In this thesis, we consider some popular stochastic differential equation models used in finance, such as the Vasicek Interest Rate model, the Heston model and a new fractional Heston model. We discuss how to perform inference about unknown quantities associated with these models in the Bayesian framework. We describe sequential importance sampling, the particle filter and the auxiliary particle filter. We apply these inference methods to the Vasicek Interest Rate model and the standard stochastic volatility model, both to sample from the posterior distribution of the underlying processes and to update the posterior distribution of the parameters sequentially, as data arrive over time. We discuss the sensitivity of our results to prior assumptions. We then consider the use of Markov chain Monte Carlo (MCMC) methodology to sample from the posterior distribution of the underlying volatility process and of the unknown model parameters in the Heston model. The particle filter and the auxiliary particle filter are also employed to perform sequential inference. Next we extend the Heston model to the fractional Heston model, by replacing the Brownian motions that drive the underlying stochastic differential equations by fractional Brownian motions, so allowing a richer dependence structure across time. Again, we use a variety of methods to perform inference. We apply our methodology to simulated and real financial data with success. We then discuss how to make forecasts using both the Heston and the fractional Heston model. We make comparisons between the models and show that using our new fractional Heston model can lead to improve forecasts for real financial data.

Functional Magnetic Resonance Imaging (fMRI) is a noninvasive imaging method that reflects local changes in brain activity. FMRI group studies involves the analysis of the functional images acquired for each of a group of subjects under the same experimental conditions. We propose a spatial marked point-process model for the activation patterns of the subjects in a group study. Each pattern is described as the sum of individual centres of activation. The marked point-process that we propose allows the researcher to enforce repulsion between all pairs of centres of an individual subject that are within a specified minimum distance of each other. It also allows the researcher to enforce attraction between similarly-located centres from different subjects. This attraction helps to compensate for the misalignment of corresponding functional areas across subjects and is a novel method of addressing the problem of imperfect inter-subject registration of functional images. We use a Bayesian framework and choose prior distributions according to current understanding of brain activity. Simulation studies and exploratory studies of our reference dataset are used to fine-tune the prior distributions. We perform inference via Markov chain Monte Carlo. The fitted model gives a summary of the activation in terms of its location, height and size. We use this summary both to identify brain regions that were activated in response to the stimuli under study and to quantify the discrepancies between the activation maps of subjects. Applied to our reference dataset, our measure is successful in separating out those subjects with activation patterns that do not agree with the overall group pattern. In addition, our measure is sensitive to subjects with a large number of activation centres relative to the other subjects in the group. The activation summary given by our model makes it possible to pursue a range of inferential questions that cannot be addressed with ease by current model-based approaches.

Emerging many-core computer architectures provide an incentive for computational methods to exhibit specific types of parallelism. Our ability to perform inference in Bayesian statistics is often dependent upon our ability to approximate expectations of functions of random variables, for which Monte Carlo methodology provides a general purpose solution using a computer. This thesis is primarily concerned with exploring the gains that can be obtained by using many-core architectures to accelerate existing population-based Monte Carlo algorithms, as well as providing a novel general framework that can be used to devise new population-based methods. Monte Carlo algorithms are often concerned with sampling random variables taking values in X whose density is known up to a normalizing constant. Population-based methods typically make use of collections of interacting auxiliary random variables, each of which is in X, in specifying an algorithm. Such methods are good candidates for parallel implementation when the collection of samples can be generated in parallel and their interaction steps are either parallelizable or negligible in cost. The first contribution of this thesis is in demonstrating the potential speedups that can be obtained for two common population-based methods, population-based Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC). The second contribution of this thesis is in the derivation of a hierarchical family of sparsity-inducing priors in regression and classification settings. Here, auxiliary variables make possible the implementation of a fast algorithm for finding local modes of the posterior density. SMC, accelerated on a many-core architecture, is then used to perform inference for a range of prior specifications to gain an understanding of sparse association signal in the context of genome-wide association studies. The third contribution is in the use of a new perspective on reversible MCMC kernels that allows for the construction of novel population-based methods. These methods differ from most existing methods in that one can make the resulting kernels define a Markov chain on X. A further development is that one can define kernels in which the number of auxiliary variables is given a distribution conditional on the values of the auxiliary variables obtained so far. This is perhaps the most important methodological contribution of the thesis, and the adaptation of the number of particles used within a particle MCMC algorithm provides a general purpose algorithm for sampling from a variety of complex distributions.

