A Bayesian Nonparametric Regression Model With Normalized Weights - A Study of Hippocampal Atrophy in Alzheimer’s Disease

Hippocampal volume is one of the best established biomarkers for Alzheimer’s disease. However, for appropriate use in clinical trials research, the evolution of hippocampal volume needs to be well understood. Recent theoretical models propose a sigmoidal pattern for its evolution. To support this theory, the use of Bayesian nonparametric regression mixture models seems particularly suitable due to the flexibility that models of this type can achieve and the unsatisfactory predictive properties of semiparametric methods. In this article, our aim is to develop an interpretable Bayesian nonparametric regression model which allows inference with combinations of both continuous and discrete covariates, as required for a full analysis of the dataset. Simple arguments regarding the interpretation of Bayesian nonparametric regression mixtures lead naturally to regression weights based on normalized sums. Difficulty in working with the intractable normalizing constant is overcome thanks to recent advances in MCMC methods and the development of a novel auxiliary variable scheme. We apply the new model and MCMC method to study the dynamics of hippocampal volume, and our results provide statistical evidence in support of the theoretical hypothesis

Bayesian inference for models with infinite-dimensionally generated intractable components

In recent years, great effort has been placed on the development of flexible statistical models, which can capture the rich and diverse structures found in real data. Complex models are often intractable, and they require non trivial techniques for inference. In the Bayesian setting, the most common intractability problem is related with nor­malizing constants which cannot be calculated directly. In this case, MCMC methods are a usefrd tool for posterior simulation of the model parameters, and many ideas have been developed to enable the con­struction of the chains with the desired stationary densities. Fre­quently, ideas applied for posterior simulation from doubly-intractable distributions involve an approximation error; general exact methods are only available for models in which both the data and the param­eters take values in fnite-dimensional spaces. In the present work we propose a novel idea, based on a series ex­pansion representation of the intractable functions, to enable MCMC simulation for models in which either the data or the parameters are infinite-dimensional. We achieve this by introducing a suitable set of latent variables with unknown and possibly infinite dimension. The MCMC construction is then made for a tractable latent model, from which the density of interest can be recovered through marginaliza­tion. We illustrate the applicability of the method in various situations. We show that the latent variable construction of the retrospective re­jection sampler commonly known as exact simulation algorithm for diffusions, is a particular case of the latent variable construction we propose. We provide an idea for an alternative exact simulation and inference scheme, through a Markov chain construction. We also present two related nonparamctric mixture models, for time series and regression analysis. Their novelty is in the construction of the mixture weights, which gives them great flexibility but introduces an intractable component generated by the infinite-dimensional parame­ters; we show how our methodology can be applied to enable MCMC inference for these models. We also show how our ideas can be used for inference when the power likelihood for nonparametric mixture models is used; a problem which is of interest in many settings and, to our knowledge, has not been solved without the introduction of some approximation error. Finally, we discuss the matter of Bayesian consistency for Markov models. Unrelated to the driving theme of the thesis, the problem naturally arises from some of the models studied. We make a first step towards a general result for strong consistency which can be used both for discretely observed diffusions and for the time series model we propose

