Person:

Stein, Nathan

Chapter 1 of this thesis focuses on accelerating perfect sampling algorithms for a Bayesian hierarchical model. A discrete data augmentation scheme together with two different parameterizations yields two Gibbs samplers for sampling from the posterior distribution of the hyperparameters of the Dirichlet-multinomial hierarchical model under a default prior distribution. The finite-state space nature of this data augmentation permits us to construct two perfect samplers using bounding chains that take advantage of monotonicity and anti-monotonicity in the target posterior distribution, but both are impractically slow. We demonstrate however that a composite algorithm that strategically alternates between the two samplers' updates can be substantially faster than either individually. We theoretically bound the expected time until coalescence for the composite algorithm, and show via simulation that the theoretical bounds can be close to actual performance. Chapters 2 and 3 introduce a strategy for constructing scientifically sensible priors in complex models. We call these priors catalytic priors to suggest that adding such prior information catalyzes our ability to use richer, more realistic models. Because they depend on observed data, catalytic priors are a tool for empirical Bayes modeling. The overall perspective is data-driven: catalytic priors have a pseudo-data interpretation, and the building blocks are alternative plausible models for observations, yielding behavior similar to hierarchical models but with a conceptual shift away from distributional assumptions on parameters. The posterior under a catalytic prior can be viewed as an optimal approximation to a target measure, subject to a constraint on the posterior distribution's predictive implications. In Chapter 3, we apply catalytic priors to several familiar models and investigate the performance of the resulting posterior distributions. We also illustrate the application of catalytic priors in a preliminary analysis of the effectiveness of a job training program, which is complicated by the need to account for noncompliance, partially defined outcomes, and missing outcome data.

Properly segmenting multiband images of the Sun by their thermal properties will help determine the thermal structure of the solar corona. However, off-the-shelf segmentation algorithms are typically inappropriate because temperature information is captured by the relative intensities in different passbands, while the absolute levels are not relevant. Input features are therefore pixel-wise proportions of photons observed in each band. To segment solar images based on these proportions, we use a modification of k-means clustering that we call the H-means algorithm because it uses the Hellinger distance to compare probability vectors. H-means has a closed-form expression for cluster centroids, so computation is as fast as k-means. Tempering the input probability vectors reveals a broader class of H-means algorithms which include spherical k-means clustering. More generally, H-means can be used anytime the input feature is a probabilistic distribution, and hence is useful beyond image segmentation applications.

Mars Reconnaissance Orbiter HiRISE images and Opportunity rover observations of the ~22 km wide Noachian age Endeavour Crater on Mars show that the rim and surrounding terrains were densely fractured during the impact crater-forming event. Fractures have also propagated upward into the overlying Burns formation sandstones. Opportunity’s observations show that the western crater rim segment, called Murray Ridge, is composed of impact breccias with basaltic compositions, as well as occasional fracture-filling calcium sulfate veins. Cook Haven, a gentle depression on Murray Ridge, and the site where Opportunity spent its sixth winter, exposes highly fractured, recessive outcrops that have relatively high concentrations of S and Cl, consistent with modest aqueous alteration. Opportunity’s rover wheels serendipitously excavated and overturned several small rocks from a Cook Haven fracture zone. Extensive measurement campaigns were conducted on two of them: Pinnacle Island and Stuart Island. These rocks have the highest concentrations of Mn and S measured to date by Opportunity and occur as a relatively bright sulfate-rich coating on basaltic rock, capped by a thin deposit of one or more dark Mn oxide phases intermixed with sulfate minerals. We infer from these unique Pinnacle Island and Stuart Island rock measurements that subsurface precipitation of sulfate-dominated coatings was followed by an interval of partial dissolution and reaction with one or more strong oxidants (e.g., O2) to produce the Mn oxide mineral(s) intermixed with sulfate-rich salt coatings. In contrast to arid regions on Earth, where Mn oxides are widely incorporated into coatings on surface rocks, our results demonstrate that on Mars the most likely place to deposit and preserve Mn oxides was in fracture zones where migrating fluids intersected surface oxidants, forming precipitates shielded from subsequent physical erosion.