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Register a new userHierarchical Bayesian Level Set Inversion
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Added by
Matthew Dunlop
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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The level set approach has proven widely successful in the study of inverse problems for interfaces, since its systematic development in the 1990s. Recently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be circumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful consideration of the development of algorithms which encode probability measure equivalences as the hierarchical parameter is varied, this leads to well-defined Gibbs based MCMC methods found by alternating Metropolis-Hastings updates of the level set function and the hierarchical parameter. These methods demonstrably outperform non-hierarchical Bayesian level set methods.
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A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of the total variation. On the other hand, sparse or noisy data often demands a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori (MAP) objective function of a suitably chosen Bayesian phase-field approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess finite perimeter and to have the ability to learn about the true perimeter.
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Dynamically rescaled Hamiltonian Monte Carlo (DRHMC) is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterisation so that the re-parameterised target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterisations of the conditional distributions specifying the hierarchical model are constant information parameterisations (CIP), the relation between the modified- and original parameterisation is bijective, explicitly computed and admit exploitation of sparsity in the numerical linear algebra involved. CIPs for a large catalogue of statistical models are presented, and from the catalogue, it is clear that many CIPs are currently routinely used in statistical computing. A relation between the proposed methodology and a class of explicitly integrated Riemann manifold Hamiltonian Monte Carlo methods is discussed. The methodology is illustrated on several example models, including a model for inflation rates with multiple levels of non-linearly dependent latent variables.
We propose a hierarchical Bayesian model to estimate the proportional contribution of source populations to a newly founded colony. Samples are derived from the first generation offspring in the colony, but mating may occur preferentially among migrants from the same source population. Genotypes of the newly founded colony and source populations are used to estimate the mixture proportions, and the mixture proportions are related to environmental and demographic factors that might affect the colonizing process. We estimate an assortative mating coefficient, mixture proportions, and regression relationships between environmental factors and the mixture proportions in a single hierarchical model. The first-stage likelihood for genotypes in the newly founded colony is a mixture multinomial distribution reflecting the colonizing process. The environmental and demographic data are incorporated into the model through a hierarchical prior structure. A simulation study is conducted to investigate the performance of the model by using different levels of population divergence and number of genetic markers included in the analysis. We use Markov chain Monte Carlo (MCMC) simulation to conduct inference for the posterior distributions of model parameters. We apply the model to a data set derived from grey seals in the Orkney Islands, Scotland. We compare our model with a similar model previously used to analyze these data. The results from both the simulation and application to real data indicate that our model provides better estimates for the covariate effects.
Studying the determinants of adverse pregnancy outcomes like stillbirth and preterm birth is of considerable interest in epidemiology. Understanding the role of both individual and community risk factors for these outcomes is crucial for planning appropriate clinical and public health interventions. With this goal, we develop geospatial mixed effects logistic regression models for adverse pregnancy outcomes. Our models account for both spatial autocorrelation and heterogeneity between neighborhoods. To mitigate the low incidence of stillbirth and preterm births in our data, we explore using class rebalancing techniques to improve predictive power. To assess the informative value of the covariates in our models, we use posterior distributions of their coefficients to gauge how well they can be distinguished from zero. As a case study, we model stillbirth and preterm birth in the city of Philadelphia, incorporating both patient-level data from electronic health records (EHR) data and publicly available neighborhood data at the census tract level. We find that patient-level features like self-identified race and ethnicity were highly informative for both outcomes. Neighborhood-level factors were also informative, with poverty important for stillbirth and crime important for preterm birth. Finally, we identify the neighborhoods in Philadelphia at highest risk of stillbirth and preterm birth.
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