FireFly: A Bayesian Approach to Source Finding in Astronomical Data

Efficient and rigorous source finding techniques are needed for the upcoming large data sets from telescopes like MeerKAT, LSST and the SKA. Most of the current source-finding algorithms lack full statistical rigor. Typically these algorithms use some form of thresholding to find sources, which leads to contamination and missed sources. Ideally we would like to use all the available information when performing source detection, including any prior knowledge we may have. Bayesian statistics is the obvious approach as it allows precise statistical interrogations of the data and the inclusion of all available information. In this thesis, we implement nested sampling and Monte Carlo Markov Chain (MCMC) techniques to develop a new Bayesian source finding technique called FireFly. FireFly employs a technique of switching ‘on’ and ‘off’ sources during sampling to deal with the fact that we don’t know how many true sources are present. It therefore tackles one of the critical questions in source finding, which is estimating the number of real sources in the image. We compare FireFly against a Bayesian evidence-based search method and show on simulated astronomical images that FireFly outperforms the evidence-based approach. We further investigate two implementations of FireFly: the first with nested sampling and the second with MCMC. Our results show that MCMC FireFly has better computational scaling than the nested sampling version FireFly but the nested sampling version of FireFly appears to perform somewhat better than MCMC FireFly. Future work should examine how best to quantify FireFly performance and extend the formalism developed here to deal with multiwavelength data

Efficient Bayesian inference via Monte Carlo and machine learning algorithms

Mención Internacional en el título de doctorIn many fields of science and engineering, we are faced with an inverse problem where we aim to recover an unobserved parameter or variable of interest from a set of observed variables. Bayesian inference is a probabilistic approach for inferring this unknown parameter that has become extremely popular, finding application in myriad problems in fields such as machine learning, signal processing, remote sensing and astronomy. In Bayesian inference, all the information about the parameter is summarized by the posterior distribution. Unfortunately, the study of the posterior distribution requires the computation of complicated integrals, that are analytically intractable and need to be approximated. Monte Carlo is a huge family of sampling algorithms for performing optimization and numerical integration that has become the main horsepower for carrying out Bayesian inference. The main idea of Monte Carlo is that we can approximate the posterior distribution by a set of samples, obtained by an iterative process that involves sampling from a known distribution. Markov chain Monte Carlo (MCMC) and importance sampling (IS) are two important groups of Monte Carlo algorithms. This thesis focuses on developing and analyzing Monte Carlo algorithms (either MCMC, IS or combination of both) under different challenging scenarios presented below. In summary, in this thesis we address several important points, enumerated (a)–(f), that currently represent a challenge in Bayesian inference via Monte Carlo. A first challenge that we address is the problematic exploration of the parameter space by off-the-shelf MCMC algorithms when there is (a) multimodality, or with (b) highly concentrated posteriors. Another challenge that we address is the (c) proposal construction in IS. Furtheremore, in recent applications we need to deal with (d) expensive posteriors, and/or we need to handle (e) noisy posteriors. Finally, the Bayesian framework also offers a way of comparing competing hypothesis (models) in a principled way by means of marginal likelihoods. Hence, a task that arises as of fundamental importance is (f) marginal likelihood computation. Chapters 2 and 3 deal with (a), (b), and (c). In Chapter 2, we propose a novel population MCMC algorithm called Parallel Metropolis-Hastings Coupler (PMHC). PMHC is very suitable for multimodal scenarios since it works with a population of states, instead of a single one, hence allowing for sharing information. PMHC combines independent exploration by the use of parallel Metropolis-Hastings algorithms, with cooperative exploration by the use of a population MCMC technique called Normal Kernel Coupler. In Chapter 3, population MCMC are combined with IS within the layered adaptive IS (LAIS) framework. The combination of MCMC and IS serves two purposes. First, an automatic proposal construction. Second, it aims at increasing the robustness, since the MCMC samples are not used directly to form the sample approximation of the posterior. The use of minibatches of data is proposed to deal with highly concentrated posteriors. Other extensions for reducing the costs with respect to the vanilla LAIS framework, based on recycling and clustering, are discussed and analyzed. Chapters 4, 5 and 6 deal with (c), (d) and (e). The use of nonparametric approximations of the posterior plays an important role in the design of efficient Monte Carlo algorithms. Nonparametric approximations of the posterior can be obtained using machine learning algorithms for nonparametric regression, such as Gaussian Processes and Nearest Neighbors. Then, they can serve as cheap surrogate models, or for building efficient proposal distributions. In Chapter 4, in the context of expensive posteriors, we propose adaptive quadratures of posterior expectations and the marginal likelihood using a sequential algorithm that builds and refines a nonparametric approximation of the posterior. In Chapter 5, we propose Regression-based Adaptive Deep Importance Sampling (RADIS), an adaptive IS algorithm that uses a nonparametric approximation of the posterior as the proposal distribution. We illustrate the proposed algorithms in applications of astronomy and remote sensing. Chapter 4 and 5 consider noiseless posterior evaluations for building the nonparametric approximations. More generally, in Chapter 6 we give an overview and classification of MCMC and IS schemes using surrogates built with noisy evaluations. The motivation here is the study of posteriors that are both costly and noisy. The classification reveals a connection between algorithms that use the posterior approximation as a cheap surrogate, and algorithms that use it for building an efficient proposal. We illustrate specific instances of the classified schemes in an application of reinforcement learning. Finally, in Chapter 7 we study noisy IS, namely, IS when the posterior evaluations are noisy, and derive optimal proposal distributions for the different estimators in this setting. Chapter 8 deals with (f). In Chapter 8, we provide with an exhaustive review of methods for marginal likelihood computation, with special focus on the ones based on Monte Carlo. We derive many connections among the methods and compare them in several simulations setups. Finally, in Chapter 9 we summarize the contributions of this thesis and discuss some potential avenues of future research.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Valero Laparra Pérez-Muelas.- Secretario: Michael Peter Wiper.- Vocal: Omer Deniz Akyildi