This work addresses issues in the study of gene-environment interaction (GxE) through research of conduct disorder (CD) among adolescents and extends the recent report of significant GxE and subsequent replication studies. A sub-sample of 1,299 individual participants/649 twin pairs and their parents from the Virginia Twin Study of Adolescent and Behavioral Development was used for whom Monoamine Oxidase A (MAOA) genotype, diagnosis of CD, maternal antisocial personality symptoms, and household neglect were obtained. This dissertation (1) tested for GxE by gender using MAOA and childhood adversity using multiple approaches to CD measurement and model assessment, (2) determined whether other mechanisms would explain differences in GxE by gender and (3) identified and assessed other genes and environments related to the interaction MAOA and childhood adversity. Using a multiple regression approach, a main effect of the low/low MAOA genotype remained after controlling other risk factors in females. However, the effects of GxE were modest and were removed by transforming the environmental measures. In contrast, there was no significant effect of the low activity MAOA allele in males although significant GxE was detected and remained after transformation. The sign of the interaction for males was opposite from females, indicating genetic sensitivity to childhood adversity may differ by gender. Upon further investigation, gender differences in GxE were due to genotype-sex interaction and may involve MAOA. A Markov Chain Monte Carlo approach including a genetic Item Response Theory modeled CD as a trait with continuous liability, since false detection of GxE may result from measurement. In males and females, the inclusion of GxE while controlling for the other covariates was appropriate, but was little improvement in model fit and effect sizes of GxE were small. Other candidate genes functioning in the serotonin and dopamine neurotransmitter systems were tested for interaction with MAOA to affect risk for CD. Main genetic effects of dopamine transporter genotype and MAOA in the presence of comorbidity were detected. No epistatic effects were detected. The use of random forests systematically assessed the environment and produced several interesting environments that will require more thoughtful consideration before incorporation into a model testing GxE.

Seismic records contain information that allows geoscientists to make inferences about the structure and properties of the Earth’s interior. Traditionally, seismic imaging and tomography methods require wavefields to be generated and recorded by identifiable sources and receivers, and use these directly-recorded signals to create models of the Earth’s subsurface. However, in recent years the method of seismic interferometry has revolutionised earthquake seismology by allowing unrecorded signals between pairs of receivers, pairs of sources, and source-receiver pairs to be constructed as Green’s functions using either cross-correlation, convolution or deconvolution of wavefields. In all of these formulations, seismic energy is recorded and emitted by surrounding boundaries of receivers and sources, which need not be active and impulsive but may even constitute continuous, naturally-occurring seismic ambient noise. In the first part of this thesis, I provide a comprehensive overview of seismic interferometry, its background theory, and examples of its application. I then test the theory and evaluate the effects of approximations that are commonly made when the interferometric formulae are applied to real datasets. Since errors resulting from some approximations can be subtle, these tests must be performed using almost error-free synthetic data produced with an exact waveform modelling method. To make such tests challenging the method and associated code must be applicable to multiply-scattering media. I developed such a modelling code specifically for interferometric tests and applications. Since virtually no errors are introduced into the results from modelling, any difference between the true and interferometric waveforms can safely be attributed to specific origins in interferometric theory. I show that this is not possible when using other, previously available methods: for example, the errors introduced into waveforms synthesised by finite-difference methods due to the modelling method itself, are larger than the errors incurred due to some (still significant) interferometric approximations; hence that modelling method can not be used to test these commonly-applied approximations. I then discuss the ability of interferometry to redatum seismic energy in both space and time, allowing virtual seismograms to be constructed at new locations where receivers may not have been present at the time of occurrence of the associated seismic source. I present the first successful application of this method to real datasets at multiple length scales. Although the results are restricted to limited bandwidths, this study demonstrates that the technique is a powerful tool in seismologists’ arsenal, paving the way for a new type of ‘retrospective’ seismology where sensors may be installed at any desired location at any time, and recordings of seismic events occurring at any other time can be constructed retrospectively – even long after their energy has dissipated. Within crustal seismology, a very common application of seismic interferometry is ambient-noise tomography (ANT). ANT is an Earth imaging method which makes use of inter-station Green’s functions constructed from cross-correlation of seismic ambient noise records. It is particularly useful in seismically quiescent areas where traditional tomography methods that rely on local earthquake sources would fail to produce interpretable results due to the lack of available data. Once constructed, interferometric Green’s functions can be analysed using standard waveform analysis techniques, and inverted for subsurface structure using more or less traditional imaging methods. In the second part of this thesis, I discuss the development and implementation of a fully non-linear inversion method which I use to perform Love-wave ANT across the British Isles. Full non-linearity is achieved by allowing both raypaths and model parametrisation to vary freely during inversion in Bayesian, Markov chain Monte Carlo tomography, the first time that this has been attempted. Since the inversion produces not only one, but a large ensemble of models, all of which fit the data to within the noise level, statistical moments of different order such as the mean or average model, or the standard deviation of seismic velocity structures across the ensemble, may be calculated: while the ensemble average map provides a smooth representation of the velocity field, a measure of model uncertainty can be obtained from the standard deviation map. In a number of real-data and synthetic examples, I show that the combination of variable raypaths and model parametrisation is key to the emergence of previously-unobserved, loop-like uncertainty topologies in the standard deviation maps. These uncertainty loops surround low- or high-velocity anomalies. They indicate that, while the velocity of each anomaly may be fairly well reconstructed, its exact location and size tend to remain uncertain; loops parametrise this location uncertainty, and hence constitute a fully non-linearised, Bayesian measure of spatial resolution. The uncertainty in anomaly location is shown to be due mainly to the location of the raypaths that were used to constrain the anomaly also only being known approximately. The emergence of loops is therefore related to the variation in raypaths with velocity structure, and hence to 2nd and higher order wave-physics. Thus, loops can only be observed using non-linear inversion methods such as the one described herein, explaining why these topologies have never been observed previously. I then present the results of fully non-linearised Love-wave group-velocity tomography of the British Isles in different frequency bands. At all of the analysed periods, the group-velocity maps show a good correlation with known geology of the region, and also robustly detect novel features. The shear-velocity structure with depth across the Irish Sea sedimentary basin is then investigated by inverting the Love-wave group-velocity maps, again fully non-linearly using Markov chain Monte Carlo inversion, showing an approximate depth to basement of 5 km. Finally, I discuss the advantages and current limitations of the fully non-linear tomography method implemented in this project, and provide guidelines and suggestions for its improvement.

