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In dieser Arbeit untersuchen wir das Critical Slowing down der Gitter-QCD Simulationen. Wir führen eine Vorstudie in der quenched Approximation durch, in der wir feststellen, dass unsere Schätzung der exponentiellen Autokorrelation wie $\tauexp(a) \sim a^{-5} $ skaliert, wobei $a$ der Gitterabstand ist. In unquenched Simulationen mit O(a)-verbesserten Wilson-Fermionen finden wir ein ähnliches Skalierungsgesetz. Die Diskussion wird von einem gro\ss{}en Satz an Ensembles sowohl in reiner Eichtheorie als auch in der Theorie mit zwei entarteten Seequarks unterstützt. Wir haben darüber hinaus die Wirkung von langsamen algorithmischen Modi in der Fehleranalyse des Erwartungswertes von typischen Gitter-QCD-Observablen (hadronische Matrixelemente und Massen) untersucht. Im Kontext der Simulationen, die durch langsame Modi betroffen sind, schlagen wir vor und testen eine Methode, um zuverlässige Schätzungen der statistischen Fehler zu bekommen. Diese Methode soll in dem typischen Simulationsbereich der Gitter-QCD helfen, nämlich dann, wenn die gesamte erfasste Statistik O(10)\tauexp ist. Dies ist der typische Fall bei Simulationen in der Nähe des Kontinuumslimes, wo der Rechenaufwand für die Erzeugung von zwei unabhängigen Datenpunkten sehr gro\ss{} sein kann. Schlie\ss{}lich diskutieren wir die Skalenbestimmung in N_f=2-Simulationen mit der Kaon Zerfallskonstante f_K als experimentellem Input. Die Methode wird zusammen mit einer gründlichen Diskussion der angewandten Fehleranalyse erklärt. Eine Beschreibung der öffentlich zugänglichen Software, die für die Fehleranalyse genutzt wurde, ist eingeschlossen. / In this work we investigate the critical slowing down of lattice QCD simulations. We perform a preliminary study in the quenched approximation where we find that our estimate of the exponential auto-correlation time scales as $\tauexp(a)\sim a^{-5}$, where $a$ is the lattice spacing. In unquenched simulations with O(a) improved Wilson fermions we do not obtain a scaling law but find results compatible with the behavior that we find in the pure gauge theory. The discussion is supported by a large set of ensembles both in pure gauge and in the theory with two degenerate sea quarks. We have moreover investigated the effect of slow algorithmic modes in the error analysis of the expectation value of typical lattice QCD observables (hadronic matrix elements and masses). In the context of simulations affected by slow modes we propose and test a method to obtain reliable estimates of statistical errors. The method is supposed to help in the typical algorithmic setup of lattice QCD, namely when the total statistics collected is of O(10)\tauexp. This is the typical case when simulating close to the continuum limit where the computational costs for producing two independent data points can be extremely large. We finally discuss the scale setting in Nf=2 simulations using the Kaon decay constant f_K as physical input. The method is explained together with a thorough discussion of the error analysis employed. A description of the publicly available code used for the error analysis is included.

[pt] A Eficiência Energética (EE) pode ser considerada sinônimo de preservação ambiental, pois a energia economizada evita a construção de novas plantas de geração e de linhas de transmissão. O Leilão de Eficiência Energética (LEE) poderia representar uma alternativa muito interessante para a dinamização e promoção de práticas de EE no Brasil. Porém, é importante mencionar que isso pressupõe uma confiança na quantidade de energia reduzida, o que só pode se tornar realidade com a implantação e desenvolvimento de um sistema de Medição e Verificação (M&V) dos consumos de energia. Neste contexto, tem-se como objetivo principal simular os preços de energia do Leilão de Eficiência Energética no ambiente regulado para conhecer se a viabilidade no Brasil poderia se concretizar. A metodologia utilizada para realizar as simulações foi a de Monte Carlo, ademais, antes se utilizou o método do Kernel com a finalidade de conseguir ajustar os dados a uma curva através de polinômios. Uma vez conseguida a curva melhor ajustada se realizou a análise de cada cenário (nas diferentes rodadas) com cada amostra (500, 1000, 5000 e 10000) para encontrar a probabilidade dos preços ficarem entre o intervalo de 110 reais e 140 reais (preços ótimos propostos no LEE). Finalmente, os resultados apresentam que a probabilidade de o preço ficar no intervalo de 110 reais e 140 reais na amostra de 500 dados é de 28,20 por cento, na amostra de 1000 é de 33,00 por cento, na amostra de 5000 é de 29,96 por cento e de 10000 é de 32,36 por cento. / [en] The Energy Efficiency (EE) is considered a synonymous of environmental preservation, because the energy saved prevents the construction of new generating plants and transmission lines. The Demand-Side Bidding (DSB) could represent a very interesting alternative for the revitalization and promotion of EE practices in Brazil. However, it is important to note that this presupposes a confidence on the amount of reduced energy, which can only take reality with the implementation and development of a measurement system and verification (M&V) the energy consumption. In this context, the main objective is to simulate of the prices of the demand-side bidding in the regulated environment to meet the viability in Brazil that could become a reality. The methodology used to perform the simulations was the Monte Carlo addition, prior to the Kernel method was used in order to be able to adjust the data to a curve, using polynomials. Once achieved the best-fitted curve was carried out through an analysis of each scenario (in different rounds) with each sample (500, 1000, 5000 and 10000) to find the probability of the price falling between the 110 real range and 140 real (great prices proposed by the DSB). Finally, the results showed that the probability of staying in the price range from 110 real nd 140 real data 500 in the sample is 28.20 percent, the sample 1000 is 33.00 percent, the sample 5000 is 29.96 percent and 10000 is 32.36 percent.