Random finite sets in multi-target tracking - efficient sequential MCMC implementation

Over the last few decades multi-target tracking (MTT) has proved to be a challenging and attractive research topic. MTT applications span a wide variety of disciplines, including robotics, radar/sonar surveillance, computer vision and biomedical research. The primary focus of this dissertation is to develop an effective and efficient multi-target tracking algorithm dealing with an unknown and time-varying number of targets. The emerging and promising Random Finite Set (RFS) framework provides a rigorous foundation for optimal Bayes multi-target tracking. In contrast to traditional approaches, the collection of individual targets is treated as a set-valued state. The intent of this dissertation is two-fold; first to assert that the RFS framework not only is a natural, elegant and rigorous foundation, but also leads to practical, efficient and reliable algorithms for Bayesian multi-target tracking, and second to provide several novel RFS based tracking algorithms suitable for the specific Track-Before-Detect (TBD) surveillance application. One main contribution of this dissertation is a rigorous derivation and practical implementation of a novel algorithm well suited to deal with multi-target tracking problems for a given cardinality. The proposed Interacting Population-based MCMC-PF algorithm makes use of several Metropolis-Hastings samplers running in parallel, which interact through genetic variation. Another key contribution concerns the design and implementation of two novel algorithms to handle a varying number of targets. The first approach exploits Reversible Jumps. The second approach is built upon the concepts of labeled RFSs and multiple cardinality hypotheses. The performance of the proposed algorithms is also demonstrated in practical scenarios, and shown to significantly outperform conventional multi-target PF in terms of track accuracy and consistency. The final contribution seeks to exploit external information to increase the performance of the surveillance system. In multi-target scenarios, kinematic constraints from the interaction of targets with their environment or other targets can restrict target motion. Such motion constraint information is integrated by using a fixed-lag smoothing procedure, named Knowledge-Based Fixed-Lag Smoother (KB-Smoother). The proposed combination IP-MCMC-PF/KB-Smoother yields enhanced tracking

