polychoric
correlation coefficient when feeding them data with varying degrees of skewness, helping demonstrate
to the reader how does changing the prior distribution for certain kinds of data helps or hinders the
estimation process. The most important results of these studies are discussed as well as future
implications for the use of Bayesian statistics in psychometrics
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Bray, Isabelle Cella. "Modelling the prevalence of Down syndrome with applications of Markov chain Monte Carlo methods." Thesis, University of Plymouth, 1998. http://hdl.handle.net/10026.1/2408.
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This thesis was motivated by applications in the epidemiology of Down syndrome and prenatal screening for Down syndrome. Methodological problems arising in these applications include under-ascertainment of cases in livebirth studies, double-sampled data with missing observations and coarsening of data. These issues are considered from a classical perspective using maximum likelihood and from a Bayesian viewpoint employing Markov chain Monte Carlo (MCMC) techniques. Livebirth prevalence studies published in the literature used a variety of data collection methods and many are of uncertain completeness. In two of the nine studies an estimate of the level of under-reporting is available. We present a meta-analysis of these studies in which maternal age-related risks and the levels of under-ascertainment in individual studies are estimated simultaneously. A modified logistic model is used to describe the relationship between Down syndrome prevalence and maternal age. The model is then extended to include data from several studies of prevalence rates observed at times of chorionic villus sampling (CVS) and amniocentesis. New estimates for spontaneous loss rates between the times" of CVS, amniocentesis and live birth are presented. The classical analysis of live birth prevalence data is then compared with an MCMC analysis which allows prior information concerning ascertainment to be incorporated. This approach is particularly attractive since the double-sampled data structure includes missing observations. The MCMC algorithm, which uses single-component Metropolis-Hastings steps to simulate model parameters and missing data, is run under three alternative prior specifications. Several convergence diagnostics are also considered and compared. Finally, MCMC techniques are used to model the distribution of fetal nuchal translucency (NT), an ultrasound marker for Down syndrome. The data are a mixture of measurements rounded to whole millimetres and measurements more accurately recorded to one decimal place. An MCMC algorithm is applied to simulate the proportion of measurements rounded to whole millimetres and parameters to describe the distribution of NT in unaffected and Down syndrome pregnancies. Predictive probabilities of Down syndrome given NT and maternal age are then calculated.
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Witte, Hugh Douglas. "Markov chain Monte Carlo and data augmentation methods for continuous-time stochastic volatility models." Diss., The University of Arizona, 1999. http://hdl.handle.net/10150/283976.
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In this paper we exploit some recent computational advances in Bayesian inference, coupled with data augmentation methods, to estimate and test continuous-time stochastic volatility models. We augment the observable data with a latent volatility process which governs the evolution of the data's volatility. The level of the latent process is estimated at finer increments than the data are observed in order to derive a consistent estimator of the variance over each time period the data are measured. The latent process follows a law of motion which has either a known transition density or an approximation to the transition density that is an explicit function of the parameters characterizing the stochastic differential equation. We analyze several models which differ with respect to both their drift and diffusion components. Our results suggest that for two size-based portfolios of U.S. common stocks, a model in which the volatility process is characterized by nonstationarity and constant elasticity of instantaneous variance (with respect to the level of the process) greater than 1 best describes the data. We show how to estimate the various models, undertake the model selection exercise, update posterior distributions of parameters and functions of interest in real time, and calculate smoothed estimates of within sample volatility and prediction of out-of-sample returns and volatility. One nice aspect of our approach is that no transformations of the data or the latent processes, such as subtracting out the mean return prior to estimation, or formulating the model in terms of the natural logarithm of volatility, are required.
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Byers, Simon. "Bayesian modeling of highly structured systems using Markov chain Monte Carlo /." Thesis, Connect to this title online; UW restricted, 1998. http://hdl.handle.net/1773/8980.
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Olsen, Andrew Nolan. "When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1433770406.
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Karawatzki, Roman, Josef Leydold, and Klaus Pötzelberger. "Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2005. http://epub.wu.ac.at/1400/1/document.pdf.