Phylogeny, the study of evolutionary relationships among species and other taxa, plays a crucial role in understanding the history of life. Bayesian analysis using Markov chain Monte Carlo (MCMC) is a widely used approach for inferring phylogenetic trees, but it suffers from slow convergence in higher dimensions and is slow to converge. This thesis focuses on exploring variational inference (VI), a methodology that is believed to lead to improved speed and accuracy of phylogenetic models. However, VI models are known to concentrate the density of the learned approximation in high-likelihood areas. This thesis evaluates the current state of Variational Inference Bayesian Phylogenetics (VBPI) and proposes a solution using a mixture of components to improve the VBPI method's performance on complex datasets and multimodal latent spaces. Additionally, we cover the basics of phylogenetics to provide a comprehensive understanding of the field. / Fylogeni, vilket är studien av evolutionära relationer mellan arter och andra taxonomiska grupper, spelar en viktig roll för att förstå livets historia. En ofta använd metod för att dra slutsatser om fylogenetiska träd är bayesiansk analys med Markov Chain Monte Carlo (MCMC), men den lider av långsam konvergens i högre dimensioner och kräver oändligt med tid. Denna uppsats fokuserar på att undersöka hur variationsinferens (VI) kan nyttjas inom fylogenetisk inferens med hög noggranhet. Vi fokuserar specifik på en modell kallad VBPI. Men VI-modeller är allmänt kända att att koncentrera sig på höga sannolikhetsområden i posteriorfördelningar. Vi utvärderar prestandan för Variatinal Inference Baysian Phylogenetics (VBPI) och föreslår en förbättring som använder mixturer av förslagsfördelningar för att förbättra VBPI-modellens förmåga att hantera mer komplexa datamängder och multimodala posteriorfördelningar. Utöver dettta går vi igenom grunderna i fylogenetik för att ge en omfattande förståelse av området.