Markov chain Monte Carlo analyses of longitudinal biomedical magnetic resonance data

Markov chain Monte Carlo simulation was used in an analysis of the data acquired in three longitudinal biomedical magnetic resonance studies. The first of these investigations uses a Bayesian nonlinear hierarchical random coefficients model to examine the longitudinal extracellular direct current (DC) potential and apparent diffusion coefficient (ADC) responses to focal ischaemia in the rat. The purpose is to perform a formal analysis of the temporal relationship between the two responses, and thus to examine the data for compatibility with a common latent (driving) process and, alternatively, the existence of an ADC threshold for anoxic depolarisation. The DC-potential and ADC transition parameter posterior probability distributions were generated, paying particular attention to the within-subject differences between the DC-potential and ADC transition characteristics. The results indicate that the DC-potential and ADC changes are not driven by a common latent process and, in addition, provide no evidence for a consistent ADC threshold associated with anoxic depolarisation. The second analysis uses data acquired in a nuclear magnetic resonance spectroscopic study into the effects of intestinal ischaemia and subsequent reperfusion on liver metabolism in the rat. The purpose of the analysis is to examine the temporal relationship between energy status [inorganic phosphate to adenosine triphosphate ratio (PAR)] and the pH response, the former of which is an indicator of liver energy failure. The posterior distribution obtained for the PAR-pH onset time difference indicates that the pH response precedes the change in PAR, suggesting that intracellular acidosis cannot be ruled out as a contributing factor to the observed liver failure. The third dataset was acquired in an electron spin resonance study of the Arrhenius behaviour of the rabbit muscle sarcoplasmic reticulum membrane. An MCMC Arrhenius plot changepoint analysis is used to estimate the order parameter 'transition' temperature

Statistical Modelling of Marine Fish Populations and Communities

Sustainable fisheries management require an understanding of the relationship between the adult population and the number of juveniles successfully added to that population each year. The process driving larval survival to enter a given stage of a fish population is highly variable and this pattern of variability reflects the strength of density-dependent mortality. Marine ecosystems are generally threatened by climate change and overfishing; the coupling of these two sources have encouraged scientists to develop end-to-end ecosystem models to study the interactions of organisms at different trophic levels and to understand their behaviours in response to climate change. Our understanding of this important and massively complex system has been constrained historically by the limited amount of data available. Recent technological advances are beginning to address this lack of data, but there is an urgent need for careful statistical methodology to synthesise this information and to make reliable predictions based upon it. In this thesis I developed methodologies specifically designed to interpret the patterns of variability in recruitment by accurately estimating the degree of heteroscedasticity in 90 published stock-recruitment datasets. To better estimate the accuracy of model parameters, I employed a Bayesian hierarchical modelling framework and applied this to multiple sets of fish populations with different model structures. Finally, I developed an end-to-end ecological model that takes into account biotic and abiotic factors, together with data on the fish communities, to assess the organisation of the marine ecosystem and to investigate the potential effects of weather or climate changes. The work developed within this thesis highlights the importance of statistical methods in estimating the patterns of variability and community structure in fish populations as well as describing the way organisms and environmental factors interact within an ecosystem