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Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. An implementation of these algorithms in C is available from the <a href="http://statmath.wu-wien.ac.at/software/hitro/">authors</a>. (author's abstract)<br>Series: Research Report Series / Department of Statistics and Mathematics
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Manrique, Garcia Aurora. "Econometric analysis of limited dependent time series." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389797.
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Li, Min. "Bayesian discovery of regulatory motifs using reversible jump Markov chain Monte Carlo /." Thesis, Connect to this title online; UW restricted, 2006. http://hdl.handle.net/1773/9529.
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Nilsson, Mats. "Building Reconstruction of Digital Height Models with the Markov Chain Monte Carlo Method." Thesis, Linköpings universitet, Datorseende, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-148886.
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Data about the earth is increasing in value and demand from customers, but itis difficult to produce accurately and cheap. This thesis examines if it is possible to take low resolution and distorted 3D data and increase the accuracy of building geometry by performing building reconstruction. Building reconstruction is performed with a Markov chain Monte Carlo method where building primitives are placed iteratively until a good fit is found. The digital height model and pixel classification used is produced by Vricon. The method is able to correctly place primitive models, but often overestimate their dimensions by about 15%.
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Fu, Jianlin. "A markov chain monte carlo method for inverse stochastic modeling and uncertainty assessment." Doctoral thesis, Universitat Politècnica de València, 2008. http://hdl.handle.net/10251/1969.
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Unlike the traditional two-stage methods, a conditional and inverse-conditional simulation approach may directly generate independent, identically distributed realizations to honor both static data and state data in one step. The Markov chain Monte Carlo (McMC) method was proved a powerful tool to perform such type of stochastic simulation. One of the main advantages of the McMC over the traditional sensitivity-based optimization methods to inverse problems is its power, flexibility and well-posedness in incorporating observation data from different sources. In this work, an improved version of the McMC method is presented to perform the stochastic simulation of reservoirs and aquifers in the framework of multi-Gaussian geostatistics.
First, a blocking scheme is proposed to overcome the limitations of the classic single-component Metropolis-Hastings-type McMC. One of the main characteristics of the blocking McMC (BMcMC) scheme is that, depending on the inconsistence between the prior model and the reality, it can preserve the prior spatial structure and statistics as users specified. At the same time, it improves the mixing of the Markov chain and hence enhances the computational efficiency of the McMC. Furthermore, the exploration ability and the mixing speed of McMC are efficiently improved by coupling the multiscale proposals, i.e., the coupled multiscale McMC method. In order to make the BMcMC method capable of dealing with the high-dimensional cases, a multi-scale scheme is introduced to accelerate the computation of the likelihood which greatly improves the computational efficiency of the McMC due to the fact that most of the computational efforts are spent on the forward simulations. To this end, a flexible-grid full-tensor finite-difference simulator, which is widely compatible with the outputs from various upscaling subroutines, is developed to solve the flow equations and a constant-displacement random-walk
particle-tracking method, which enhances the com<br>Fu, J. (2008). A markov chain monte carlo method for inverse stochastic modeling and uncertainty assessment [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/1969<br>Palancia
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Lindahl, John, and Douglas Persson. "Data-driven test case design of automatic test cases using Markov chains and a Markov chain Monte Carlo method." Thesis, Malmö universitet, Fakulteten för teknik och samhälle (TS), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mau:diva-43498.
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Large and complex software that is frequently changed leads to testing challenges. It is well established that the later a fault is detected in software development, the more it costs to fix. This thesis aims to research and develop a method of generating relevant and non-redundant test cases for a regression test suite, to catch bugs as early in the development process as possible. The research was executed at Axis Communications AB with their products and systems in mind. The approach utilizes user data to dynamically generate a Markov chain model and with a Markov chain Monte Carlo method, strengthen that model. The model generates test case proposals, detects test gaps, and identifies redundant test cases based on the user data and data from a test suite. The sampling in the Markov chain Monte Carlo method can be modified to bias the model for test coverage or relevancy. The model is generated generically and can therefore be implemented in other API-driven systems. The model was designed with scalability in mind and further implementations can be made to increase the complexity and further specialize the model for individual needs.
