A recent wave of research has attempted to define fairness quantitatively. In particular, this work has explored what fairness might mean in the context of decisions based on the predictions of statistical and machine learning models. The rapid growth of this new field has led to wildly inconsistent motivations, terminology, and notation, presenting a serious challenge for cataloging and comparing definitions. This article attempts to bring much-needed order. First, we explicate the various choices and assumptions made—often implicitly—to justify the use of prediction-based decision-making. Next, we show how such choices and assumptions can raise fairness concerns and we present a notationally consistent catalog of fairness definitions from the literature. In doing so, we offer a concise reference for thinking through the choices, assumptions, and fairness considerations of prediction-based decision-making. Expected final online publication date for the Annual Review of Statistics, Volume 8 is March 8, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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This prognostic study validates a machine learning (random forest) prediction model of elevated blood lead levels by comparing with a parsimonious logistic regression among children in a Women, Infants, and Children cohort.

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer n as the sum of k triangular numbers is asymptotically equivalent to the modified divisor function σ 2k−1 (2n + k). 2 for 0 (4) to study the functions r k (n). Ono, Robins, and Wahl [1995] defined an analogous modular form to study triangular numbers. We begin by defining triangular numbers.

This paper studies the evaluation of methods for targeting the allocation of limited resources to a high-risk subpopulation. We consider a randomized controlled trial to measure the difference in efficiency between two targeting methods and show that it is biased. An alternative, survey-based design is shown to be unbiased. Both designs are simulated for the evaluation of a policy to target lead hazard investigations using a predictive model. Based on our findings, we advised the Chicago Department of Public Health to use the survey design for their field trial. Our work anticipates further developments in economics that will be important as predictive modeling becomes an increasingly common policy tool.

Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.