Nonparametric Bayes for Big Data

<p>Classical asymptotic theory deals with models in which the sample size $n$ goes to infinity with the number of parameters $p$ being fixed. However, rapid advancement of technology has empowered today's scientists to collect a huge number of explanatory variables</p><p>to predict a response. Many modern applications in science and engineering belong to the ``big data" regime in which both $p$ and $n$ may be very large. A variety of genomic applications even have $p$ substantially greater than $n$. With the advent of MCMC, Bayesian approaches exploded in popularity. Bayesian inference often allows easier interpretability than frequentist inference. Therefore, it becomes important to understand and evaluate</p><p>Bayesian procedures for ``big data" from a frequentist perspective.</p><p>In this dissertation, we address a number of questions related to solving large-scale statistical problems via Bayesian nonparametric methods.</p><p>It is well-known that classical estimators can be inconsistent in the high-dimensional regime without any constraints on the model. Therefore, imposing additional low-dimensional structures on the high-dimensional ambient space becomes inevitable. In the first two chapters of the thesis, we study the prediction performance of high-dimensional nonparametric regression from a minimax point of view. We consider two different low-dimensional constraints: 1. the response depends only on a small subset of the covariates; 2. the covariates lie on a low dimensional manifold in the original high dimensional ambient space. We also provide Bayesian nonparametric methods based on Gaussian process priors that are shown to be adaptive to unknown smoothness or low-dimensional manifold structure by attaining minimax convergence rates up to log factors. In chapter 3, we consider high-dimensional classification problems where all data are of categorical nature. We build a parsimonious model based on Bayesian tensor factorization for classification while doing inferences on the important predictors.</p><p>It is generally believed that ensemble approaches, which combine multiple algorithms or models, can outperform any single algorithm at machine learning tasks, such as prediction. In chapter 5, we propose Bayesian convex and linear aggregation approaches motivated by regression applications. We show that the proposed approach is minimax optimal when the true data-generating model is a convex or linear combination of models in the list. Moreover, the method can adapt to sparsity structure in which certain models should receive zero weights, and the method is tuning parameter free unlike competitors. More generally, under an M-open view when the truth falls outside the space of all convex/linear combinations, our theory suggests that the posterior measure tends to concentrate on the best approximation of the truth at the minimax rate.</p><p>Chapter 6 is devoted to sequential Markov chain Monte Carlo algorithms for Bayesian on-line learning of big data. The last chapter attempts to justify the use of posterior distribution to conduct statistical inferences for semiparametric estimation problems (the semiparametric Bernstein von-Mises theorem) from a frequentist perspective.</p>Dissertatio

Essays on Numerical Integration in Hamiltonian Monte Carlo

This thesis considers a variety of topics broadly unified under the theme of geometric integration for Riemannian manifold Hamiltonian Monte Carlo. In chapter 2, we review fundamental topics in numerical computing (section 2.1), classical mechanics (section 2.2), integration on manifolds (section 2.3), Riemannian geometry (section 2.5), stochastic differential equations (section 2.4), information geometry (section 2.6), and Markov chain Monte Carlo (section 2.7). The purpose of these sections is to present the topics discussed in the thesis within a broader context. The subsequent chapters are self-contained to an extent, but contain references back to this foundational material where appropriate. Chapter 3 gives a formal means of conceptualizing the Markov chains corresponding to Riemannian manifold Hamiltonian Monte Carlo and related methods; this formalism is useful for understanding the significance of reversibility and volume-preservation for maintaining detailed balance in Markov chain Monte Carlo. Throughout the remainder of the thesis, we investigate alternative methods of geometric numerical integration for use in Riemannian manifold Hamiltonian Monte Carlo, discuss numerical issues involving violations of reversibility and detailed balance, and propose new algorithms with superior theoretical foundations. In chapter 4, we evaluate the implicit midpoint integrator for Riemannian manifold Hamiltonian Monte Carlo, presenting the first time that this integrator has been deployed and assessed within this context. We discuss attributes of the implicit midpoint integrator that make it preferable, and inferior, to alternative methods of geometric integration such as the generalized leapfrog procedure. In chapter 5, we treat an empirical question as to what extent convergence thresholds play a role in geometric numerical integration in Riemannian manifold Hamiltonian Monte Carlo. If the convergence threshold is too large, then the Markov chain transition kernel will fail to maintain detailed balance, whereas a convergence threshold that is very small will incur computational penalties. We investigate these phenomena and suggest two mechanisms, based on stochastic approximation and higher-order solvers for non-linear equations, which can aid in identifying convergence thresholds or suppress its significance. In chapter 6, we consider a numerical integrator for Markov chain Monte Carlo based on the Lagrangian, rather than Hamiltonian, formalism in classical mechanics. Our contributions include clarifying the order of accuracy of this numerical integrator, which has been misunderstood in the literature, and evaluating a simple change that can accelerate the implementation of the method, but which comes at the cost of producing more serially auto-correlated samples. We also discuss robustness properties of the Lagrangian numerical method that do not materialize in the Hamiltonian setting. Chapter 7 examines theories of geometric ergodicity for Riemannian manifold Hamiltonian Monte Carlo and Lagrangian Monte Carlo, and proposes a simple modification to these Markov chain methods that enables geometric ergodicity to be inherited from the manifold Metropolis-adjusted Langevin algorithm. In chapter 8, we show how to revise an explicit integration using a theory of Lagrange multipliers so that the resulting numerical method satisfies the properties of reversibility and volume-preservation. Supplementary content in chapter E investigates topics in the theory of shadow Hamiltonians of the implicit midpoint method in the case of non-canonical Hamiltonian mechanics and chapter F, which treats the continual adaptation of a parameterized proposal distribution in the independent Metropolis-Hastings sampler