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Fischer, Alexander. "An Uncoupling Coupling method for Markov chain Monte Carlo simulations with an application to biomolecules." [S.l. : s.n.], 2003. http://www.diss.fu-berlin.de/2003/234/index.html.
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Yang, Chao. "ON PARTICLE METHODS FOR UNCERTAINTY QUANTIFICATION IN COMPLEX SYSTEMS." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1511967797285962.
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Lewis, John Robert. "Bayesian Restricted Likelihood Methods." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1407505392.
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Walker, Matthew James. "Methods for Bayesian inversion of seismic data." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10504.
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The purpose of Bayesian seismic inversion is to combine information derived from seismic data and prior geological knowledge to determine a posterior probability distribution over parameters describing the elastic and geological properties of the subsurface. Typically the subsurface is modelled by a cellular grid model containing thousands or millions of cells within which these parameters are to be determined. Thus such inversions are computationally expensive due to the size of the parameter space (being proportional to the number of grid cells) over which the posterior is to be determined. Therefore, in practice approximations to Bayesian seismic inversion must be considered. A particular, existing approximate workflow is described in this thesis: the so-called two-stage inversion method explicitly splits the inversion problem into elastic and geological inversion stages. These two stages sequentially estimate the elastic parameters given the seismic data, and then the geological parameters given the elastic parameter estimates, respectively. In this thesis a number of methodologies are developed which enhance the accuracy of this approximate workflow. To reduce computational cost, existing elastic inversion methods often incorporate only simplified prior information about the elastic parameters. Thus a method is introduced which transforms such results, obtained using prior information specified using only two-point geostatistics, into new estimates containing sophisticated multi-point geostatistical prior information. The method uses a so-called deep neural network, trained using only synthetic instances (or `examples') of these two estimates, to apply this transformation. The method is shown to improve the resolution and accuracy (by comparison to well measurements) of elastic parameter estimates determined for a real hydrocarbon reservoir. It has been shown previously that so-called mixture density network (MDN) inversion can be used to solve geological inversion analytically (and thus very rapidly and efficiently) but only under certain assumptions about the geological prior distribution. A so-called prior replacement operation is developed here, which can be used to relax these requirements. It permits the efficient MDN method to be incorporated into general stochastic geological inversion methods which are free from the restrictive assumptions. Such methods rely on the use of Markov-chain Monte-Carlo (MCMC) sampling, which estimate the posterior (over the geological parameters) by producing a correlated chain of samples from it. It is shown that this approach can yield biased estimates of the posterior. Thus an alternative method which obtains a set of non-correlated samples from the posterior is developed, avoiding the possibility of bias in the estimate. The new method was tested on a synthetic geological inversion problem; its results compared favourably to those of Gibbs sampling (a MCMC method) on the same problem, which exhibited very significant bias. The geological prior information used in seismic inversion can be derived from real images which bear similarity to the geology anticipated within the target region of the subsurface. Such so-called training images are not always available from which this information (in the form of geostatistics) may be extracted. In this case appropriate training images may be generated by geological experts. However, this process can be costly and difficult. Thus an elicitation method (based on a genetic algorithm) is developed here which obtains the appropriate geostatistics reliably and directly from a geological expert, without the need for training images. 12 experts were asked to use the algorithm (individually) to determine the appropriate geostatistics for a physical (target) geological image. The majority of the experts were able to obtain a set of geostatistics which were consistent with the true (measured) statistics of the target image.
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Hörmann, Wolfgang, and Josef Leydold. "Monte Carlo Integration Using Importance Sampling and Gibbs Sampling." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 2005. http://epub.wu.ac.at/1642/1/document.pdf.