<div>Mixture modeling is a classical unsupervised learning method with applications to clustering and density estimation. This dissertation studies two challenges in modeling data with mixture models. The first part addresses problems that arise when modeling observations lying on constrained spaces, such as the boundaries of a city or a landmass. It is often desirable to model such data through the use of mixture models, especially nonparametric mixture models. Specifying the component distributions and evaluating normalization constants raise modeling and computational challenges. In particular, the likelihood forms an intractable quantity, and Bayesian inference over the parameters of these models results in posterior distributions that are doubly-intractable. We address this problem via a model based on rejection sampling and an algorithm based on data augmentation. Our approach is to specify such models as restrictions of standard, unconstrained distributions to the constraint set, with measurements from the model simulated by a rejection sampling algorithm. Posterior inference proceeds by Markov chain Monte Carlo, first imputing the rejected samples given mixture parameters and then resampling parameters given all samples. We study two modeling approaches: mixtures of truncated Gaussians and truncated mixtures of Gaussians, along with Markov chain Monte Carlo sampling algorithms for both. We also discuss variations of the models, as well as approximations to improve mixing, reduce computational cost, and lower variance.</div><div><br></div><div>The second part of this dissertation explores the application of mixture models to estimate contamination rates in matched tumor and normal samples. Bulk sequencing of tumor samples are prone to contaminations from normal cells, which lead to difficulties and inaccuracies in determining the mutational landscape of the cancer genome. In such instances, a matched normal sample from the same patient can be used to act as a control for germline mutations. Probabilistic models are popularly used in this context due to their flexibility. We propose a hierarchical Bayesian model to denoise the contamination in such data and detect somatic mutations in tumor cell populations. We explore the use of a Dirichlet prior on the contamination level and extend this to a framework of Dirichlet processes. We discuss MCMC schemes to sample from the joint posterior distribution and evaluate its performance on both synthetic experiments and publicly available data.</div>

