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.

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.

The spectrum has become one of the most important and scarce resources for future wireless communications. However, the current static spectrum policy cannot meet the increasing demands for spectrum access. To improve spectrum efficiency, dynamic spectrum access (DSA) attempts to allocate the spectrum to users in an intelligent manner. Cognitive radio (CR) is an enabling technology for DSA, and can maximize spectrum utilization by introducing unlicensed or secondary users (SUs) to the primary system. The key component of DSA is dynamic spectrum sharing (DSS), which is responsible for providing efficient and fair spectrum allocation or scheduling solutions among licensed or primary users (PUs) and SUs. This thesis focuses on the design of efficient DSS schemes for the future wireless communication networks. Firstly, based on the coordinated DSS model, this thesis proposes a heterogeneous-prioritized spectrum sharing policy for coordinated dynamic spectrum access networks. Secondly, based on the uncoordinated DSS model, a novel partial spectrum sharing strategy and the cross-layer optimization method have been proposed to achieve efficient spectrum sharing between two licensed networks. Then, a hybrid strategy which combines the overlay and underlay schemes is proposed under uncoordinated DSS model. The proposed analytical methods can provide efficient and accurate modeling to predict the behaviors of the PUs and SUs in DSS systems. This thesis presents the performance prediction of the proposed novel DSS schemes that achieve efficient spectrum sharing for coordinated and uncoordinated future wireless networks.

The study on the longest stretch of consecutive successes in \random" trials dates back to 1916 when the German philosopher Karl Marbe wrote a paper concerning the longest stretch of consecutive births of children of the same sex as appearing in the birth register of a Bavarian town. The result was actually used by parents to \predict" the sex of their children. The longest stretch of same-sex births during that time in 200 thousand birth registrations was actually 17 t log2(200 103): During the past century, the research of longest stretch of consecutive successes (longest runs) has found applications in various areas, especially in the theory of reliability. The aim of this thesis is to study large deviations on longest runs in the setting of Markov chains. More precisely, we establish a general large deviation principle for the longest success run in a two-state (success or failure) Markov chain. Our tool is based on a recent result regarding a general large deviation for the longest success run in Bernoulli trails. It turns out that the main ingredient in the proof is to implement several global and local estimates of the cumulative distribution function of the longest success run.

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

This report describes a method to control the density distribution of a large number of autonomous agents. The approach is based on the fact that there are a large number of agents in the system, and hence the time evolution of the probabilistic density distribution of agents can be described as a Markov chain. The main contribution of this paper is the synthesis of a Markov matrix which will guide the multi-agent system density to a desired steady-state density distribution, in a probabilistic sense, while satisfying some motion and safety constraints. Also, an adaptive density control method based on real time density feedback is introduced to synthesize a time-varying Markov ma- trix, which leads to better convergence to the desired density distribution. Finally, a decentralized density computation method is described. This method guarantees that all agents will have a best, and common, density estimate in a finite, with an explicit bound, number of communication updates. / text

馬可夫鏈鎖(Markov Chain)理論係於一九０七年首由A.A. MarKov提出，初僅論及有限型，多年來經數學專家之研究，業經建立無限型之理論體系。近年來作業研究學者不斷研究，已將有限馬可夫鏈鎖在管理科學方面作廣泛的應用。然在國內，對此理論尚少討論，實際應用更未論及。 本文主要在對此理論作一簡介，並探討其在應收賬款估計備抵壞賬問題之實際應用方法，期能藉此引起注意，對此一新興應用領域，鑽研學理，並使企業界能夠實際引用。 本文共分六章，第一章緒論，敘述撰寫本文動機及搜集資料經過。第二章簡介理論，敘述有限馬可夫鏈鎖之理論體系，由於實例應用採用吸收馬可夫鏈鎖，故對其討論較多。第三章列舉馬可夫鏈鎖理論幾個重要的應用領域，諸如市場預測、決策問題等，第四章說明在動態規劃上的應用，即所謂馬可夫決策過程。第五章實例，首先簡介某公司業務及應收賬款概況，然後應用第二章之理論進行寶例計算並解釋其結果。第六章結論及建議，說明實例應用之理論模型的主要功能，並對某公司之業務及賬務處理提供建議。 本文所需資料均自某公司原始資料歸併求算，在實際資料與理論模型如何配合方面，遭遇甚多困難，幸蒙吾師陸民仁教授、鄭子昊教授悉心指導及其公司之協助；在實例計算時，復蒙政大講師周汝及先生惠予協助，本文方能如期完成，在此謹致由衷謝忱。至於本文內容，因個人學識及時間之限制，誤漏之處，在所難免，敬祈閱卷教授指正。

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.