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To evaluate the expectation of a simple function with respect to a complicated multivariate density Monte Carlo integration has become the main technique. Gibbs sampling and importance sampling are the most popular methods for this task. In this contribution we propose a new simple general purpose importance sampling procedure. In a simulation study we compare the performance of this method with the performance of Gibbs sampling and of importance sampling using a vector of independent variates. It turns out that the new procedure is much better than independent importance sampling; up to dimension five it is also better than Gibbs sampling. The simulation results indicate that for higher dimensions Gibbs sampling is superior. (author's abstract)<br>Series: Preprint Series / Department of Applied Statistics and Data Processing
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Niederberger, Theresa [Verfasser], and Patrick [Akademischer Betreuer] Cramer. "Markov chain Monte Carlo methods for parameter identification in systems biology models / Theresa Niederberger. Betreuer: Patrick Cramer." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2012. http://d-nb.info/1036101029/34.
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Matsumoto, Nobuyuki. "Geometry of configuration space in Markov chain Monte Carlo methods and the worldvolume approach to the tempered Lefschetz thimble method." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263464.
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Allchin, Lorraine Doreen May. "Statistical methods for mapping complex traits." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:65f392ba-1b64-4b00-8871-7cee98809ce1.
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The first section of this thesis addresses the problem of simultaneously identifying multiple loci that are associated with a trait, using a Bayesian Markov Chain Monte Carlo method. It is applicable to both case/control and quantitative data. I present simulations comparing the methods to standard frequentist methods in human case/control and mouse QTL datasets, and show that in the case/control simulations the standard frequentist method out performs my model for all but the highest effect simulations and that for the mouse QTL simulations my method performs as well as the frequentist method in some cases and worse in others. I also present analysis of real data and simulations applying my method to a simulated epistasis data set. The next section was inspired by the challenges involved in applying a Markov Chain Monte Carlo method to genetic data. It is an investigation into the performance and benefits of the Matlab parallel computing toolbox, specifically its implementation of the Cuda programing language to Matlab's higher level language. Cuda is a language which allows computational calculations to be carried out on the computer's graphics processing unit (GPU) rather than its central processing unit (CPU). The appeal of this tool box is its ease of use as few code adaptions are needed. The final project of this thesis was to develop an HMM for reconstructing the founders of sparsely sequenced inbred populations. The motivation here, that whilst sequencing costs are rapidly decreasing, it is still prohibitively expensive to fully sequence a large number of individuals. It was proposed that, for populations descended from a known number of founders, it would be possible to sequence these individuals with a very low coverage, use a hidden Markov model (HMM) to represent the chromosomes as mosaics of the founders, then use these states to impute the missing data. For this I developed a Viterbi algorithm with a transition probability matrix based on recombination rate which changes for each observed state.
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Tiboaca, Oana D. "On the application of the reversible jump Markov chain Monte Carlo method within structural dynamics." Thesis, University of Sheffield, 2016. http://etheses.whiterose.ac.uk/17126/.