Functional Connectivity Analysis of FMRI Time-Series Data

The term ``functional connectivity' is used to denote correlations in activation among spatially-distinct brain regions, either in a resting state or when processing external stimuli. Functional connectivity has been extensively evaluated with several functional neuroimaging methods, particularly PET and fMRI. Yet these relationships have been quantified using very different measures and the extent to which they index the same constructs is unclear. We have implemented a variety of these functional connectivity measures in a new freely-available MATLAB toolbox. These measures are categorized into two groups: whole time-series and trial-based approaches. We evaluate these measures via simulations with different patterns of functional connectivity and provide recommendations for their use. We also apply these measures to a previously published fMRI dataset in which activity in dorsal anterior cingulate cortex (dACC) and dorsolateral prefrontal cortex (DLPFC) was evaluated in 32 healthy subjects during a digit sorting task. Though all implemented measures demonstrate functional connectivity between dACC and DLPFC activity during event-related tasks, different participants appeared to display qualitatively different relationships.We also propose a new methodology for exploring functional connectivity in slow event-related designs, where stimuli are presented at a sufficient separation to examine the dynamic responses in brain regions. Our methodology simultaneously determines the level of smoothing to obtain the underlying noise-free BOLD response and the functional connectivity among several regions. Smoothing is accomplished through an empirical basis via functional principal components analysis. The coefficients of the basis are assumed to be correlated across regions, and the nature and strength of functional connectivity is derived from this correlation matrix. The model is implemented within a Bayesian framework by specifying priors on the parameters and using a Markov Chain Monte Carlo (MCMC) Gibbs sampling algorithm. We demonstrate this new approach on a sample of clinically depressed subjects and healthy controls in examining relationships among three brain regions implicated in depression and emotion during emotional information processing. The results show that depressed subjects display decreased coupling between left amygdala and DLPFC compared to healthy subjects and this may potentially be due to inefficient functioning in mediating connectivity from the rostral portion Brodmann's area24 (BA24)

Probabilistic Modelling of Uncertainty with Bayesian nonparametric Machine Learning

This thesis addresses the use of probabilistic predictive modelling and machine learning for quantifying uncertainties. Predictive modelling makes inferences of a process from observations obtained using computational modelling, simulation, or experimentation. This is often achieved using statistical machine learning models which predict the outcome as a function of variable predictors and given process observations. Towards this end Bayesian nonparametric regression is used, which is a highly flexible and probabilistic type of statistical model and provides a natural framework in which uncertainties can be included. The contributions of this thesis are threefold. Firstly, a novel approach to quantify parametric uncertainty in the Gaussian process latent variable model is presented, which is shown to improve predictive performance when compared with the commonly used variational expectation maximisation approach. Secondly, an emulator using manifold learning (local tangent space alignment) is developed for the purpose of dealing with problems where outputs lie in a high dimensional manifold. Using this, a framework is proposed to solve the forward problem for uncertainty quantification and applied to two fluid dynamics simulations. Finally, an enriched clustering model for generalised mixtures of Gaussian process experts is presented, which improves clustering, scaling with the number of covariates, and prediction when compared with what is known as the alternative model. This is then applied to a study of Alzheimer’s disease, with the aim of improving prediction of disease progression

Special Topics in Latent Variable Models with Spatially and Temporally Correlated Latent Variables

The term latent variable model (LVM) refers to any statistical procedure that utilizes information contained in a set of observed variables to construct a set of underlying latent variables that drive the observed values and associations. Independent component analysis (ICA) is a LVM that separates recorded mixtures of signals into independent source signals, called independent components (ICs). ICA is popular tool for separating brain signals of interest from artifacts and noise in electroencephalogram (EEG) data. Due to challenges in the estimation of uncertainties in ICA, standard errors are not generally estimated alongside ICA estimates and thus ICs representing brain signals of interest cannot be distinguished through a statistical hypothesis testing framework. In Chapter 2 of this dissertation, we propose a bootstrapping algorithm for ICA that produces bootstrap samples that retain critical correlation structures in the data. These are used to compute uncertainties for ICA parameter estimates and to construct a hypothesis test to identify ICs representing brain activity, which we demonstrate in the context of EEG functional connectivity. In Chapter 3, we extend this bootstrapping approach to accommodate pre-ICA dimension reduction procedures, and we use the resulting method to compare popular strategies for pre-ICA dimension reduction in EEG research. In the final chapter, we turn our attention to another LVM, factor analysis, which utilizes the covariance structure of a set of correlated observed variables to model a smaller number of unmeasured underlying variables. A spatial factor analysis (SFA) model can be used to quantify the social vulnerability of communities based on a set of observed social variables. Current SFA methodology is ill-equipped to handle spatial misalignment in the observed variables. We propose a joint spatial factor analysis model that identifies a common set of latent variables underlying spatially misaligned observed variables and produces results at the level of the smallest spatial units, thereby minimizing loss of information. We apply this model to spatially misaligned data to construct an index of community social vulnerability for Louisiana, which we integrate with Louisiana flood data to identify communities at high risk during natural disasters, based on both social and geographic features.Doctor of Philosoph