Wherever wildlife share space with boaters, collisions are a potential source of mortality. Establishing protection and speed zones are the primary actions taken to mitigate collision risk. However, creation of protection zones may be a point of contention with stakeholders as new zones can have significant socioeconomic impacts. The Florida Manatee is a prime example of a species whose abundance and viability are constrained by this balance between the needs of humans and wildlife on a shared landscape. The goal of this work is to help further understand the risk to manatees by quantifying the probability of lethal collisions. I hypothesized that higher boat speeds increase the probability of lethal injury to manatee during a collision and aimed to characterize the relationship between vessel speed and the probability of lethal injury to manatee. I used a logistic regression model implemented with a Bayesian approach and fitted to citizen reported collision data as a feasible method for examining the influence of vessel speed in contributing to lethal injury to a manatee when struck. I also present a method for dealing with uncertainty in data used to report collisions. To conduct this analysis, I used citizen reported collision data. These data are typically collected opportunistically, suffer from low sample sizes, and have uncertainty in reported vessel speeds. Uncertainty associated with speed values in reported collision events was assessed by performing a multiple imputation to replace qualitative vessel speed – in other words, “missing data” – with quantitative values. This procedure involves fitting log-normal distributions to radar data that contained precise vessel speeds along with a physical description like ‘planing’, ‘plowing’, or ‘idle’. For each imputation of the data, a quantitative value was selected randomly from that distribution and used in place of its initial corresponding speed category. I evaluated issues related to quasi-separation and model fit using simulated data sets to explore the importance of sample size and evaluated the effect of key sources of error. The prediction that greater strike speed increases the probability of lethal injury was supported by the data that I analyze

This thesis consists of four papers and the main theme present is dependence, through time as in serial correlation, and across individuals, as in random effects. The individual papers may be grouped in many different ways. As is, the first two are concerned with autoregressive dynamics in a single time series and then a panel context, while the subject of the final two papers is parametric covariance modeling. Though set in a panel context the results in the latter are generally applicable. The approach taken is Bayesian. This choice is prompted by the coherent framework that the Bayesian principle offers for quantifying uncertainty and subsequently making inference in the presence of it. Recent advances in numerical methods have also made the Bayesian choice simpler. In the first paper an existing model to conduct inference directly on the roots of the autoregressive polynomial is extended to include seasonal components and to allow for a polynomial trend of arbitrary degree. The resulting highly flexible model robustifies against misspecification by implicitly averaging over different lag lengths, number of unit roots and specifications for the deterministic trend. An application to the Swedish real GDP illustrates the rich set of information about the dynamics of a time series that can be extracted using this modeling framework. The second paper offers an extension to a panel of time series. Limiting the scope, but at the same time simplifying matters considerably, the mean model is dropped restricting the applicability to non-trending panels. The main motivation of the extension is the construction of a flexible panel unit root test. The proposed approach circumvents the classical confusing problem of stating a relevant null hypothesis. It offers the possibility of more distinct inference with respect to unit root composition in the collection of time series. It also addresses the two important issues of model uncertainty and cross-section correlation. The model is illustrated using a panel of real exchange rates to investigate the purchasing power parity hypothesis. Many interesting panel models imply a structure on the covariance matrix in terms of a small number of parameters. In the third paper, exploiting this structure it is demonstrated how common panel data models lend themselves to direct sampling of the variance parameters. Not necessarily always practical, the implementation can be described by a simple and generally applicable template. For the method to be practical, simple to program and quick to execute, it is essential that the inverse of the covariance matrix can be written as a reasonably simple function of the parameters of interest. Also preferable but in no way necessary is the availability of a computationally convenient expression for the determinant of the covariance matrix as well as a bounded support for the parameters. Using the template, the computations involved in direct sampling and effect selection are illustrated in the context of a one- and two-way random effects model respectively. Having established direct sampling as a viable alternative in the previous paper, the generic template is applied to panel models with serial correlation in the fourth paper. In the case of pure serial correlation, with no random effects present, applying the template and using a Jeffreys type prior leads to very simple computations. In the very general setting of a mixed effects model with autocorrelated errors direct sampling of all variance parameters does not appear to be possible or at least not obviously practical. One important special case is identified in the model with the random individual effects model with autocorrelation. / <p>Diss. Stockholm : Handelshögskolan i Stockholm, 2004 viii s., s. 1-9: sammanfattning, s. 10-116: 4 uppsatser</p>