In this work we investigate the use of perfect sampling methods within the context of Bayesian linear regression. We focus on inference problems related to the marginal posterior model probabilities. Model averaged inference for the response and Bayesian variable selection are considered. Perfect sampling is an alternate form of Markov chain Monte Carlo that generates exact sample points from the posterior of interest. This approach removes the need for burn-in assessment faced by traditional MCMC methods. For model averaged inference, we find the monotone Gibbs coupling from the past (CFTP) algorithm is the preferred choice. This requires the predictor matrix be orthogonal, preventing variable selection, but allowing model averaging for prediction of the response. Exploring choices of priors for the parameters in the Bayesian linear model, we investigate sufficiency for monotonicity assuming Gaussian errors. We discover that a number of other sufficient conditions exist, besides an orthogonal predictor matrix, for the construction of a monotone Gibbs Markov chain. Requiring an orthogonal predictor matrix, we investigate new methods of orthogonalizing the original predictor matrix. We find that a new method using the modified Gram-Schmidt orthogonalization procedure performs comparably with existing transformation methods, such as generalized principal components. Accounting for the effect of using an orthogonal predictor matrix, we discover that inference using model averaging for in-sample prediction of the response is comparable between the original and orthogonal predictor matrix. The Gibbs sampler is then investigated for sampling when using the original predictor matrix and the orthogonal predictor matrix. We find that a hybrid method, using a standard Gibbs sampler on the orthogonal space in conjunction with the monotone CFTP Gibbs sampler, provides the fastest computation and convergence to the posterior distribution. We conclude the hybrid approach should be used when the monotone Gibbs CFTP sampler becomes impractical, due to large backwards coupling times. We demonstrate large backwards coupling times occur when the sample size is close to the number of predictors, or when hyper-parameter choices increase model competition. The monotone Gibbs CFTP sampler should be taken advantage of when the backwards coupling time is small. For the problem of variable selection we turn to the exact version of the independent Metropolis-Hastings (IMH) algorithm. We reiterate the notion that the exact IMH sampler is redundant, being a needlessly complicated rejection sampler. We then determine a rejection sampler is feasible for variable selection when the sample size is close to the number of predictors and using Zellner’s prior with a small value for the hyper-parameter c. Finally, we use the example of simulating from the posterior of c conditional on a model to demonstrate how the use of an exact IMH view-point clarifies how the rejection sampler can be adapted to improve efficiency.

802.16與802.11e均有提供服務品質(QoS)，但是其MAC並不相同，為了達到QoS的保證，我們使用馬可夫鍊(Markov Chain)模型分析在不同連線數量時802.11e EDCA的延遲時間(delay time)。然後，我們可以再利用允入控制(CAC)機制限制連線的數量以保證延遲時間的需求，並使用令牌桶(Token Bucket)機制，在滿足延遲及頻寬的需求下控制輸出流量，在我們的令牌桶機制中可以依照頻寬需求的變化自動調整令牌(Token)產生速率，最後使用封包丟棄機制提升吞吐量(throughput)。 　　在提出我們的方法後，我們使用Qualnet模擬器驗證延遲時間、封包丟棄率及吞吐量，結果表示我們所提出的方法在三方面都有明顯的改進。 / IEEE 802.16 and 802.11e both provide Quality of Service (QoS), but the MAC of betweens is different. Ensuring the QoS guarantee, we use a Markov Chain model to analyze the 802.11e EDCA delay time under variance number of connections. Therefore, we can employ a CAC mechanism constraining the number of connections to guarantee the delay requirement. Further, considering the delay requirement and the bandwidth, we use a Token Bucket mechanism to throttle the traffic output that ensures the delay and bandwidth to be satisfied. And our Token Bucket mechanism can tune the token rate automatically by bandwidth requirement. Finally, we use the Packet Drop mechanism to improve throughput. After my methodology, we validate the delay, packet drop rate and throughput by simulator Qualnet. We have significant improvement in delay, drop rate, and throughput.