Bayesian generative learning of brain and spinal cord templates from neuroimaging datasets

In the field of neuroimaging, Bayesian modelling techniques have been largely adopted and recognised as powerful tools for the purpose of extracting quantitative anatomical and functional information from medical scans. Nevertheless the potential of Bayesian inference has not yet been fully exploited, as many available tools rely on point estimation techniques, such as maximum likelihood estimation, rather than on full Bayesian inference. The aim of this thesis is to explore the value of approximate learning schemes, for instance variational Bayes, to perform inference from brain and spinal cord MRI data. The applications that will be explored in this work mainly concern image segmentation and atlas construction, with a particular emphasis on the problem of shape and intensity prior learning, from large training data sets of structural MR scans. The resulting computational tools are intended to enable integrated brain and spinal cord morphometric analyses, as opposed to the approach that is most commonly adopted in neuroimaging, which consists in optimising separate tools for brain and spine morphometrics

Functional Organization of the Human Brain: How We See, Feel, and Decide.

The human brain is responsible for constructing how we perceive, think, and act in the world around us. The organization of these functions is intricately distributed throughout the brain. Here, I discuss how functional magnetic resonance imaging (fMRI) was employed to understand three broad questions: how do we see, feel, and decide? First, high-resolution fMRI was used to measure the polar angle representation of saccadic eye movements in the superior colliculus. We found that eye movements along the superior-inferior visual field are mapped across the medial-lateral anatomy of a subcortical midbrain structure, the superior colliculus (SC). This result is consistent with the topography in monkey SC. Second, we measured the empathic responses of the brain as people watched a hand get painfully stabbed with a needle. We found that if the hand was labeled as belonging to the same religion as the observer, the empathic neural response was heightened, creating a strong ingroup bias that could not be readily manipulated. Third, we measured brain activity in individuals as they made free decisions (i.e., choosing randomly which of two buttons to press) and found the activity within fronto-thalamic networks to be significantly decreased compared to being instructed (forced) to press a particular button. I also summarize findings from several other projects ranging from addiction therapies to decoding visual imagination to how corporations are represented as people. Together, these approaches illustrate how functional neuroimaging can be used to understand the organization of the human brain

Functional Organization of the Human Brain: How We See, Feel, and Decide.

The human brain is responsible for constructing how we perceive, think, and act in the world around us. The organization of these functions is intricately distributed throughout the brain. Here, I discuss how functional magnetic resonance imaging (fMRI) was employed to understand three broad questions: how do we see, feel, and decide? First, high-resolution fMRI was used to measure the polar angle representation of saccadic eye movements in the superior colliculus. We found that eye movements along the superior-inferior visual field are mapped across the medial-lateral anatomy of a subcortical midbrain structure, the superior colliculus (SC). This result is consistent with the topography in monkey SC. Second, we measured the empathic responses of the brain as people watched a hand get painfully stabbed with a needle. We found that if the hand was labeled as belonging to the same religion as the observer, the empathic neural response was heightened, creating a strong ingroup bias that could not be readily manipulated. Third, we measured brain activity in individuals as they made free decisions (i.e., choosing randomly which of two buttons to press) and found the activity within fronto-thalamic networks to be significantly decreased compared to being instructed (forced) to press a particular button. I also summarize findings from several other projects ranging from addiction therapies to decoding visual imagination to how corporations are represented as people. Together, these approaches illustrate how functional neuroimaging can be used to understand the organization of the human brain