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System Identification (SID) is an important area of structural dynamics and is concerned with constructing a functional relationship between the inputs and the outputs of a system. Furthermore, it estimates the parameters that the studied system depends upon. This aspect of structural dynamics has been studied for many years and computational methods have been developed in order to deal with the system identification of real structures, with the aim of getting a better understanding of their dynamic behaviour. The most straightforward classification of structures is into structures that behave linearly and structures that behave nonlinearly. Even so, one needs to keep in mind that no structure is indefinitely linear. During its service, a structure can behave nonlinearly at any given point, under the right working and environmental conditions. A key challenge in applying SID to real systems is in handling the uncertainty inherent in the process. Uncertainty arises from various sources such as modelling error and measurement error (noisy data), resulting in uncertainty in the parameter estimates. One of the ways in which one can deal with uncertainty is by adopting a probabilistic framework. In this way one admits the limitations in the process of SID through providing probability distributions over the models and parameters of interest, rather than a simple 'best estimate'. Throughout this work a Bayesian probabilistic framework is adopted as it covers the uncertainty issue without over-fitting (it provides the simplest, least complex solutions to the issue at hand). Of great interest when working within a Bayesian framework are Markov Chain Monte Carlo(MCMC) sampling methods. Of relevance to this research are the Metropolis-Hastings(MH) algorithm and the Reversible Jump Markov Chain Monte Carlo(RJMCMC) algorithm. Markov Chain Monte Carlo(MCMC) methods and algorithms have been extensively investigated for linear dynamical systems. One of the advantages of these methods being used in a Bayesian framework is that they handle uncertainty in a principled way. In recent years, increasing attention has been paid to the role nonlinearity plays in engineering problems. As a result, there is an increasing focus on developing computational tools that may be applied to nonlinear systems as well as linear systems, with the objective that they should provide reliable results at reasonable computational costs. The application of MCMC methods in nonlinear system identi cation(NLSID) has focused on parameter estimation. However, often enough, the model form of systems is assumed known which is not the case in many contexts(such as NLSID when the nonlinearity is hard to identify and model, or Structural Health Monitoring when the damage extent or number of damage sites is unknown). The current thesis is concerned with the development of computational tools for performing System Identification in the context of structural dynamics, for both linear and nonlinear systems. The research presented within this work will demonstrate how the Reversible Jump Markov Chain Monte Carlo algorithm, within a Bayesian framework, can be used in the area of SID in a structural dynamics context for doing both parameter estimation and model selection. The performance of the RJMCMC algorithm will be benchmarked throughout against the MH algorithm. Several numerical case studies will be introduced to demonstrate how the RJMCMC algorithm may be applied in linear and nonlinear SID; and one numerical case study to demonstrate application to a SHM problem. These will be followed by experimental case studies to evaluate linear and nonlinear SID performance for a real structure.
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Dhulipala, Lakshmi Narasimha Somayajulu. "Bayesian Methods for Intensity Measure and Ground Motion Selection in Performance-Based Earthquake Engineering." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/88493.
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The objective of quantitative Performance-Based Earthquake Engineering (PBEE) is designing buildings that meet the specified performance objectives when subjected to an earthquake. One challenge to completely relying upon a PBEE approach in design practice is the open-ended nature of characterizing the earthquake ground motion by selecting appropriate ground motions and Intensity Measures (IM) for seismic analysis. This open-ended nature changes the quantified building performance depending upon the ground motions and IMs selected. So, improper ground motion and IM selection can lead to errors in structural performance prediction and thus to poor designs. Hence, the goal of this dissertation is to propose methods and tools that enable an informed selection of earthquake IMs and ground motions, with the broader goal of contributing toward a robust PBEE analysis. In doing so, the change of perspective and the mechanism to incorporate additional information provided by Bayesian methods will be utilized.
Evaluation of the ability of IMs towards predicting the response of a building with precision and accuracy for a future, unknown earthquake is a fundamental problem in PBEE analysis. Whereas current methods for IM quality assessment are subjective and have multiple criteria (hence making IM selection challenging), a unified method is proposed that enables rating the numerous IMs. This is done by proposing the first quantitative metric for assessing IM accuracy in predicting the building response to a future earthquake, and then by investigating the relationship between precision and accuracy. This unified metric is further expected to provide a pathway toward improving PBEE analysis by allowing the consideration of multiple IMs.
Similar to IM selection, ground motion selection is important for PBEE analysis. Consensus on the "right" input motions for conducting seismic response analyses is often varied and dependent on the analyst. Hence, a general and flexible tool is proposed to aid ground motion selection. General here means the tool encompasses several structural types by considering their sensitivities to different ground motion characteristics. Flexible here means the tool can consider additional information about the earthquake process when available with the analyst. Additionally, in support of this ground motion selection tool, a simplified method for seismic hazard analysis for a vector of IMs is developed.
