Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham's-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains. © Institute of Mathematical Statistics, 2010.

Probabilistic Fréchet means for time varying persistence diagrams

© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (Dp)^N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

Studies in Aetiology and Historical Methodology in Herodotus

This dissertation interrogates existing scholarly paradigms regarding aetiology in the Histories of Herodotus in order to open up new avenues to approach a complex and varied topic. Since aetiology has mostly been treated as the study of cause and effect in the Histories, this work expands the purview of aetiology to include Herodotus’ explanations of origins more generally. The overarching goal in examining the methodological principles of Herodotean aetiology is to show the extent to which they resonate across the Histories according to their initial development in the proem, especially in those places that seem to deviate from the work’s driving force (i.e. the Persian Wars). Though the focus is on correlating the principles espoused in the proem with their deployment in Herodotus’ ethnographies and other seemingly divergent portions of his work, the dissertation also demonstrates the influence of these principles on some of the more “historical” aspects of the Histories where the struggle between Greeks and barbarians is concerned. The upshot is to make a novel case not only for the programmatic significance of the proem, but also for the cohesion of Herodotean methodology from cover to cover, a perennial concern for scholars of Greek history and historiography.

Chapter One illustrates how the proem to the Histories (1.1.0-1.5.3) prefigures Herodotus’ engagement with aetiological discussions throughout the Histories. Chapter Two indicates how the reading of the proem laid out in Chapter One allows for Herodotus’ deployment of aetiology in the Egyptian logos (especially where the pharaoh Psammetichus’ investigation of the origins of Egyptian language, nature, and custom are concerned) to be viewed within the methodological continuum of the Histories at large. Chapter Three connects Herodotus’ programmatic interest in the origins of erga (i.e. “works” or “achievements” manifested as monuments and deeds of abstract and concrete sorts) with the patterns addressed in Chapters One and Two. Chapter Four examines aetiological narratives in the Scythian logos and argues through them that this logos is as integral to the Histories as the analogous Egyptian logos studied in Chapter Two. Chapter Five demonstrates how the aetiologies associated with the Greeks’ collaboration with the Persians (i.e. medism) in the lead-up to the battle of Thermopylae recapitulate programmatic patterns isolated in previous chapters and thereby extend the methodological continuum of the Histories beyond the “ethnographic” logoi to some of the most representative “historical” logoi of Herodotus’ work. Chapter Six concludes the dissertation and makes one final case for methodological cohesion by showing the inextricability of the end of the Histories from its beginning.