An important task in brain modeling is that of estimating model parameters and quantifying their uncertainty. In this thesis we tackle this problem from a Bayesian perspective: we use experimental data to update the prior information about model parameters, in order to obtain their posterior distribution. Uncertainty quantification via a direct computation of the posterior has a prohibitive computational cost in high dimensions. An alternative to a direct computation is offered by Markov chain Monte Carlo (MCMC) methods. The aim of this project is to analyse some of the methods within this class and improve their convergence. In this thesis we describe the following MCMC methods: Metropolis-Hastings (MH) algorithm, Metropolis adjusted Langevin algorithm (MALA), simplified manifold MALA (smMALA) and Approximate Bayesian Computation MCMC (ABCMCMC). SmMALA is further analysed in Paper A, where we propose an algorithm to approximate a key component of this algorithm (the Fisher Information) when applied to ODE models, with the purpose of reducing the computational cost of the method. A theoretical analysis of MCMC methods is carried out in Paper B and relies on tools from the theory of large deviations. In particular, we analyse the convergence of the MH algorithm by stating and proving a large deviation principle (LDP) for the empirical measures produced by the algorithm. Some of the methods analysed in this thesis are implemented in an R package, available on GitHub as “icpm-kth/uqsa” and presented in Paper C, and are applied to subcellular pathway models within neurons in the context of uncertainty quantification of the model parameters. / En viktig uppgift inom hjärnmodellering är att uppskatta parametrar i modellen och kvantifiera deras osäkerhet. I denna avhandling hanterar vi detta problem från ett Bayesianskt perspektiv: vi använder experimentell data för att uppdatera a priori kunskap av modellparametrar, för att erhålla deras posteriori-fördelning. Osäkerhetskvantifiering (UQ) via direkt beräkning av posteriorfördelningen har en hög beräkningskostnad vid höga dimensioner. Ett alternativ till direkt beräkning ges av Markov chain Monte Carlo (MCMC) metoder. Syftet med det här projektet är att analysera några MCMC metoder och förbättra deras konvergens. I denna avhandling beskriver vi följande MCMC algoritmer: “Metropolis-Hastings” (MH), “Metropolis adjusted Langevin” (MALA), “Simplified Manifold MALA” (smMALA) och “Approximate Bayesian Computation MCMC” (ABCMCMC). SmMALA analyseras i artikel A. Där presenterar vi en algoritm för att approximera en nyckelkomponent av denna algoritm (Fisher informationen) när den tillämpas på ODE modeller i syfte att minska metodens beräkningskostnad. En teoretisk analys av MCMC metoder behandlas i artikel B och bygger på verktyg från teorin av stora avvikelser. Mer specifikt, vi analyserar MH algoritmens konvergens genom att formulera och bevisa en stora avvikelser princip (LDP) för de empiriska mått som produceras av algoritmen. Några av metoderna analyserade i den här avhandlingen har implementerats i ett R paket som finns på GitHub som “icpm-kth/uqsa” och presenteras i artikel C. Metoderna tillämpas på subcellulära vägmodeller inom neuroner i sammanhanget av osäkerhetskvantifieringen av modellparametrar. / <p>QC 2023-08-21</p>

The Rasch sampler is an efficient algorithm to sample binary matrices with given marginal sums. It is a Markov chain Monte Carlo (MCMC) algorithm. The program can handle matrices of up to 1024 rows and 64 columns. A special option allows to sample square matrices with given marginals and fixed main diagonal, a problem prominent in social network analysis. In all cases the stationary distribution is uniform. The user has control on the serial dependency. (authors' abstract)

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.

Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Proteins are the fundamental unit of computation and signal processing in biological systems. A quantitative understanding of protein biophysics is of paramount importance, since even slight malfunction of proteins can lead to diverse and severe disease states. However, developing accurate and useful mechanistic models of protein function can be strikingly elusive. I demonstrate that the adoption of Bayesian statistical methods can greatly aid in modeling protein systems. I first discuss the pitfall of parameter non-identifiability and how a Bayesian approach to modeling can yield reliable and meaningful models of molecular systems. I then delve into a particular case of non-identifiability within the context of an emerging experimental technique called single molecule photobleaching. I show that the interpretation of this data is non-trivial and provide a rigorous inference model for the analysis of this pervasive experimental tool. Finally, I introduce the use of nonparametric Bayesian inference for the analysis of single molecule time series. These methods aim to circumvent problems of model selection and parameter identifiability and are demonstrated with diverse applications in single molecule biophysics. The adoption of sophisticated inference methods will lead to a more detailed understanding of biophysical systems. / text

We propose using statistical regular pavings (SRPs) as an efficient and adaptive statistical data structure for processing massive, multi-dimensional data. A regular paving (RP) is an ordered binary tree that recursively bisects a box in $\Rz^{d}$ along the first widest side. An SRP is extended from an RP by allowing mutable caches of recursively computable statistics of the data. In this study we use SRPs for two major applications: estimating histogram densities and summarising large spatio-temporal datasets. The SRP histograms produced are $L_1$-consistent density estimators driven by a randomised priority queue that adaptively grows the SRP tree, and formalised as a Markov chain over the space of SRPs. A way to select an estimate is to run a Markov chain over the space of SRP trees, also initialised by the randomised priority queue, but here the SRP tree either shrinks or grows adaptively through pruning or splitting operations. The stationary distribution of the Markov chain is then the posterior distribution over the space of all possible histograms. We then take advantage of the recursive nature of SRPs to make computationally efficient arithmetic averages, and take the average of the states sampled from the stationary distribution to obtain the posterior mean histogram estimate. We also show that SRPs are capable of summarizing large datasets by working with a dataset containing high frequency aircraft position information. Recursively computable statistics can be stored for variable-sized regions of airspace. The regions themselves can be created automatically to reflect the varying density of aircraft observations, dedicating more computational resources and providing more detailed information in areas with more air traffic. In particular, SRPs are able to very quickly aggregate or separate data with different characteristics so that data describing individual aircraft or collected using different technologies (reflecting different levels of precision) can be stored separately and yet also very quickly combined using standard arithmetic operations.