This dissertation addresses four critical issues in IM and ground motion selection for PBEE by proposing: (1) a simplified method for performing vector hazard analysis given multiple IMs; (2) a Bayesian framework to aid ground motion selection which is flexible and general to incorporate preferences of the analyst; (3) a unified metric to aid IM quality assessment for seismic fragility and demand hazard assessment; (4) Bayesian models for capturing heteroscedasticity (non-constant standard deviation) in seismic response analyses which may further influence IM selection.<br>Doctor of Philosophy<br>Earthquake ground shaking is a complex phenomenon since there is no unique way to assess its strength. Yet, the strength of ground motion (shaking) becomes an integral part for predicting the future earthquake performance of buildings using the Performance-Based Earthquake Engineering (PBEE) framework. The PBEE framework predicts building performance in terms of expected financial losses, possible downtime, the potential of the building to collapse under a future earthquake. Much prior research has shown that the predictions made by the PBEE framework are heavily dependent upon how the strength of a future earthquake ground motion is characterized. This dependency leads to uncertainty in the predicted building performance and hence its seismic design. The goal of this dissertation therefore is to employ Bayesian reasoning, which takes into account the alternative explanations or perspectives of a research problem, and propose robust quantitative methods that aid IM selection and ground motion selection in PBEE The fact that the local intensity of an earthquake can be characterized in multiple ways using Intensity Measures (IM; e.g., peak ground acceleration) is problematic for PBEE because it leads to different PBEE results for different choices of the IM. While formal procedures for selecting an optimal IM exist, they may be considered as being subjective and have multiple criteria making their use difficult and inconclusive. Bayes rule provides a mechanism called change of perspective using which a problem that is difficult to solve from one perspective could be tackled from a different perspective. This change of perspective mechanism is used to propose a quantitative, unified metric for rating alternative IMs. The immediate application of this metric is aiding the selection of the best IM that would predict the building earthquake performance with least bias. Structural analysis for performance assessment in PBEE is conducted by selecting ground motions which match a target response spectrum (a representation of future ground motions). The definition of a target response spectrum lacks general consensus and is dependent on the analysts’ preferences. To encompass all these preferences and requirements of analysts, a Bayesian target response spectrum which is general and flexible is proposed. While the generality of this Bayesian target response spectrum allow analysts select those ground motions to which their structures are the most sensitive, its flexibility permits the incorporation of additional information (preferences) into the target response spectrum development. This dissertation addresses four critical questions in PBEE: (1) how can we best define ground motion at a site?; (2) if ground motion can only be defined by multiple metrics, how can we easily derive the probability of such shaking at a site?; (3) how do we use these multiple metrics to select a set of ground motion records that best capture the site’s unique seismicity; (4) when those records are used to analyze the response of a structure, how can we be sure that a standard linear regression technique accurately captures the uncertainty in structural response at low and high levels of shaking?
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Wang, Yinglu. "A Markov Chain Based Method for Time Series Data Modeling and Prediction." University of Cincinnati / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1592395278430805.
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Chen, Yuting. "Inférence bayésienne dans les modèles de croissance de plantes pour la prévision et la caractérisation des incertitudes." Thesis, Châtenay-Malabry, Ecole centrale de Paris, 2014. http://www.theses.fr/2014ECAP0040/document.