New methods for infinite and high-dimensional approximate Bayesian computation

The remarkable complexity of modern applied problems often requires the use of probabilistic models where the likelihood is intractable -- in the sense that it cannot be numerically evaluated, not even up to a normalizing constant. The statistical literature provides an extensive array of methods designed to bypass this constraint. Still, inference in this context remains computationally challenging, particularly for high-dimensional models. We focus on the important class of Approximation Bayesian Computation (ABC) methods. Various state-of-the-art ABC techniques are combined to fit an intractable model that describes the epidemiological dynamics of multidrug-resistant tuberculosis. This study addresses a number of important biological questions in a principled manner, providing useful insights to this extraordinarily relevant research topic. We propose a functional regression adjustment ABC procedure that permits the estimation of infinite-dimensional parameters, which effectively launches ABC into the non-parametric framework. Two likelihood-free algorithms are also introduced. The first exploits the principles of ABC and the so-called coverage property to recalibrate an auxiliary approximate posterior estimator. This approach further strengthens the links between ABC and indirect inference, allowing a more comprehensive use of the auxiliary estimator. The second algorithm employs the ABC machinery to build approximate samplers for the intractable full conditional distributions. These samplers are then combined to form a likelihood-free approximate Gibbs sampler. The granular nature of our approach (that comes from breaking down the problem into small pieces) makes it suitable for highly-structured problems. We demonstrate this property by fitting an intractable and very high-dimensional state space model

Bayesian Approaches to Emulation for a Complex Computer Crop Yield Simulator with Mixed Inputs

Agriculture is one area where the simulation of crop growth, nutrition, soil condition and pollution could be invaluable in any land management decisions. The Environmental Policy Integrated Climate Model (EPIC) is a simulation model to investigate the behaviour of crop yield in response to changes in inputs such as fertiliser levels, soil, steepness, and other environmental covariates. We build a model for crop yield around a non-linear Mitscherlich Baule growth model to make inferences about crop yield response to changes in continuous input and factor variables. A Bayesian hierarchical approach to the modelling was taken for mixed inputs, requiring Markov Chain Monte Carlo simulations to obtain samples from the posterior distributions, to validate and illustrate the results, and to carry out model selection. The emulation of complex computer simulations has become an effective tool in exploring this high-dimensional simulated process's behaviour. Initially, we built a Bayes linear emulator to efficiently emulate crop yield as a function of the simulator's continuous inputs only. We explore emulator diagnostics and present the results from the emulation of a subset of the simulated EPIC data output. Computer models with quantitative inputs are used widely, but the challenge is incorporating the factors. We propose a framework for solving this issue considering the Bayes linear emulation approach. We explore a variety of correlation structures to represent the mixed inputs and combine this with the Bayes linear approach to construct an emulator. Finally, we developed a method to make an optimal decision for the farmers to gain maximum utility considering yield and pollutants, accounting for weather factors, land characteristics and fertiliser use