This thesis presents Bayesian solutions to inference problems for three types of social network data structures: a single observation of a social network, repeated observations on the same social network, and repeated observations on a social network developing through time. A social network is conceived as being a structure consisting of actors and their social interaction with each other. A common conceptualisation of social networks is to let the actors be represented by nodes in a graph with edges between pairs of nodes that are relationally tied to each other according to some definition. Statistical analysis of social networks is to a large extent concerned with modelling of these relational ties, which lends itself to empirical evaluation. The first paper deals with a family of statistical models for social networks called exponential random graphs that takes various structural features of the network into account. In general, the likelihood functions of exponential random graphs are only known up to a constant of proportionality. A procedure for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods is presented. The algorithm consists of two basic steps, one in which an ordinary Metropolis-Hastings up-dating step is used, and another in which an importance sampling scheme is used to calculate the acceptance probability of the Metropolis-Hastings step. In paper number two a method for modelling reports given by actors (or other informants) on their social interaction with others is investigated in a Bayesian framework. The model contains two basic ingredients: the unknown network structure and functions that link this unknown network structure to the reports given by the actors. These functions take the form of probit link functions. An intrinsic problem is that the model is not identified, meaning that there are combinations of values on the unknown structure and the parameters in the probit link functions that are observationally equivalent. Instead of using restrictions for achieving identification, it is proposed that the different observationally equivalent combinations of parameters and unknown structure be investigated a posteriori. Estimation of parameters is carried out using Gibbs sampling with a switching devise that enables transitions between posterior modal regions. The main goal of the procedures is to provide tools for comparisons of different model specifications. Papers 3 and 4, propose Bayesian methods for longitudinal social networks. The premise of the models investigated is that overall change in social networks occurs as a consequence of sequences of incremental changes. Models for the evolution of social networks using continuos-time Markov chains are meant to capture these dynamics. Paper 3 presents an MCMC algorithm for exploring the posteriors of parameters for such Markov chains. More specifically, the unobserved evolution of the network in-between observations is explicitly modelled thereby avoiding the need to deal with explicit formulas for the transition probabilities. This enables likelihood based parameter inference in a wider class of network evolution models than has been available before. Paper 4 builds on the proposed inference procedure of Paper 3 and demonstrates how to perform model selection for a class of network evolution models.

<p>This thesis focuses on the development of ABC methods for statistical modeling in complex dynamic systems. Motivated by real applications in biology, I propose computational strategies for Bayesian inference in contexts where standard Monte Carlo methods cannot be directly applied due to the high complexity of the dynamic model and/or data limitations.</p><p> Chapter 2 focuses on stochastic bionetwork models applied to data generated from the marginal distribution of a few network nodes at snapshots in time. I present a Bayesian computational strategy, coupled with an approach to summarizing and numerically characterizing biological phenotypes that are represented in terms of the resulting sample distributions of cellular markers. ABC and mixture modeling are used to define the approach to linking mechanistic mathematical models of network dynamics to snapshot data, using a toggle switch example integrating simulated and real data as context. </p><p> Chapter 3 focuses on the application of the methodology presented in Chapter 2 to the Myc/Rb/E2F network. This network involves a relatively high number of parameters and stochastic equations in the model specification and, thus, is substantially more complex than the toggle switch example. The analysis of the Myc/Rb/E2F network is performed with simulated and real data. I demonstrate that the proposed method can indicate which parameters can be learned about using the marginal data. </p><p> In Chapter 4, I present an ABC SMC method that uses data-based adaptive weights. This easily implemented and computationally trivial extension of ABC SMC can substantially improve acceptance rates. This is demonstrated through a series of examples with simulated and real data, including the toggle switch example. Theoretical justification is also provided to explain why this method is expected to improve the effectiveness of ABC SMC.</p><p> In Chapter 5, I present an integrated Bayesian computational strategy for fitting complex dynamic models to sparse time-series data. This is applied to experimental data from an immunization response study with Indian Rhesus macaques. The computational strategy consists of two stages: first, MCMC is implemented based on simplified sampling steps, and then, the resulting approximate output is used to generate a proposal distribution for the parameters that results in an efficient ABC procedure. The incorporation of ABC as a correction tool improves the model fit, as is demonstrated through predictive posterior analysis on the data sets of the study.</p><p> Chapter 6 presents additional discussion and comments on potential future research directions.</p> / Dissertation