Full textAbstract:
La croissance des plantes en interaction avec l'environnement peut être décrite par des modèles mathématiques. Ceux-ci présentent des perspectives prometteuses pour un nombre considérable d'applications telles que la prévision des rendements ou l'expérimentation virtuelle dans le contexte de la sélection variétale. Dans cette thèse, nous nous intéressons aux différentes solutions capables d'améliorer les capacités prédictives des modèles de croissance de plantes, en particulier grâce à des méthodes statistiques avancées. Notre contribution se résume en quatre parties.Tout d'abord, nous proposons un nouveau modèle de culture (Log-Normal Allocation and Senescence ; LNAS). Entièrement construit dans un cadre probabiliste, il décrit seulement les processus écophysiologiques essentiels au bilan de la biomasse végétale afin de contourner les problèmes d'identification et d'accentuer l'évaluation des incertitudes. Ensuite, nous étudions en détail le paramétrage du modèle. Dans le cadre Bayésien, nous mettons en œuvre des méthodes Monte-Carlo Séquentielles (SMC) et des méthodes de Monte-Carlo par Chaînes de Markov (MCMC) afin de répondre aux difficultés soulevées lors du paramétrage des modèles de croissance de plantes, caractérisés par des équations dynamiques non-linéaires, des données rares et un nombre important de paramètres. Dans les cas où la distribution a priori est peu informative, voire non-informative, nous proposons une version itérative des méthodes SMC et MCMC, approche équivalente à une variante stochastique d'un algorithme de type Espérance-Maximisation, dans le but de valoriser les données d'observation tout en préservant la robustesse des méthodes Bayésiennes. En troisième lieu, nous soumettons une méthode d'assimilation des données en trois étapes pour résoudre le problème de prévision du modèle. Une première étape d'analyse de sensibilité permet d'identifier les paramètres les plus influents afin d'élaborer une version plus robuste de modèle par la méthode de sélection de modèles à l'aide de critères appropriés. Ces paramètres sélectionnés sont par la suite estimés en portant une attention particulière à l'évaluation des incertitudes. La distribution a posteriori ainsi obtenue est considérée comme information a priori pour l'étape de prévision, dans laquelle une méthode du type SMC telle que le filtrage par noyau de convolution (CPF) est employée afin d'effectuer l'assimilation de données. Dans cette étape, les estimations des états cachés et des paramètres sont mis à jour dans l'objectif d'améliorer la précision de la prévision et de réduire l'incertitude associée. Finalement, d'un point de vue applicatif, la méthodologie proposée est mise en œuvre et évaluée avec deux modèles de croissance de plantes, le modèle LNAS pour la betterave sucrière et le modèle STICS pour le blé d'hiver. Quelques pistes d'utilisation de la méthode pour l'amélioration du design expérimental sont également étudiées, dans le but d'améliorer la qualité de la prévision. Les applications aux données expérimentales réelles montrent des performances prédictives encourageantes, ce qui ouvre la voie à des outils d'aide à la décision en agriculture<br>Plant growth models aim to describe plant development and functional processes in interaction with the environment. They offer promising perspectives for many applications, such as yield prediction for decision support or virtual experimentation inthe context of breeding. This PhD focuses on the solutions to enhance plant growth model predictive capacity with an emphasis on advanced statistical methods. Our contributions can be summarized in four parts. Firstly, from a model design perspective, the Log-Normal Allocation and Senescence (LNAS) crop model is proposed. It describes only the essential ecophysiological processes for biomass budget in a probabilistic framework, so as to avoid identification problems and to accentuate uncertainty assessment in model prediction. Secondly, a thorough research is conducted regarding model parameterization. In a Bayesian framework, both Sequential Monte Carlo (SMC) methods and Markov chain Monte Carlo (MCMC) based methods are investigated to address the parameterization issues in the context of plant growth models, which are frequently characterized by nonlinear dynamics, scarce data and a large number of parameters. Particularly, whenthe prior distribution is non-informative, with the objective to put more emphasis on the observation data while preserving the robustness of Bayesian methods, an iterative version of the SMC and MCMC methods is introduced. It can be regarded as a stochastic variant of an EM type algorithm. Thirdly, a three-step data assimilation approach is proposed to address model prediction issues. The most influential parameters are first identified by global sensitivity analysis and chosen by model selection. Subsequently, the model calibration is performed with special attention paid to the uncertainty assessment. The posterior distribution obtained from this estimation step is consequently considered as prior information for the prediction step, in which a SMC-based on-line estimation method such as Convolution Particle Filtering (CPF) is employed to perform data assimilation. Both state and parameter estimates are updated with the purpose of improving theprediction accuracy and reducing the associated uncertainty. Finally, from an application point of view, the proposed methodology is implemented and evaluated with two crop models, the LNAS model for sugar beet and the STICS model for winter wheat. Some indications are also given on the experimental design to optimize the quality of predictions. The applications to real case scenarios show encouraging predictive performances and open the way to potential tools for yield prediction in agriculture